diff --git a/README.md b/README.md index aca7685..2398fb8 100644 --- a/README.md +++ b/README.md @@ -32,7 +32,14 @@ - **Indexing** — basic-indexing `Slice` views plus **boolean/fancy indexing**: `MaskSelect` (`a[mask]`), `Nonzero` (`flatnonzero`), and `Take`. - **Manipulation** — `Concatenate`/`Stack`/`VStack`/`HStack`. -- **Linear algebra** — `MatMul`/`Dot`/`Inner`/`Outer`. +- **Linear algebra** — `MatMul`/`Dot`/`Inner`/`Outer`; package + [`linalg`](linalg) is `numpy.linalg` in pure Go: `Solve`, `Inv`, `Det`, + `Slogdet`, `Cholesky`, `QR` (all four modes), `SVD`/`SVDVals`, + `Eigh`/`Eigvalsh`, `Eig`/`Eigvals`, `Lstsq`, `Pinv`, `MatrixRank`, `Norm`, + `Cond`, `MatrixPower`, `MultiDot` — stacked `(..., M, N)` inputs (run in + parallel), every dtype, LAPACK's algorithms (divide and conquer for the + SVD and Hermitian eigenproblems), checked against numpy 2.5.3 on 1,040 + cases. See [docs/linalg.md](docs/linalg.md) for accuracy and speed. - **Fourier transforms** — package [`fft`](fft) is `numpy.fft` on arrays (`FFT`/`RFFT`/`FFT2`/`FFTN`/… with `n`/`s`, `axes` and `norm`, single precision kept), on [go-fft](https://github.com/go-fft/fft). @@ -229,6 +236,8 @@ u, _ := random.Rand(3) // [0.5488135 0.71518937 0.6 b, _ := ndarray.Arange(0, 6, 1) b, _ = b.Reshape(3, 2) prod, _ := m.MatMul(b) // (2,3)·(3,2) -> (2,2) +inv, _ := linalg.Inv(prod) // np.linalg.inv +left, sv, vh, _ := linalg.SVD(m, false) // np.linalg.svd(m, full_matrices=False) ``` ## License diff --git a/docs/bench_linalg_numpy.py b/docs/bench_linalg_numpy.py new file mode 100644 index 0000000..5fcfa46 --- /dev/null +++ b/docs/bench_linalg_numpy.py @@ -0,0 +1,32 @@ +# numpy side of docs/linalg.md: uv run --with numpy==2.5.3 python docs/bench_linalg_numpy.py +import time, sys, numpy as np, numpy.linalg as la +def mat(n, kind): + r = np.random.default_rng(n) + a = r.standard_normal((n, n)) + if kind == "spd": return a @ a.T + n * np.eye(n) + if kind == "sym": return a + a.T + return a +ops = [("solve", "", lambda a: la.solve(a, np.ones(a.shape[0]))), ("inv", "", la.inv), + ("cholesky", "spd", la.cholesky), ("qr", "", la.qr), ("svd", "", lambda a: la.svd(a, full_matrices=False)), + ("eigh", "sym", la.eigh), ("eig", "", la.eig)] +sizes = [int(x) for x in sys.argv[1].split(",")] if len(sys.argv) > 1 else [64, 256, 1024] +for name, kind, f in ops: + for n in sizes: + a = mat(n, kind) + f(a) + reps = max(1, int(0.3 / max(1e-6, (lambda t0: (f(a), time.perf_counter() - t0)[1])(time.perf_counter())))) + reps = min(reps, 200) + ts = [] + for _ in range(reps): + t0 = time.perf_counter(); f(a); ts.append(time.perf_counter() - t0) + print(f"{name}/{n} {np.median(ts)*1e3:.3f} ms") + +# Stacks of small matrices. +rs=np.random.default_rng(9) +for cnt,n in [(10000,4),(1000,32)]: + a=rs.standard_normal((cnt,n,n)); s=a+np.swapaxes(a,-1,-2) + for name,f in [("inv",lambda: la.inv(a)),("det",lambda: la.det(a)),("eigh",lambda: la.eigh(s)),("svd",lambda: la.svd(a,full_matrices=False)),("eig",lambda: la.eig(a))]: + f(); ts=[] + for _ in range(7): + t=time.perf_counter(); f(); ts.append(time.perf_counter()-t) + print(f"{name}/{cnt}x{n} {np.median(ts)*1e3:.3f} ms") diff --git a/docs/linalg.md b/docs/linalg.md new file mode 100644 index 0000000..134db19 --- /dev/null +++ b/docs/linalg.md @@ -0,0 +1,256 @@ +# linalg — numpy.linalg in pure Go + +Package [`linalg`](../linalg) gives ndarray arrays the functions of +`numpy.linalg`, written in Go without cgo or assembly of its own: the +matrix products it relies on are the ndarray package's packed SIMD GEMM. + +| numpy.linalg | linalg | +|---|---| +| `solve(a, b)` | `Solve(a, b)` | +| `inv(a)` | `Inv(a)` | +| `det(a)`, `slogdet(a)` | `Det(a)`, `Slogdet(a)` | +| `cholesky(a, upper=…)` | `Cholesky(a, Lower \| Upper)` | +| `qr(a, mode)` | `QR(a, ModeReduced \| ModeComplete \| ModeR \| ModeRaw)` | +| `svd(a, full_matrices)`, `svdvals(a)` | `SVD(a, full)`, `SVDVals(a)` | +| `eigh(a, UPLO)`, `eigvalsh(a, UPLO)` | `Eigh(a, uplo)`, `Eigvalsh(a, uplo)` | +| `eig(a)`, `eigvals(a)` | `Eig(a)`, `Eigvals(a)` | +| `lstsq(a, b, rcond)` | `Lstsq(a, b)`, `LstsqRcond(a, b, rcond)` → `LstsqResult{X, Residuals, Rank, S}` | +| `pinv(a, rcond)` | `Pinv(a)`, `PinvRcond(a, rcond)` | +| `matrix_rank(A, tol)` | `MatrixRank(a)`, `MatrixRankTol(a, tol)` | +| `norm(x, ord, axis, keepdims)` | `Norm(x, ord, axes, keepdims)`; `ord` is `Ord{}` (None), `P(p)`, `Fro`, `Nuc` | +| `cond(x, p)` | `Cond(x, p)` | +| `matrix_power(a, n)` | `MatrixPower(a, n)` | +| `multi_dot(arrays)` | `MultiDot(arrays...)` | +| `LinAlgError` | `ErrLinAlg`, wrapped by `ErrSingular`, `ErrNotPosDef`, `ErrNoConvergence` | + +Shape problems give `ErrShape` (numpy raises `LinAlgError` for some of them +and `ValueError` for others), bad options `ErrArgs`. + +## Semantics + +- **Stacks.** The last two axes are the matrix; leading axes are a stack, + processed one matrix per goroutine. `Solve` broadcasts the stack + dimensions of `a` and `b` as numpy's gufunc does; a 1-D `b` is one + vector, any other `b` a stack of matrices (numpy 2's rule). +- **Dtypes.** As in numpy, everything is computed in double precision: + float64 and complex128 directly, integers and bools promoted to float64, + float32 and complex64 widened. Results have numpy's dtypes: single + precision in, single precision out (rounded from the double result, which + is what numpy does too — its linalg gufuncs also run in double); singular + values, Hermitian eigenvalues, `logabsdet` and norms are real; `Eig` is + always complex (numpy 2). `MatrixPower` with n ≥ 0 keeps an integer dtype. +- **Conventions.** LAPACK's: Householder reflectors with β = −sign(α)‖x‖, so + R's diagonal can be negative as in numpy; ascending Hermitian eigenvalues; + decreasing singular values; `Eig` vectors with unit norm and their largest + component real, conjugate vectors for conjugate eigenvalues of a real + matrix (dgeev's normalization). +- **Not unique.** Signs and phases of eigenvectors and singular vectors, and + any orthonormal basis of a repeated eigenvalue's space, may differ from + numpy's. Compare `A·v = λ·v`, `A = U·S·Vh`, not the vectors. + +## Algorithms, and why + +All are LAPACK's, re-expressed for row-major storage. Each is written once, +generic over `float64 | complex128`; Go 1.27 stencils the two and folds the +type switches of the helpers (`conj`, `abs`, …), so the real code pays +nothing for the complex case (measured: same speed as a hand-specialised +loop). + +| operation | algorithm | where the time goes | +|---|---|---| +| `Solve`, `Inv`, `Det`, `Slogdet` | LU with partial pivoting, recursive (getrf2): halves the columns, so the trailing update and the triangular solves are GEMMs | GEMM | +| `Cholesky` | recursive potrf: `L21 = A21·L11⁻ᴴ`, `A22 −= L21·L21ᴴ` | GEMM | +| `QR` | blocked Householder (geqrf/orgqr, panels of 32, compact WY `I − V·T·Vᴴ`) | GEMM | +| `Eigh` | blocked tridiagonalization (latrd, panels of 32: half the flops in GEMM, half in matrix-vector products) + divide and conquer (stedc: Cuppen's split, deflation, secular equation, Gu–Eisenstat's recomputed vector) above order 25, implicit QL/QR (steqr) below | tridiagonalization | +| `Eigvalsh` | the same reduction + QL/QR without vectors | tridiagonalization | +| `SVD` | blocked Golub–Kahan bidiagonalization (labrd) + divide and conquer on the bidiagonal (bdsdc: Gu and Eisenstat's arrow matrix and secular equation) above order 25, implicit zero-shift/shifted QR (bdsqr, Demmel–Kahan) below; wide matrices factor Aᴴ | bidiagonalization | +| `SVDVals`, `MatrixRank`, `Cond` (2-norm), `Norm` (2, nuc) | the same reduction + bdsqr without vectors | bidiagonalization | +| `Eig`, `Eigvals` | balancing (gebal: permutations and powers of two), Hessenberg reduction (gehd2), Francis double-shift QR (dlahqr; single-shift zlahqr for complex input), eigenvectors by back-substitution on the Schur form with overflow scaling (trevc), back-transformed by one GEMM, gebak, dgeev's normalization | QR iteration | +| `Lstsq` | SVD, `x = V·S⁺·Uᴴ·b` over singular values above `rcond·σ₁`; the residuals from the Householder QR of `a` (the tail of `Qᴴb`, free of the cancellation in `b − a·x`) | SVD | +| `Pinv` | SVD, `V·S⁺·Uᴴ` | SVD | +| `Norm`, `MatrixPower`, `MultiDot` | direct; 2-norms rescaled against overflow; repeated squaring in numpy's order; numpy's matrix-chain dynamic programme | — | + +**The SVD: why Golub–Kahan with QR, and divide and conquer.** The choice +was between Golub–Kahan bidiagonalization followed by the implicit-shift QR +iteration, and one-sided Jacobi. One-sided Jacobi is the more accurate in a +strong sense — it computes the small singular values of a matrix whose +columns are badly scaled to high relative accuracy (Demmel–Veselić) — but it +needs 5–10 sweeps of O(n³) rotations, several times the work, and it is not +what numpy computes. Golub–Kahan reduction is backward stable: the computed +singular values are those of `A + E` with `‖E‖ ≈ ε‖A‖`, so each is within +about `ε·σ₁` of the truth — numpy's (gesdd's) accuracy, exactly. The QR +iteration (bdsqr, with Demmel and Kahan's zero shift) then finds the +singular values of the bidiagonal to high relative accuracy, so it adds +nothing to that error. It is used at the leaves of the divide and conquer +and whenever the vectors are not wanted. With vectors, the QR iteration's +plane rotations cost O(n³) of non-vectorized work (5.8 s at n = 1024 in the +first version); divide and conquer, LAPACK's gesdd algorithm and the one +numpy calls, puts that work into GEMMs (0.26 s now). Its singular vectors +stay orthogonal because the updating vector is recomputed from the computed +roots (Gu and Eisenstat), and the secular equation is solved with each +distance `dᵢ − σ` formed from the nearest pole, in the squared form +`(dᵢ − σ)(dᵢ + σ)`. + +**Hermitian eigenproblems** follow the same pattern: the tridiagonal QL/QR +iteration (steqr) is used without vectors and for orders up to 25, divide +and conquer (stedc) above, which took n = 1024 from 3.6 s to 0.13 s. + +**General eigenproblems** use the classical double-shift QR (dlahqr), not +LAPACK's multishift QR with aggressive early deflation (dlaqr0–5). That is +the largest gap to numpy, at large orders (below). + +## Accuracy and verification + +**Against numpy.** `linalg/testdata/cases.py` generates 1,040 cases with +numpy 2.5.3 (`uv run --with numpy python linalg/testdata/cases.py > +linalg/testdata/cases.json`; the JSON is embedded in the test binary, so the +emulated CI architectures read it from anywhere). They cover every +function; float64, float32, complex128, complex64, int32/int64 and bool +inputs; 0×0, 1×1, tall, wide, stacked and broadcast shapes; Hilbert +matrices (κ up to 10¹⁶), graded and rank-deficient matrices, exactly +singular ones, non-symmetric matrices with complex eigenvalues, Jordan +blocks and other defective matrices, NaN and Inf inputs; and every error +numpy raises on them (124 cases). The test checks, for each case: + +- errors: an error of the same kind (`ErrSingular`, `ErrNotPosDef`, + `ErrNoConvergence`, `ErrLinAlg` for non-finite input to `Eig`, + `ErrShape`, `ErrArgs`); +- dtype and shape of every result, exactly; +- unique values (solutions, inverses, determinants, Cholesky factors, + singular values, eigenvalues, least-squares solutions and residuals, + pseudo-inverses, ranks, norms, condition numbers, powers, products) + elementwise, within `32·n·ε·κ` of the largest entry of the result, with + `κ` the condition number of the problem (1 for singular values, Hermitian + eigenvalues and norms; `cond(A)` for solves and inverses; the effective + condition over the kept singular values for `Lstsq` and `Pinv`, squared for + the residuals), plus one float32 rounding for single-precision results; +- factors that are not unique through their identities: `‖QᴴQ − I‖`, + `‖A − QR‖/‖A‖`, R upper triangular; `‖UᴴU − I‖`, `‖VhVhᴴ − I‖`, + `‖A − U·S·Vh‖/‖A‖`; `‖VᴴV − I‖`, `‖AV − VΛ‖/‖A‖` with ascending + eigenvalues; for `Eig`, `‖Av − λv‖/‖A‖` per vector, unit norm, largest + component real, conjugate pairs exactly conjugate — all within + `32·n·ε` (`ε` of the result's dtype); +- general eigenvalues matched to numpy's nearest unused one, within 100 + times how far numpy's own eigenvalues move when each entry is perturbed by + 10⁻¹⁴ of itself (an empirical condition number, recorded in the fixture), + so a Jordan block's eigenvalue, accurate only to `ε^(1/4)`, is judged by + what is achievable. + +914 cases are value-checked and 124 are errors; the 2 deliberate departures +below are logged. + +**Beyond the fixture** (`properties_test.go`, `kernels_test.go`): the same +identities at orders 100 and 131 (above the blocking and divide-and-conquer +thresholds), real and complex, tall and wide; a stack of 16 matrices run in +parallel equals the matrices run one by one; repeated eigenvalues and +singular values, rank-deficient and zero matrices (the deflation paths); +cyclic permutations, Grcar and Frank matrices, orthogonal matrices, +high-order Jordan blocks and coupled rotation blocks (exceptional shifts, +overflow rescaling in the back-substitution); balancing at the limits of the +exponent range; every iteration forced to fail (`ErrNoConvergence` from +each function); and the 2×2 kernels (laev2, las2, lasv2, lanv2, lartg, +solve2) checked through their defining identities on the inputs that reach +their special cases. + +**Negative controls.** Dropping the row interchange in the LU leaf (no +partial pivoting) fails 64 fixture cases — 23 `solve`, 17 `inv`, 10 `det`, +10 `slogdet`, 4 `matrix_power` — and the order-100 identity test. Loosening +the secular-equation solver's stopping test by 10⁸ fails the four order-34 +`eigh` cases (eigenvalues off by ~10⁻¹¹ relative) and the zero-diagonal +bidiagonal divide-and-conquer test (`‖B − USVᵀ‖` of 10⁻⁸). + +**Coverage** is 100% of statements under the CI gate (`go test -race +-coverpkg=./... ./...`). + +## Speed against numpy + +Median wall time of `BenchmarkLinalg` against numpy 2.5.3 (`la.solve(a, +ones)`, `la.inv`, `la.cholesky` of `A·Aᵀ + nI`, `la.qr` reduced, +`la.svd(full_matrices=False)`, `la.eigh` of `A + Aᵀ`, `la.eig`), n×n +standard normal matrices, Apple M-series (16 cores, arm64), three +interleaved rounds. numpy's wheel uses **Apple Accelerate**. The host was +shared and heavily loaded (load average 16–23 on 16 cores during the runs), +so absolute numbers are noisy; the ratios held across rounds. + +| op | n | Go (ms) | numpy (ms) | Go / numpy | +|---|---|---|---|---| +| solve | 64 | 0.046 | 0.022 | 2.1 | +| solve | 256 | 0.99 | 0.25 | 3.9 | +| solve | 1024 | 21.2 | 10.9 | 1.9 | +| inv | 64 | 0.11 | 0.029 | 3.7 | +| inv | 256 | 2.19 | 0.58 | 3.8 | +| inv | 1024 | 39.4 | 23.7 | 1.7 | +| cholesky | 64 | 0.030 | 0.009 | 3.3 | +| cholesky | 256 | 0.54 | 0.24 | 2.2 | +| cholesky | 1024 | 10.9 | 6.6 | 1.7 | +| qr | 64 | 0.15 | 0.083 | 1.8 | +| qr | 256 | 2.32 | 1.57 | 1.5 | +| qr | 1024 | 54.7 | 37.3 | 1.5 | +| svd | 64 | 0.78 | 0.30 | 2.6 | +| svd | 256 | 13.3 | 5.25 | 2.5 | +| svd | 1024 | 263 | 123 | 2.1 | +| eigh | 64 | 0.41 | 0.22 | 1.9 | +| eigh | 256 | 6.54 | 3.12 | 2.1 | +| eigh | 1024 | 130 | 78 | 1.7 | +| eig | 64 | 1.07 | 1.06 | 1.0 | +| eig | 256 | 55 | 19.4 | 2.8 | +| eig | 1024 | 5060 | 712 | 7.1 | + +**numpy is faster on single matrices**, by 1.5–4× (7× for `eig` at 1024). +Most of that is the matrix product underneath: on the same host the ndarray +GEMM takes 18 ms for a 1024² product where Accelerate (which uses the +matrix coprocessor) takes 3.8 ms, 0.37 ms against 0.10 ms at 256. The +blocked algorithms here spend most of their time in that GEMM, so their gap +to numpy is smaller than the GEMM's. The exceptions: + +- `eig` at large n: the classical double-shift QR is O(10n³) of scalar + work; LAPACK's multishift QR with aggressive early deflation moves it into + GEMMs. This is the main follow-up. +- small orders (64): fixed costs (conversion of the input, allocation, + the packing of a small GEMM) weigh more than the arithmetic. + +**Stacks of small matrices run in parallel** — numpy's gufuncs loop over +them on one core — and there this package is faster (median of 3, same +host): + +| op | stack | Go (ms) | numpy (ms) | numpy / Go | +|---|---|---|---|---| +| inv | 10000 × 4×4 | 0.94 | 3.83 | 4.1 | +| det | 10000 × 4×4 | 0.81 | 2.12 | 2.6 | +| eigh | 10000 × 4×4 | 3.0 | 13.8 | 4.6 | +| svd | 10000 × 4×4 | 5.3 | 29.5 | 5.6 | +| eig | 10000 × 4×4 | 4.9 | 55.7 | 11 | +| inv | 1000 × 32×32 | 3.4 | 8.5 | 2.5 | +| det | 1000 × 32×32 | 1.6 | 4.0 | 2.5 | +| eigh | 1000 × 32×32 | 21.3 | 50.4 | 2.4 | +| svd | 1000 × 32×32 | 44.3 | 75.9 | 1.7 | +| eig | 1000 × 32×32 | 29.7 | 232 | 7.8 | + +Reproduce: `go test -run XXX -bench 'Linalg|Stacked' ./linalg`, and the +numpy side with `uv run --with numpy==2.5.3 python docs/bench_linalg_numpy.py` +(the same sizes and distributions; the values differ, being from another +generator). + +## Differences from numpy + +| numpy | here | why | +|---|---|---| +| `det` is `sign·exp(logabsdet)` | the product of the pivots, accumulated as mantissa and exponent | exact to rounding, no spurious overflow or underflow of intermediate products, and no `|logdet|·ε` error from `exp` | +| `norm` of a vector with entries near 10²⁰⁰ overflows to `inf` | the 2-norms are rescaled (`1.414e200` for `[1e200, 1e200]`) | the true value is representable | +| float32 norms accumulate in float32 | in float64, then rounded | more accurate; within a float32 rounding of numpy | +| `cond(singular, 'nuc')` raises `LinAlgError` (the SVD of its NaN "inverse" fails) | `+Inf`, as for every other order | a singular matrix has infinite condition number | +| `eigh` of a matrix with a NaN returns NaN eigenvalues | `ErrNoConvergence` | the iteration cannot proceed; numpy's `svd` and `eig` refuse NaN too | +| `LinAlgError` for a wrong rank or a non-square matrix | `ErrShape` | a shape error, not a numerical failure | +| `svd(…, hermitian=True)`, `pinv(…, hermitian=True)`, `matrix_rank(…, hermitian=True)`, `matrix_rank(rtol=)`, `pinv(rtol=)` | not provided (`PinvRcond(a, max(M,N)·ε)` is `rtol=None`) | follow-ups | +| `tensorsolve`, `tensorinv`, `cross`, `outer`, `trace`, `diagonal`, `matmul`, `vecdot`, `matrix_norm`, `vector_norm`, `matrix_transpose` | not in this package (`ndarray` has `MatMul`, `Outer`, `Transpose`; `Norm` covers both norms) | follow-ups | +| `qr` modes `'full'` and `'economic'` (deprecated) | not provided | deprecated in numpy | + +## What is left + +- Multishift QR with aggressive early deflation for `Eig` (the 7× gap at + n = 1024), and a blocked Hessenberg reduction. +- A faster small-matrix path (n ≤ 64): unblocked kernels without the GEMM + packing, fewer copies. +- The `hermitian` options and `rtol` arguments above, and the tensor + functions. +- `Eigh` on NaN input returning NaNs like numpy, if that is preferred. diff --git a/internal/kernels/gemm.go b/internal/kernels/gemm.go index bfc0afa..18b585b 100644 --- a/internal/kernels/gemm.go +++ b/internal/kernels/gemm.go @@ -434,10 +434,15 @@ func dotRange(a, b []float64) float64 { // is pure unit-stride streaming — far cheaper than routing a single-column GEMM // through the packer. dst need not be pre-zeroed (each entry is written, not // accumulated). Rows are disjoint, so the parallel result equals the serial one. -func MatVecP(dst, a, v []float64, m, k int) { +func MatVecP(dst, a, v []float64, m, k int) { MatVecStridedP(dst, a, k, v, m, k) } + +// MatVecStridedP is MatVecP on rows lda elements apart: row i of a is +// a[i*lda : i*lda+k], so a sub-block of a larger row-major matrix is +// multiplied in place. +func MatVecStridedP(dst, a []float64, lda int, v []float64, m, k int) { body := func(lo, hi int) { for i := lo; i < hi; i++ { - dst[i] = dotSIMD(a[i*k:i*k+k], v) + dst[i] = dotSIMD(a[i*lda:i*lda+k], v) } } if m*k < matVecThreshold { diff --git a/internal/kernels/gemm_test.go b/internal/kernels/gemm_test.go index 60774d9..be421fd 100644 --- a/internal/kernels/gemm_test.go +++ b/internal/kernels/gemm_test.go @@ -287,6 +287,24 @@ func TestMatVecP(t *testing.T) { withMaxProcs(128, func() { check(1) }) // parallel + w>m clamp } +// TestMatVecStridedP checks mat·vec on a sub-block of a wider matrix: only +// the first k of every lda elements are read. +func TestMatVecStridedP(t *testing.T) { + m, k, lda := 5, 6, 11 + a, v := randVec(m*lda, 5), randVec(k, 6) + got := make([]float64, m) + MatVecStridedP(got, a, lda, v, m, k) + for i := 0; i < m; i++ { + want := 0.0 + for p := 0; p < k; p++ { + want += a[i*lda+p] * v[p] + } + if math.Abs(got[i]-want) > 1e-9*(1+math.Abs(want)) { + t.Fatalf("MatVecStridedP [%d]: %v != %v", i, got[i], want) + } + } +} + // TestVecMatP checks vec·mat (1-D · 2-D) against a column-accumulation reference. func TestVecMatP(t *testing.T) { k, n := 11, 9 diff --git a/linalg/bdc.go b/linalg/bdc.go new file mode 100644 index 0000000..0e6e3f9 --- /dev/null +++ b/linalg/bdc.go @@ -0,0 +1,262 @@ +package linalg + +import ( + "math" + "sort" +) + +// The divide-and-conquer SVD of a bidiagonal matrix (Gu and Eisenstat's +// method, as LAPACK's dbdsdc/dlasd0-4, which numpy's gesdd calls). A +// bidiagonal B is split at a middle row into two smaller bidiagonals; from +// their SVDs, B's is that of an "arrow" matrix — a first row z over a +// diagonal — whose singular values are the roots of a secular equation and +// whose singular vectors are explicit. Products with the halves' vectors +// are GEMMs, which is where the work is; the QR iteration it replaces +// spends O(n³) in plane rotations instead. + +// bdsdc returns the SVD B = U·diag(d)·VT of the n×n upper bidiagonal (d, e): +// singular values decreasing into d, U and VT n×n row-major. e is +// destroyed. +func bdsdc(n int, d, e []float64) (u, vt []float64, ok bool) { + return bdRec(n, 0, d, e) +} + +// bdRec is the SVD of the n×(n+sqre) upper bidiagonal (d, e) — e has +// n-1+sqre entries — as U (n×n) and VT ((n+sqre)×(n+sqre)), the singular +// values decreasing into d. With sqre = 1, the last row of VT spans the +// null space. +func bdRec(n, sqre int, d, e []float64) ([]float64, []float64, bool) { + if n <= dcLeaf { + return bdLeaf(n, sqre, d, e) + } + k := n / 2 + nl, nr := k, n-k-1 + alpha, beta := d[k], e[k] + u1, vt1, ok1 := bdRec(nl, 1, d[:nl], e[:nl]) + u2, vt2, ok2 := bdRec(nr, sqre, d[k+1:], e[k+1:]) + if !ok1 || !ok2 { + return nil, nil, false + } + u, vt := bdMerge(n, sqre, k, d, alpha, beta, u1, vt1, u2, vt2) + return u, vt, true +} + +// bdLeaf solves a small bidiagonal by implicit QR (bdsqr). A non-square +// one is first made square: plane rotations on the columns chase its last +// entry up and out (LAPACK dlasdq), and only VT sees them. +func bdLeaf(n, sqre int, d, e []float64) ([]float64, []float64, bool) { + m := n + sqre + w := make([]float64, m*m) + for i := 0; i < m; i++ { + w[i*m+i] = 1 + } + if sqre == 1 && n > 0 { + f := e[n-1] // the entry to remove, at (i, n) + for i := n - 1; i >= 0; i-- { + c, s, r := lartg(d[i], f) + d[i] = r + rot(w[i*m:(i+1)*m], w[n*m:(n+1)*m], c, s) + if i > 0 { + f = -s * e[i-1] + e[i-1] *= c + } + } + } + ut := make([]float64, n*n) + for i := 0; i < n; i++ { + ut[i*n+i] = 1 + } + if !bdsqr(n, d, e, w, m, ut, n) { + return nil, nil, false + } + return transposed(n, n, ut), w, true +} + +// bdMerge forms the SVD of the n×(n+sqre) bidiagonal split at row k (dlasd1): +// d[:k] and d[k+1:] hold the halves' singular values, (u1, vt1) and +// (u2, vt2) their vectors, alpha and beta row k's two entries. +func bdMerge(n, sqre, k int, d []float64, alpha, beta float64, u1, vt1, u2, vt2 []float64) ([]float64, []float64) { + m := n + sqre + nl, nr := k, n-k-1 + m1, m2 := nl+1, nr+sqre + // Scale to avoid overflow and underflow. + orgnrm := math.Max(math.Abs(alpha), math.Abs(beta)) + for i := 0; i < n; i++ { + if i != k { + orgnrm = math.Max(orgnrm, math.Abs(d[i])) + } + } + if orgnrm == 0 { + orgnrm = 1 + } + alpha /= orgnrm + beta /= orgnrm + // The bases, transposed (one vector per row), in the arrow's order: row + // 0 is the arrow's first row (left: e_k; right: the halves' null + // vectors), then the halves' singular pairs. + ut := make([]float64, n*n) + vtf := make([]float64, m*m) + pole := make([]float64, n) + z := make([]float64, n) + ut[k] = 1 + for i := 0; i < nl; i++ { + row := ut[(1+i)*n:] + for r := 0; r < nl; r++ { + row[r] = u1[r*nl+i] + } + copy(vtf[(1+i)*m:(1+i)*m+m1], vt1[i*m1:(i+1)*m1]) + pole[1+i] = d[i] / orgnrm + z[1+i] = alpha * vt1[i*m1+nl] + } + for i := 0; i < nr; i++ { + row := ut[(1+nl+i)*n:] + for r := 0; r < nr; r++ { + row[k+1+r] = u2[r*nr+i] + } + copy(vtf[(1+nl+i)*m+m1:(2+nl+i)*m], vt2[i*m2:(i+1)*m2]) + pole[1+nl+i] = d[k+1+i] / orgnrm + z[1+nl+i] = beta * vt2[i*m2] + } + null1 := vt1[nl*m1 : (nl+1)*m1] + a1 := alpha * null1[nl] + if sqre == 1 { + // Two null vectors: rotate them into one that meets row k and one + // that stays null. + null2 := vt2[nr*m2 : (nr+1)*m2] + a2 := beta * null2[0] + r0 := lapy2(a1, a2) + c, s := 1.0, 0.0 + if r0 != 0 { + c, s = a1/r0, a2/r0 + } + for i, v := range null1 { + vtf[i] = c * v + vtf[(m-1)*m+i] = -s * v + } + for i, v := range null2 { + vtf[m1+i] = s * v + vtf[(m-1)*m+m1+i] = c * v + } + z[0] = r0 + } else { + copy(vtf[:m1], null1) + z[0] = a1 + } + // Sort the poles ascending; the arrow's 0 stays first. + idx := make([]int, n) + for i := range idx { + idx[i] = i + } + sort.SliceStable(idx[1:], func(a, b int) bool { return pole[idx[1+a]] < pole[idx[1+b]] }) + // Deflation (dlasd2). + tol := 8 * eps * math.Max(math.Max(math.Abs(alpha), math.Abs(beta)), pole[idx[n-1]]) + urow := func(g int) []float64 { return ut[g*n : (g+1)*n] } + vrow := func(g int) []float64 { return vtf[g*m : (g+1)*m] } + keep := []int{0} + var defl []int + pj := -1 + for _, g := range idx[1:] { + if math.Abs(z[g]) <= tol { + defl = append(defl, g) + continue + } + if pj < 0 { + pj = g + continue + } + if math.Abs(pole[g]-pole[pj]) <= tol { + // Close poles: a rotation zeroes z at pj. + s, c := z[pj], z[g] + tau := lapy2(c, s) + c /= tau + s = -s / tau + z[g], z[pj] = tau, 0 + rot(urow(pj), urow(g), c, s) + rot(vrow(pj), vrow(g), c, s) + defl = append(defl, pj) + } else { + keep = append(keep, pj) + } + pj = g + } + if pj >= 0 { + keep = append(keep, pj) + } + if math.Abs(z[0]) <= tol { + z[0] = tol + } + kk := len(keep) + sk, zk := make([]float64, kk), make([]float64, kk) + for i, g := range keep { + sk[i], zk[i] = pole[g], z[g] + } + if kk > 1 && sk[1] <= tol/2 { + sk[1] = tol / 2 + } + // The secular equation in the squared poles. + p := make([]float64, kk) + for i, v := range sk { + p[i] = v * v + } + diff := func(i, o int) float64 { return (sk[i] - sk[o]) * (sk[i] + sk[o]) } + delta := make([]float64, kk*kk) + sigma := make([]float64, kk) + for j := 0; j < kk; j++ { + sigma[j] = math.Sqrt(secular(p, zk, 1, j, diff, delta[j*kk:(j+1)*kk])) + } + zh := zHat(p, zk, delta, diff) + // The arrow's singular vectors: right ∝ zhat_i/(s_i² - σ²), left ∝ + // (-1, s_i·zhat_i/(s_i² - σ²)). + vr := make([]float64, kk*kk) + ul := make([]float64, kk*kk) + for j := 0; j < kk; j++ { + r, l := vr[j*kk:(j+1)*kk], ul[j*kk:(j+1)*kk] + for i := range r { + r[i] = zh[i] / delta[j*kk+i] + l[i] = sk[i] * r[i] + } + l[0] = -1 + sr, sl := 1/nrm2(kk, r, 1), 1/nrm2(kk, l, 1) + for i := range r { + r[i] *= sr + l[i] *= sl + } + } + gu, gv := make([]float64, kk*n), make([]float64, kk*m) + for i, g := range keep { + copy(gu[i*n:(i+1)*n], urow(g)) + copy(gv[i*m:(i+1)*m], vrow(g)) + } + outU := make([]float64, n*n) + outV := make([]float64, m*m) + gemm(kk, n, kk, 1, rm(ul, 0, kk), rm(gu, 0, n), 0, outU, n) + gemm(kk, m, kk, 1, rm(vr, 0, kk), rm(gv, 0, m), 0, outV, m) + vals := append(sigma, make([]float64, len(defl))...) + for i, g := range defl { + vals[kk+i] = pole[g] + copy(outU[(kk+i)*n:(kk+i+1)*n], urow(g)) + copy(outV[(kk+i)*m:(kk+i+1)*m], vrow(g)) + } + if sqre == 1 { + copy(outV[(m-1)*m:], vtf[(m-1)*m:]) + } + // Sort decreasing and unscale. + ord := make([]int, n) + for i := range ord { + ord[i] = i + } + sort.SliceStable(ord, func(a, b int) bool { return vals[ord[a]] > vals[ord[b]] }) + u := make([]float64, n*n) + vt := make([]float64, m*m) + for i, o := range ord { + d[i] = vals[o] * orgnrm + for r := 0; r < n; r++ { + u[r*n+i] = outU[o*n+r] + } + copy(vt[i*m:(i+1)*m], outV[o*m:(o+1)*m]) + } + if sqre == 1 { + copy(vt[(m-1)*m:], outV[(m-1)*m:]) + } + return u, vt +} diff --git a/linalg/bench_test.go b/linalg/bench_test.go new file mode 100644 index 0000000..d32e631 --- /dev/null +++ b/linalg/bench_test.go @@ -0,0 +1,117 @@ +package linalg + +import ( + "fmt" + "math/rand/v2" + "testing" + + "github.com/go-ndarray/ndarray" +) + +// benchMatrix is a seeded n×n standard normal matrix; spd makes it +// symmetric positive definite (A·Aᵀ + n·I), sym symmetric. +func benchMatrix(n int, kind string) *ndarray.Array { + r := rand.New(rand.NewPCG(uint64(n), 7)) + d := make([]float64, n*n) + for i := range d { + d[i] = r.NormFloat64() + } + a, _ := ndarray.FromSlice(d, n, n) + switch kind { + case "spd": + s, _ := a.MatMul(a.Transpose()) + for i := 0; i < n; i++ { + s.Set(s.At(i, i)+float64(n), i, i) + } + return s + case "sym": + s, _ := a.Add(a.Transpose()) + return s + } + return a +} + +var benchSizes = []int{64, 256, 1024} + +// The operations measured against numpy in docs/linalg.md. +var benchOps = []struct { + name, kind string + f func(a *ndarray.Array) error +}{ + {"solve", "", func(a *ndarray.Array) error { + b, _ := ndarray.Ones(a.Shape()[0]) + _, err := Solve(a, b) + return err + }}, + {"inv", "", func(a *ndarray.Array) error { _, err := Inv(a); return err }}, + {"cholesky", "spd", func(a *ndarray.Array) error { _, err := Cholesky(a, Lower); return err }}, + {"qr", "", func(a *ndarray.Array) error { _, _, err := QR(a, ModeReduced); return err }}, + {"svd", "", func(a *ndarray.Array) error { _, _, _, err := SVD(a, false); return err }}, + {"eigh", "sym", func(a *ndarray.Array) error { _, _, err := Eigh(a, Lower); return err }}, + {"eig", "", func(a *ndarray.Array) error { _, _, err := Eig(a); return err }}, +} + +func BenchmarkLinalg(b *testing.B) { + for _, op := range benchOps { + for _, n := range benchSizes { + a := benchMatrix(n, op.kind) + b.Run(fmt.Sprintf("%s/%d", op.name, n), func(b *testing.B) { + for b.Loop() { + if err := op.f(a); err != nil { + b.Fatal(err) + } + } + }) + } + } +} + +// BenchmarkStacked measures stacks of small matrices, which run one matrix +// per goroutine (numpy's gufuncs loop over them on one core). +func BenchmarkStacked(b *testing.B) { + r := rand.New(rand.NewPCG(9, 9)) + for _, sh := range [][2]int{{10000, 4}, {1000, 32}} { + cnt, n := sh[0], sh[1] + d := make([]float64, cnt*n*n) + for i := range d { + d[i] = r.NormFloat64() + } + a, _ := ndarray.FromSlice(d, cnt, n, n) + s, _ := a.Add(swapLast(a)) + for _, op := range []struct { + name string + f func() error + }{ + {"inv", func() error { _, err := Inv(a); return err }}, + {"det", func() error { _, err := Det(a); return err }}, + {"eigh", func() error { _, _, err := Eigh(s, Lower); return err }}, + {"svd", func() error { _, _, _, err := SVD(a, false); return err }}, + {"eig", func() error { _, _, err := Eig(a); return err }}, + } { + b.Run(fmt.Sprintf("%s/%dx%d", op.name, cnt, n), func(b *testing.B) { + for b.Loop() { + if err := op.f(); err != nil { + b.Fatal(err) + } + } + }) + } + } +} + +// swapLast transposes the last two axes of a 3-D array. +func swapLast(a *ndarray.Array) *ndarray.Array { + s := a.Shape() + d, _ := ndarray.Data[float64](a) + out := make([]float64, len(d)) + m := s[1] + for b := 0; b < s[0]; b++ { + for i := 0; i < m; i++ { + for j := 0; j < m; j++ { + out[b*m*m+j*m+i] = d[b*m*m+i*m+j] + } + } + } + t, _ := ndarray.FromSlice(out, s...) + return t +} diff --git a/linalg/blas.go b/linalg/blas.go new file mode 100644 index 0000000..8b7688a --- /dev/null +++ b/linalg/blas.go @@ -0,0 +1,569 @@ +package linalg + +import ( + "math" + "math/cmplx" + + "github.com/go-ndarray/ndarray/internal/kernels" +) + +// The algorithms are written once, generic over the two computation types. +// The helpers below switch on the type parameter; the compiler stencils +// float64 and complex128 separately and folds the switch away, so the real +// instantiation pays nothing for the complex case. +type scalar interface{ float64 | complex128 } + +// Machine constants, as LAPACK's dlamch reports them for IEEE double. +const ( + eps = 0x1p-53 // relative machine precision, dlamch('E') + ulp = 0x1p-52 // eps * base, dlamch('P') + safmin = 0x1p-1022 // smallest normal number, dlamch('S') +) + +func isCplx[T scalar]() bool { + var z T + _, ok := any(z).(complex128) + return ok +} + +func conj[T scalar](x T) T { + if v, ok := any(x).(complex128); ok { + return any(complex(real(v), -imag(v))).(T) + } + return x +} + +func re[T scalar](x T) float64 { + if v, ok := any(x).(complex128); ok { + return real(v) + } + return any(x).(float64) +} + +func im[T scalar](x T) float64 { + if v, ok := any(x).(complex128); ok { + return imag(v) + } + return 0 +} + +func fromReal[T scalar](r float64) T { + var z T + if _, ok := any(z).(complex128); ok { + return any(complex(r, 0)).(T) + } + return any(r).(T) +} + +// abs is the modulus |x|. +func abs[T scalar](x T) float64 { + if v, ok := any(x).(complex128); ok { + return cmplx.Abs(v) + } + return math.Abs(any(x).(float64)) +} + +// abs1 is |re x| + |im x|, LAPACK's cabs1 (|x| for a real x). +func abs1[T scalar](x T) float64 { + if v, ok := any(x).(complex128); ok { + return math.Abs(real(v)) + math.Abs(imag(v)) + } + return math.Abs(any(x).(float64)) +} + +// rscale is x*r for a real r, without the cross terms (and NaNs from 0*Inf) +// of a complex product. +func rscale[T scalar](x T, r float64) T { + if v, ok := any(x).(complex128); ok { + return any(complex(real(v)*r, imag(v)*r)).(T) + } + return any(any(x).(float64) * r).(T) +} + +// absSq is |x|². +func absSq[T scalar](x T) float64 { + if v, ok := any(x).(complex128); ok { + return real(v)*real(v) + imag(v)*imag(v) + } + f := any(x).(float64) + return f * f +} + +// nrm2 is the Euclidean norm of x[0], x[inc], ..., n elements, safe from +// overflow and underflow. +func nrm2[T scalar](n int, x []T, inc int) float64 { + var ss float64 + for i := 0; i < n; i++ { + ss += absSq(x[i*inc]) + } + if ss > 0x1p-900 && ss < 0x1p900 { + return math.Sqrt(ss) + } + if math.IsNaN(ss) { + return ss + } + // Zero, underflowed or overflowed: rescale by the largest component. + var big float64 + for i := 0; i < n; i++ { + big = math.Max(big, math.Max(math.Abs(re(x[i*inc])), math.Abs(im(x[i*inc])))) + } + if big == 0 || math.IsInf(big, 1) { + return big + } + ss = 0 + for i := 0; i < n; i++ { + r, c := re(x[i*inc])/big, im(x[i*inc])/big + ss += r*r + c*c + } + return big * math.Sqrt(ss) +} + +// lapy2 is sqrt(x²+y²) without destructive overflow. +func lapy2(x, y float64) float64 { return math.Hypot(x, y) } + +// lapy3 is sqrt(x²+y²+z²) without destructive overflow. +func lapy3(x, y, z float64) float64 { + w := math.Max(math.Abs(x), math.Max(math.Abs(y), math.Abs(z))) + if w == 0 { + return math.Abs(x) + math.Abs(y) + math.Abs(z) + } + x, y, z = x/w, y/w, z/w + return w * math.Sqrt(x*x+y*y+z*z) +} + +// mview is a strided matrix operand: element (i, j) is d[off+i*rs+j*cs], +// conjugated when cj is set. Transposition swaps the strides. +type mview[T scalar] struct { + d []T + off int + rs, cs int + conjVal bool +} + +// rm views a row-major matrix with leading dimension ld starting at d[off]. +func rm[T scalar](d []T, off, ld int) mview[T] { return mview[T]{d: d, off: off, rs: ld, cs: 1} } + +func (v mview[T]) at(i, j int) T { + x := v.d[v.off+i*v.rs+j*v.cs] + if v.conjVal { + return conj(x) + } + return x +} + +// c is the conjugate, t the transpose, h the conjugate transpose. +func (v mview[T]) c() mview[T] { v.conjVal = !v.conjVal; return v } +func (v mview[T]) t() mview[T] { v.rs, v.cs = v.cs, v.rs; return v } +func (v mview[T]) h() mview[T] { + v.rs, v.cs = v.cs, v.rs + v.conjVal = !v.conjVal + return v +} + +// gemmMin is the smallest m*n*k for which gemm hands the product to the +// packed SIMD GEMM of the ndarray kernels; below it a plain loop is faster +// than the packing. +var gemmMin = 1 << 13 + +// gemm computes C = beta*C + alpha*A*B for an m×k A, a k×n B and an m×n +// row-major C (leading dimension ldc, from c[0]); beta is 0 or 1. +func gemm[T scalar](m, n, k int, alpha T, a, b mview[T], beta T, c []T, ldc int) { + if m == 0 || n == 0 { + return + } + if beta == 0 { + for i := 0; i < m; i++ { + clear(c[i*ldc : i*ldc+n]) + } + } + if k == 0 { + return + } + if m*n*k >= gemmMin && min(m, n, k) >= 4 { + fastGemm(m, n, k, alpha, a, b, c, ldc) + return + } + for i := 0; i < m; i++ { + ci := c[i*ldc : i*ldc+n] + for p := 0; p < k; p++ { + aip := alpha * a.at(i, p) + if b.cs == 1 && !b.conjVal { + bp := b.d[b.off+p*b.rs : b.off+p*b.rs+n] + for j, x := range bp { + ci[j] += aip * x + } + continue + } + for j := range ci { + ci[j] += aip * b.at(p, j) + } + } + } +} + +// fastGemm adds alpha*A*B to C through the float64 packed GEMM: directly +// for real operands, as four real products for complex ones. +func fastGemm[T scalar](m, n, k int, alpha T, a, b mview[T], c []T, ldc int) { + tmp := make([]float64, m*n) + if cf, ok := any(c).([]float64); ok { + av, bv := any(a).(mview[float64]), any(b).(mview[float64]) + kernels.MatMulStridedP(tmp, kmat(av), kmat(bv), m, k, n) + al := any(alpha).(float64) + for i := 0; i < m; i++ { + row := cf[i*ldc : i*ldc+n] + t := tmp[i*n : i*n+n] + for j := range row { + row[j] += al * t[j] + } + } + return + } + ar, ai := split(m, k, a) + br, bi := split(k, n, b) + ti := make([]float64, m*n) + ra, rb := kernels.Mat{Data: ar, RS: k, CS: 1}, kernels.Mat{Data: br, RS: n, CS: 1} + ia, ib := kernels.Mat{Data: ai, RS: k, CS: 1}, kernels.Mat{Data: bi, RS: n, CS: 1} + kernels.MatMulStridedP(tmp, ra, rb, m, k, n) // Re·Re + kernels.MatMulStridedP(ti, ia, ib, m, k, n) // Im·Im + for i := range tmp { + tmp[i] -= ti[i] + } + t3 := make([]float64, m*n) + clear(ti) + kernels.MatMulStridedP(ti, ra, ib, m, k, n) // Re·Im + kernels.MatMulStridedP(t3, ia, rb, m, k, n) // Im·Re + cz := any(c).([]complex128) + al := any(alpha).(complex128) + for i := 0; i < m; i++ { + row := cz[i*ldc : i*ldc+n] + for j := range row { + row[j] += al * complex(tmp[i*n+j], ti[i*n+j]+t3[i*n+j]) + } + } +} + +func kmat(v mview[float64]) kernels.Mat { + return kernels.Mat{Data: v.d, Off: v.off, RS: v.rs, CS: v.cs} +} + +// split copies an r×c complex operand into contiguous real and imaginary +// parts. +func split[T scalar](r, c int, v mview[T]) (re, im []float64) { + re, im = make([]float64, r*c), make([]float64, r*c) + for i := 0; i < r; i++ { + for j := 0; j < c; j++ { + z := any(v.at(i, j)).(complex128) + re[i*c+j], im[i*c+j] = real(z), imag(z) + } + } + return re, im +} + +// trsmLeaf is the order at which the recursive triangular solves and +// factorizations switch to their unblocked loops. +var trsmLeaf = 32 + +// trsmLLU solves L·X = B in place of B, L n×n unit lower triangular (row-major +// at l, leading dimension ldl), B n×nrhs. +func trsmLLU[T scalar](n, nrhs int, l []T, ldl int, b []T, ldb int) { + if n <= trsmLeaf { + for i := 1; i < n; i++ { + bi := b[i*ldb : i*ldb+nrhs] + for k := 0; k < i; k++ { + lik := l[i*ldl+k] + bk := b[k*ldb : k*ldb+nrhs] + for j, x := range bk { + bi[j] -= lik * x + } + } + } + return + } + n1 := n / 2 + trsmLLU(n1, nrhs, l, ldl, b, ldb) + gemm(n-n1, nrhs, n1, -1, rm(l, n1*ldl, ldl), rm(b, 0, ldb), 1, b[n1*ldb:], ldb) + trsmLLU(n-n1, nrhs, l[n1*ldl+n1:], ldl, b[n1*ldb:], ldb) +} + +// trsmLUN solves U·X = B in place of B, U n×n upper triangular with a +// nonzero diagonal, B n×nrhs. +func trsmLUN[T scalar](n, nrhs int, u []T, ldu int, b []T, ldb int) { + if n <= trsmLeaf { + for i := n - 1; i >= 0; i-- { + bi := b[i*ldb : i*ldb+nrhs] + for k := i + 1; k < n; k++ { + uik := u[i*ldu+k] + bk := b[k*ldb : k*ldb+nrhs] + for j, x := range bk { + bi[j] -= uik * x + } + } + d := u[i*ldu+i] + for j := range bi { + bi[j] /= d + } + } + return + } + n1 := n / 2 + trsmLUN(n-n1, nrhs, u[n1*ldu+n1:], ldu, b[n1*ldb:], ldb) + gemm(n1, nrhs, n-n1, -1, rm(u, n1, ldu), rm(b, n1*ldb, ldb), 1, b, ldb) + trsmLUN(n1, nrhs, u, ldu, b, ldb) +} + +// trsmRLC solves X·Lᴴ = B in place of B, L n×n lower triangular with a +// nonzero diagonal, B m×n. +func trsmRLC[T scalar](m, n int, l []T, ldl int, b []T, ldb int) { + if n <= trsmLeaf { + for r := 0; r < m; r++ { + x := b[r*ldb : r*ldb+n] + for j := 0; j < n; j++ { + s := x[j] + lj := l[j*ldl : j*ldl+j] + for k, v := range lj { + s -= x[k] * conj(v) + } + x[j] = s / conj(l[j*ldl+j]) + } + } + return + } + n1 := n / 2 + trsmRLC(m, n1, l, ldl, b, ldb) + gemm(m, n-n1, n1, -1, rm(b, 0, ldb), rm(l, n1*ldl, ldl).h(), 1, b[n1:], ldb) + trsmRLC(m, n-n1, l[n1*ldl+n1:], ldl, b[n1:], ldb) +} + +// herkLower computes the lower triangle of C -= A·Aᴴ, A n×k, C n×n; the +// strictly upper triangle may be overwritten with garbage. +func herkLower[T scalar](n, k int, a []T, lda int, c []T, ldc int) { + if n <= 2*trsmLeaf { + gemm(n, n, k, -1, rm(a, 0, lda), rm(a, 0, lda).h(), 1, c, ldc) + return + } + n1 := n / 2 + herkLower(n1, k, a, lda, c, ldc) + gemm(n-n1, n1, k, -1, rm(a, n1*lda, lda), rm(a, 0, lda).h(), 1, c[n1*ldc:], ldc) + herkLower(n-n1, k, a[n1*lda:], lda, c[n1*ldc+n1:], ldc) +} + +// rot applies the plane rotation [c s; -s c] to the rows x and y: +// x' = c x + s y, y' = c y - s x. +func rot[T scalar](x, y []T, c, s float64) { + y = y[:len(x)] + for i, xv := range x { + yv := y[i] + x[i] = rscale(xv, c) + rscale(yv, s) + y[i] = rscale(yv, c) - rscale(xv, s) + } +} + +// lartg generates a plane rotation with c real: [c s; -s c]·[f; g] = [r; 0] +// (LAPACK dlartg, 3.10 and later). +func lartg(f, g float64) (c, s, r float64) { + const rtmin = 0x1p-511 // sqrt(safmin) + const rtmax = 0x1p+511 // sqrt(safmax/2) + switch { + case g == 0: + return 1, 0, f + case f == 0: + return 0, math.Copysign(1, g), math.Abs(g) + } + f1, g1 := math.Abs(f), math.Abs(g) + if f1 > rtmin && f1 < rtmax && g1 > rtmin && g1 < rtmax { + d := math.Sqrt(f*f + g*g) + c = f1 / d + r = math.Copysign(d, f) + return c, g / r, r + } + u := math.Min(1/safmin, math.Max(safmin, math.Max(f1, g1))) + fs, gs := f/u, g/u + d := math.Sqrt(fs*fs + gs*gs) + c = math.Abs(fs) / d + r = math.Copysign(d, f) + return c, gs / r, r * u +} + +// larfg generates an elementary reflector H = I - tau·v·vᴴ with +// Hᴴ·[alpha; x] = [beta; 0], beta real, v = [1; x'] (LAPACK zlarfg): x (n-1 +// elements, stride inc) is overwritten by x', and the new alpha (= beta) and +// tau are returned. +func larfg[T scalar](n int, alpha T, x []T, inc int) (T, T) { + xnorm := nrm2(n-1, x, inc) + alphr, alphi := re(alpha), im(alpha) + if xnorm == 0 && alphi == 0 { + return alpha, 0 + } + beta := -math.Copysign(lapy3(alphr, alphi, xnorm), alphr) + const sfm = safmin / eps + knt := 0 + if math.Abs(beta) < sfm { + // beta may be inaccurate: scale x and recompute. + for { + knt++ + for i := 0; i < n-1; i++ { + x[i*inc] = rscale(x[i*inc], 1/sfm) + } + beta /= sfm + alphi /= sfm + alphr /= sfm + if math.Abs(beta) >= sfm || knt >= 20 { + break + } + } + xnorm = nrm2(n-1, x, inc) + beta = -math.Copysign(lapy3(alphr, alphi, xnorm), alphr) + } + var tau, scal T + if isCplx[T]() { + tau = any(complex((beta-alphr)/beta, -alphi/beta)).(T) + scal = any(1 / (complex(alphr, alphi) - complex(beta, 0))).(T) + } else { + tau = fromReal[T]((beta - alphr) / beta) + scal = fromReal[T](1 / (alphr - beta)) + } + for i := 0; i < n-1; i++ { + x[i*inc] *= scal + } + for ; knt > 0; knt-- { + beta *= sfm + } + return fromReal[T](beta), tau +} + +// larfL applies H = I - tau·v·vᴴ from the left to the m×n matrix C (v has +// m elements, contiguous): C -= tau·v·(vᴴC). w is scratch of length n. +func larfL[T scalar](m, n int, v []T, tau T, c []T, ldc int, w []T) { + if tau == 0 || n == 0 { + return + } + w = w[:n] + clear(w) + for i := 0; i < m; i++ { + vi := conj(v[i]) + if vi == 0 { + continue + } + row := c[i*ldc : i*ldc+n] + for j, x := range row { + w[j] += vi * x + } + } + for i := 0; i < m; i++ { + f := tau * v[i] + if f == 0 { + continue + } + row := c[i*ldc : i*ldc+n] + for j := range row { + row[j] -= f * w[j] + } + } +} + +// larfR applies H = I - tau·v·vᴴ from the right to the m×n matrix C (v has +// n elements): C -= tau·(Cv)·vᴴ. +func larfR[T scalar](m, n int, v []T, tau T, c []T, ldc int) { + if tau == 0 || m == 0 { + return + } + v = v[:n] + for i := 0; i < m; i++ { + row := c[i*ldc : i*ldc+n] + var s T + for j, x := range row { + s += x * v[j] + } + s *= tau + if s == 0 { + continue + } + for j := range row { + row[j] -= s * conj(v[j]) + } + } +} + +// blockReflector is the compact WY form H1·H2···Hk = I - V·T·Vᴴ of k +// reflectors (LAPACK larft, forward, columnwise): V is m×k unit lower +// trapezoidal, T k×k upper triangular, both dense row-major. +type blockReflector[T scalar] struct { + m, k int + v, t []T +} + +// newBlockReflector gathers the k reflectors stored below the diagonal of +// the m×k panel at a (leading dimension lda), with scalars tau. +func newBlockReflector[T scalar](m, k int, a []T, lda int, tau []T) blockReflector[T] { + v := make([]T, m*k) + for i := 0; i < m; i++ { + for j := 0; j < k && j <= i; j++ { + if i == j { + v[i*k+j] = 1 + } else { + v[i*k+j] = a[i*lda+j] + } + } + } + return blockReflectorFromV(m, k, v, tau) +} + +func blockReflectorFromV[T scalar](m, k int, v []T, tau []T) blockReflector[T] { + g := make([]T, k*k) // g = Vᴴ V + gemm(k, k, m, 1, rm(v, 0, k).h(), rm(v, 0, k), 0, g, k) + t := make([]T, k*k) + for i := 0; i < k; i++ { + ti := tau[i] + // T(0:i, i) = -tau_i · T(0:i, 0:i) · (Vᴴ V)(0:i, i) + for j := 0; j < i; j++ { + var s T + for p := j; p < i; p++ { + s += t[j*k+p] * g[p*k+i] + } + t[j*k+i] = -ti * s + } + t[i*k+i] = ti + } + return blockReflector[T]{m: m, k: k, v: v, t: t} +} + +// applyLeft overwrites the m×n matrix C with H·C (conjT false) or Hᴴ·C +// (conjT true), H = I - V·T·Vᴴ. +func (br blockReflector[T]) applyLeft(n int, c []T, ldc int, conjT bool) { + m, k := br.m, br.k + w := make([]T, k*n) // W = Vᴴ C + gemm(k, n, m, 1, rm(br.v, 0, k).h(), rm(c, 0, ldc), 0, w, n) + tw := make([]T, k*n) // op(T) W + tv := rm(br.t, 0, k) + if conjT { + tv = tv.h() + } + gemm(k, n, k, 1, tv, rm(w, 0, n), 0, tw, n) + gemm(m, n, k, -1, rm(br.v, 0, k), rm(tw, 0, n), 1, c, ldc) +} + +// mulMixed returns the m×n product A·B of an m×k A of either type and a real +// k×n B, through the float64 GEMM (twice, for the parts of a complex A). +func mulMixed[T scalar](m, n, k int, a mview[T], b mview[float64]) []T { + out := make([]T, m*n) + if af, ok := any(a).(mview[float64]); ok { + gemm(m, n, k, 1, af, b, 0, any(out).([]float64), n) + return out + } + ar, ai := split(m, k, a) + re, im := make([]float64, m*n), make([]float64, m*n) + gemm(m, n, k, 1, rm(ar, 0, k), b, 0, re, n) + gemm(m, n, k, 1, rm(ai, 0, k), b, 0, im, n) + oz := any(out).([]complex128) + for i := range oz { + oz[i] = complex(re[i], im[i]) + } + return out +} + +// mulMixedLeft returns the m×n product A·B of a real m×k A and a k×n B of +// either type. +func mulMixedLeft[T scalar](m, n, k int, a mview[float64], b mview[T]) []T { + // (A·B)ᵀ = Bᵀ·Aᵀ + return transposed(n, m, mulMixed(n, m, k, b.t(), a.t())) +} diff --git a/linalg/cholesky.go b/linalg/cholesky.go new file mode 100644 index 0000000..4fe811b --- /dev/null +++ b/linalg/cholesky.go @@ -0,0 +1,100 @@ +package linalg + +import ( + "math" + + "github.com/go-ndarray/ndarray" +) + +// potrf factors the Hermitian positive definite n×n matrix whose lower +// triangle is at a (leading dimension lda) as L·Lᴴ, L overwriting the lower +// triangle (LAPACK potrf, recursive). The strictly upper triangle is used as +// scratch. It returns -1, or the index of the leading minor that is not +// positive definite. +func potrf[T scalar](n int, a []T, lda int) int { + if n <= trsmLeaf { + for j := 0; j < n; j++ { + rj := a[j*lda : j*lda+j] + d := re(a[j*lda+j]) + for _, v := range rj { + d -= absSq(v) + } + if !(d > 0) { // also catches NaN + return j + } + d = math.Sqrt(d) + a[j*lda+j] = fromReal[T](d) + for i := j + 1; i < n; i++ { + ri := a[i*lda : i*lda+j] + s := a[i*lda+j] + for k, v := range rj { + s -= ri[k] * conj(v) + } + a[i*lda+j] = rscale(s, 1/d) + } + } + return -1 + } + n1 := n / 2 + if info := potrf(n1, a, lda); info >= 0 { + return info + } + trsmRLC(n-n1, n1, a, lda, a[n1*lda:], lda) + herkLower(n-n1, n1, a[n1*lda:], lda, a[n1*lda+n1:], lda) + if info := potrf(n-n1, a[n1*lda+n1:], lda); info >= 0 { + return info + n1 + } + return -1 +} + +// Cholesky is numpy.linalg.cholesky: for each Hermitian positive definite +// matrix A of a (..., M, M), the lower triangular L with A = L·Lᴴ (uplo +// Lower), or the upper triangular U with A = Uᴴ·U (uplo Upper, numpy's +// upper=True). Only that triangle of A is read. A matrix that is not +// positive definite gives ErrNotPosDef. +func Cholesky(a *ndarray.Array, uplo UPLO) (*ndarray.Array, error) { + o, err := square(a, "Cholesky") + if err != nil { + return nil, err + } + if o.cplx { + return choleskyT[complex128](o, uplo) + } + return choleskyT[float64](o, uplo) +} + +func choleskyT[T scalar](o operand, uplo UPLO) (*ndarray.Array, error) { + n := o.n + d := load[T](o.a) + var fe firstErr + parallelFor(o.count(), n*n*n/3, func(i int) { + x := d[i*n*n : (i+1)*n*n] + if uplo == Upper { + conjTranspose(n, x) + } + if potrf(n, x, n) >= 0 { + fe.set(i, ErrNotPosDef) + return + } + for r := 0; r < n; r++ { + clear(x[r*n+r+1 : r*n+n]) + } + if uplo == Upper { + conjTranspose(n, x) + } + }) + if fe.err != nil { + return nil, fe.err + } + return wrap(d, shapeOf(o.batch, n, n), o.single), nil +} + +// conjTranspose replaces the n×n matrix x by its conjugate transpose. +func conjTranspose[T scalar](n int, x []T) { + for r := 0; r < n; r++ { + x[r*n+r] = conj(x[r*n+r]) + for c := r + 1; c < n; c++ { + x[r*n+c], x[c*n+r] = conj(x[c*n+r]), conj(x[r*n+c]) + } + } +} diff --git a/linalg/dc.go b/linalg/dc.go new file mode 100644 index 0000000..53e97dd --- /dev/null +++ b/linalg/dc.go @@ -0,0 +1,290 @@ +package linalg + +import ( + "math" + "sort" +) + +// The divide-and-conquer eigensolver for symmetric tridiagonal matrices +// (Cuppen's method in the form of LAPACK dstedc/dlaed0-4, with Gu and +// Eisenstat's recomputation of the updating vector, which keeps the +// eigenvectors numerically orthogonal). Its work is dominated by matrix +// products, which run on the packed GEMM; the implicit QL/QR iteration it +// replaces spends O(n³) in plane rotations instead. + +// dcLeaf is the order below which subproblems are solved by steqr. +var dcLeaf = 25 + +// stedc returns the eigenvalues of the symmetric tridiagonal (d, e), +// ascending (overwriting d), and the transpose of its eigenvector matrix: +// row k of the n×n result is the eigenvector of d[k]. e is destroyed. +func stedc(n int, d, e []float64) ([]float64, bool) { + if n <= dcLeaf { + zt := make([]float64, n*n) + for i := 0; i < n; i++ { + zt[i*n+i] = 1 + } + return zt, steqr(n, d, e, zt, n) + } + m := n / 2 + rho := e[m-1] + d[m-1] -= math.Abs(rho) + d[m] -= math.Abs(rho) + qt1, ok1 := stedc(m, d[:m], e[:m-1]) + qt2, ok2 := stedc(n-m, d[m:], e[m:]) + if !ok1 || !ok2 { + return nil, false + } + return dcMerge(n, m, d, rho, qt1, qt2), true +} + +// dcMerge solves T = diag(Q1·D1·Q1ᵀ, Q2·D2·Q2ᵀ) + |rho|·v·vᵀ (dlaed1): the +// eigenvalues D of the halves are in d[:m] and d[m:], ascending, and the +// transposed eigenvector matrices in qt1, qt2. +func dcMerge(n, m int, d []float64, rho float64, qt1, qt2 []float64) []float64 { + n2 := n - m + // Q, transposed, and the updating vector z = Qᵀv, of norm one. + qt := make([]float64, n*n) + z := make([]float64, n) + sgn := math.Copysign(1, rho) + for g := 0; g < m; g++ { + copy(qt[g*n:g*n+m], qt1[g*m:(g+1)*m]) + z[g] = qt1[g*m+m-1] / math.Sqrt2 + } + for g := 0; g < n2; g++ { + copy(qt[(m+g)*n+m:(m+g+1)*n], qt2[g*n2:(g+1)*n2]) + z[m+g] = sgn * qt2[g*n2] / math.Sqrt2 + } + rho = 2 * math.Abs(rho) + // Sort the poles (each half is ascending already). + idx := make([]int, n) + for i := range idx { + idx[i] = i + } + sort.SliceStable(idx, func(a, b int) bool { return d[idx[a]] < d[idx[b]] }) + ds, zs := make([]float64, n), make([]float64, n) + for i, g := range idx { + ds[i], zs[i] = d[g], z[g] + } + row := func(i int) []float64 { g := idx[i]; return qt[g*n : g*n+n] } + // Deflation (dlaed2): a negligible component of z, or two poles close + // enough that a rotation zeroes one component, leave an eigenpair of the + // halves unchanged. + zmax, dmax := 0.0, 0.0 + for i := range ds { + zmax = math.Max(zmax, math.Abs(zs[i])) + dmax = math.Max(dmax, math.Abs(ds[i])) + } + tol := 8 * eps * math.Max(dmax, zmax) + var keep, defl []int // positions in the sorted order + pj := -1 + for j := 0; j < n; j++ { + if rho*math.Abs(zs[j]) <= tol { + defl = append(defl, j) + continue + } + if pj < 0 { + pj = j + continue + } + s, c := zs[pj], zs[j] + tau := lapy2(c, s) + t := ds[j] - ds[pj] + c /= tau + s = -s / tau + if math.Abs(t*c*s) <= tol { + zs[j], zs[pj] = tau, 0 + rot(row(pj), row(j), c, s) + t = ds[pj]*c*c + ds[j]*s*s + ds[j] = ds[pj]*s*s + ds[j]*c*c + ds[pj] = t + defl = append(defl, pj) + } else { + keep = append(keep, pj) + } + pj = j + } + if pj >= 0 { + keep = append(keep, pj) + } + k := len(keep) + dk, zk := make([]float64, k), make([]float64, k) + for i, p := range keep { + dk[i], zk[i] = ds[p], zs[p] + } + // The secular equation, one root at a time; delta[j*k+i] = dk[i] - lambda_j. + lam := make([]float64, k) + delta := make([]float64, k*k) + diff := func(i, o int) float64 { return dk[i] - dk[o] } + for j := 0; j < k; j++ { + lam[j] = secular(dk, zk, rho, j, diff, delta[j*k:(j+1)*k]) + } + // Gu–Eisenstat: the z for which the computed roots are exact, then the + // eigenvectors (zhat_i / (d_i - lambda_j)), normalized. + zhat := zHat(dk, zk, delta, diff) + u := make([]float64, k*k) // row j: the coefficients of eigenvector j + for j := 0; j < k; j++ { + uj := u[j*k : (j+1)*k] + for i := range uj { + uj[i] = zhat[i] / delta[j*k+i] + } + s := 1 / nrm2(k, uj, 1) + for i := range uj { + uj[i] *= s + } + } + // New eigenvectors: rows of U·Q[keep], then the deflated ones. + gath := make([]float64, k*n) + for i, p := range keep { + copy(gath[i*n:(i+1)*n], row(p)) + } + out := make([]float64, n*n) + gemm(k, n, k, 1, rm(u, 0, k), rm(gath, 0, n), 0, out, n) + vals := append(lam, make([]float64, len(defl))...) + for i, p := range defl { + vals[k+i] = ds[p] + copy(out[(k+i)*n:(k+i+1)*n], row(p)) + } + // Sort ascending. + ord := make([]int, n) + for i := range ord { + ord[i] = i + } + sort.SliceStable(ord, func(a, b int) bool { return vals[ord[a]] < vals[ord[b]] }) + res := make([]float64, n*n) + for i, o := range ord { + d[i] = vals[o] + copy(res[i*n:(i+1)*n], out[o*n:(o+1)*n]) + } + return res +} + +// secular returns the j-th smallest root lambda of the secular equation +// 1/rho + Σ z_i²/(p_i - lambda) = 0 (poles p ascending and distinct, z +// without zeros, rho > 0), and stores p_i - lambda into delta, each +// computed from the pole nearest the root so that it is accurate to a few +// ulps even when the root is very close to a pole (LAPACK dlaed4's and +// dlasd4's requirement). diff(i, o) must return p_i - p_o accurately: for +// the squared poles of the SVD, (d_i - d_o)(d_i + d_o), not a difference of +// rounded squares. +// +// The iteration is the rational two-pole model of Bunch, Nielsen and +// Sorensen — the sum over the poles left of the root and the sum over those +// right of it are each replaced by a constant plus one pole, matching value +// and slope — safeguarded by a bracket that bisection falls back on. +func secular(p, z []float64, rho float64, j int, diff func(i, o int) float64, delta []float64) float64 { + k := len(p) + dd := delta // the poles relative to the origin + var origin int + var lo, hi float64 + if j == k-1 { + origin = k - 1 + for _, v := range z { + hi += v * v + } + hi *= rho + } else { + gap := diff(j+1, j) + f := 1 / rho + for i := range p { + f += z[i] * z[i] / (diff(i, j) - gap/2) + } + if f >= 0 { + origin, lo, hi = j, 0, gap/2 + } else { + origin, lo, hi = j+1, -gap/2, 0 + } + } + for i := range p { + dd[i] = diff(i, origin) + } + var p1, p2 float64 // the poles bounding the root + if j == k-1 { + p1 = dd[j] + } else { + p1, p2 = dd[j], dd[j+1] + } + tau := (lo + hi) / 2 + for it := 0; it < 200; it++ { + var psi, dpsi, phi, dphi float64 + for i := 0; i <= j; i++ { + t := z[i] / (dd[i] - tau) + psi += z[i] * t + dpsi += t * t + } + for i := j + 1; i < k; i++ { + t := z[i] / (dd[i] - tau) + phi += z[i] * t + dphi += t * t + } + f := 1/rho + psi + phi + if f < 0 { + lo = tau + } else { + hi = tau + } + erretm := 8*(math.Abs(psi)+math.Abs(phi)+1/rho) + math.Abs(tau)*(dpsi+dphi) + if math.Abs(f) <= eps*erretm || hi-lo <= 2*eps*math.Max(math.Abs(lo), math.Abs(hi)) { + break + } + // The two-pole model: c + b1/(p1-x) + b2/(p2-x) = 0. + b1 := dpsi * (p1 - tau) * (p1 - tau) + c := 1/rho + psi - dpsi*(p1-tau) + var x float64 + if j == k-1 { + c += phi + x = p1 + b1/c + } else { + b2 := dphi * (p2 - tau) * (p2 - tau) + c += phi - dphi*(p2-tau) + x = quadRoot(c, b1, b2, p1, p2, lo, hi) + } + if !(x > lo && x < hi) { + x = (lo + hi) / 2 + } + tau = x + } + for i := range dd { + dd[i] -= tau + } + return p[origin] + tau +} + +// zHat returns Gu and Eisenstat's updating vector: the z for which the +// computed roots are the exact eigenvalues of diag(p) + rho·z·zᵀ, with the +// signs of z. delta[j*k+i] is p_i - lambda_j. +func zHat(p, z, delta []float64, diff func(i, o int) float64) []float64 { + k := len(p) + zh := make([]float64, k) + for i := 0; i < k; i++ { + w := delta[i*k+i] + for j := 0; j < k; j++ { + if j != i { + w *= delta[j*k+i] / diff(i, j) + } + } + zh[i] = math.Copysign(math.Sqrt(math.Abs(w)), z[i]) + } + return zh +} + +// quadRoot solves c·(p1-x)(p2-x) + b1·(p2-x) + b2·(p1-x) = 0 for the root +// between lo and hi, or returns NaN. +func quadRoot(c, b1, b2, p1, p2, lo, hi float64) float64 { + a := c + b := -(c*(p1+p2) + b1 + b2) + cc := c*p1*p2 + b1*p2 + b2*p1 + if a == 0 { + return -cc / b + } + disc := b*b - 4*a*cc + if disc < 0 { + return math.NaN() + } + q := -0.5 * (b + math.Copysign(math.Sqrt(disc), b)) + r1, r2 := q/a, cc/q + if r1 > lo && r1 < hi { + return r1 + } + return r2 +} diff --git a/linalg/eig.go b/linalg/eig.go new file mode 100644 index 0000000..35c7ad5 --- /dev/null +++ b/linalg/eig.go @@ -0,0 +1,1111 @@ +package linalg + +import ( + "fmt" + "math" + "math/cmplx" + + "github.com/go-ndarray/ndarray" +) + +// errNonFinite is numpy's "Array must not contain infs or NaNs", raised by +// eig and eigvals before any work. +var errNonFinite = fmt.Errorf("%w: array must not contain infs or NaNs", ErrLinAlg) + +// gebal balances the n×n matrix a in place (LAPACK gebal, job 'B', 3.12): +// rows and columns that isolate eigenvalues are permuted to the ends, +// leaving the block ilo..ihi, which is then scaled by powers of two so its +// rows and columns have comparable norms. perm and scale record the +// permutation and scaling for gebak. +func gebal[T scalar](n int, a []T) (ilo, ihi int, perm []int, scale []float64) { + perm, scale = make([]int, n), make([]float64, n) + for i := range perm { + perm[i], scale[i] = i, 1 + } + at := func(i, j int) T { return a[i*n+j] } + swapCols := func(i, j, rows int) { + for r := 0; r < rows; r++ { + a[r*n+i], a[r*n+j] = a[r*n+j], a[r*n+i] + } + } + swapRowsFrom := func(i, j, c0 int) { + for c := c0; c < n; c++ { + a[i*n+c], a[j*n+c] = a[j*n+c], a[i*n+c] + } + } + k, l := 0, n-1 + // Rows isolating an eigenvalue go to the bottom. + for noconv := true; noconv; { + noconv = false + for i := l; i >= 0; i-- { + canSwap := true + for j := 0; j <= l; j++ { + if i != j && at(i, j) != 0 { + canSwap = false + break + } + } + if !canSwap { + continue + } + perm[l] = i + if i != l { + swapCols(i, l, l+1) + swapRowsFrom(i, l, k) + } + noconv = true + if l == 0 { + return 0, 0, perm, scale + } + l-- + } + } + // Columns isolating an eigenvalue go to the left. + for noconv := true; noconv; { + noconv = false + for j := k; j <= l; j++ { + canSwap := true + for i := k; i <= l; i++ { + if i != j && at(i, j) != 0 { + canSwap = false + break + } + } + if !canSwap { + continue + } + perm[k] = j + if j != k { + swapCols(j, k, l+1) + swapRowsFrom(j, k, k) + } + noconv = true + k++ + } + } + // Scale the block k..l. + const radix, factor = 2.0, 0.95 + sfmin1 := safmin / ulp + sfmax1 := 1 / sfmin1 + sfmin2 := sfmin1 * radix + sfmax2 := 1 / sfmin2 + for noconv := true; noconv; { + noconv = false + for i := k; i <= l; i++ { + c := nrm2(l-k+1, a[k*n+i:], n) + r := nrm2(l-k+1, a[i*n+k:], 1) + ca, ra := 0.0, 0.0 + bi, bv := 0, -1.0 + for row := 0; row <= l; row++ { + if v := abs1(at(row, i)); v > bv { + bi, bv = row, v + } + } + ca = abs(at(bi, i)) + bi, bv = k, -1.0 + for col := k; col < n; col++ { + if v := abs1(at(i, col)); v > bv { + bi, bv = col, v + } + } + ra = abs(at(i, bi)) + // c and r are nonzero: a row or column of the block without an + // off-diagonal entry would have been permuted out above. + g := r / radix + f := 1.0 + s := c + r + for c < g && math.Max(f, math.Max(c, ca)) < sfmax2 && math.Min(r, math.Min(g, ra)) > sfmin2 { + f *= radix + c *= radix + ca *= radix + r /= radix + g /= radix + ra /= radix + } + g = c / radix + for g >= r && math.Max(r, ra) < sfmax2 && math.Min(math.Min(f, c), math.Min(g, ca)) > sfmin2 { + f /= radix + c /= radix + g /= radix + ca /= radix + r *= radix + ra *= radix + } + if c+r >= factor*s { + continue + } + if f < 1 && scale[i] < 1 && f*scale[i] <= sfmin1 { + continue + } + if f > 1 && scale[i] > 1 && scale[i] >= sfmax1/f { + continue + } + scale[i] *= f + noconv = true + for col := k; col < n; col++ { + a[i*n+col] = rscale(a[i*n+col], 1/f) + } + for row := 0; row <= l; row++ { + a[row*n+i] = rscale(a[row*n+i], f) + } + } + } + return k, l, perm, scale +} + +// gebak undoes gebal on the right eigenvectors, the columns of the n×n v. +func gebak[T scalar](n, ilo, ihi int, perm []int, scale []float64, v []T) { + for i := ilo; i <= ihi; i++ { + if s := scale[i]; s != 1 { + row := v[i*n : i*n+n] + for c := range row { + row[c] = rscale(row[c], s) + } + } + } + for ii := 0; ii < n; ii++ { + i := ii + if i >= ilo && i <= ihi { + continue + } + if i < ilo { + i = ilo - 1 - ii + } + if k := perm[i]; k != i { + swapRows(v, n, i, k) + } + } +} + +// gehrd reduces rows and columns ilo..ihi of the n×n a to upper Hessenberg +// form H = Qᴴ·A·Q (LAPACK gehd2); with wantQ it returns Q, row-major. The +// entries below the subdiagonal are zeroed. +func gehrd[T scalar](n, ilo, ihi int, a []T, wantQ bool) []T { + nh := ihi - ilo // the number of reflectors + taus := make([]T, max(nh, 0)) + v := make([]T, n) + w := make([]T, n) + for i := ilo; i < ihi; i++ { + cnt := ihi - i // rows i+1..ihi + var x []T + if cnt > 1 { + x = a[(i+2)*n+i:] + } + beta, tau := larfg(cnt, a[(i+1)*n+i], x, n) + taus[i-ilo] = tau + v[0] = 1 + for r := 1; r < cnt; r++ { + v[r] = a[(i+1+r)*n+i] + } + larfR(ihi+1, cnt, v, tau, a[i+1:], n) + larfL(cnt, n-i-1, v[:cnt], conj(tau), a[(i+1)*n+i+1:], n, w) + a[(i+1)*n+i] = beta + } + var q []T + if wantQ { + q = make([]T, n*n) + for i := 0; i < n; i++ { + q[i*n+i] = 1 + } + if nh > 0 { + refl := make([]T, nh*nh) + for r := 1; r < nh; r++ { + for c := 0; c < r; c++ { + refl[r*nh+c] = a[(ilo+1+r)*n+ilo+c] + } + } + qs := orgqr(nh, nh, nh, refl, nh, taus) + for r := 0; r < nh; r++ { + copy(q[(ilo+1+r)*n+ilo+1:], qs[r*nh:(r+1)*nh]) + } + } + } + for r := 2; r < n; r++ { + clear(a[r*n : r*n+r-1]) + } + return q +} + +// lanv2 is the Schur factorization of the real 2×2 [a b; c d] in standard +// form (LAPACK dlanv2): it returns the new entries (either c = 0, or a = d +// and b·c < 0 for a complex pair), the eigenvalues and the rotation. +func lanv2(a, b, c, d float64) (aa, bb, cc, dd, rt1r, rt1i, rt2r, rt2i, cs, sn float64) { + const multpl = 4 + const safmn2 = 0x1p-485 // base**int(log(safmin/ulp)/log(base)/2) + const safmx2 = 1 / safmn2 + switch { + case c == 0: + cs, sn = 1, 0 + case b == 0: + // Swap rows and columns. + cs, sn = 0, 1 + a, d = d, a + b, c = -c, 0 + case a-d == 0 && math.Signbit(b) != math.Signbit(c): + cs, sn = 1, 0 + default: + temp := a - d + p := 0.5 * temp + bcmax := math.Max(math.Abs(b), math.Abs(c)) + bcmis := math.Min(math.Abs(b), math.Abs(c)) * math.Copysign(1, b) * math.Copysign(1, c) + scale := math.Max(math.Abs(p), bcmax) + z := (p/scale)*p + (bcmax/scale)*bcmis + if z >= multpl*ulp { + // Real eigenvalues: compute a and d. + z = p + math.Copysign(math.Sqrt(scale)*math.Sqrt(z), p) + a = d + z + d -= (bcmax / z) * bcmis + tau := lapy2(c, z) + cs, sn = z/tau, c/tau + b -= c + c = 0 + break + } + // Complex eigenvalues, or real (almost) equal ones: make the + // diagonal entries equal. + sigma := b + c + for count := 0; count < 20; count++ { + scale = math.Max(math.Abs(temp), math.Abs(sigma)) + if scale >= safmx2 { + sigma *= safmn2 + temp *= safmn2 + continue + } + if scale <= safmn2 { + sigma *= safmx2 + temp *= safmx2 + continue + } + break + } + p = 0.5 * temp + tau := lapy2(sigma, temp) + cs = math.Sqrt(0.5 * (1 + math.Abs(sigma)/tau)) + sn = -(p / (tau * cs)) * math.Copysign(1, sigma) + // [aa bb; cc dd] = [a b; c d]·[cs -sn; sn cs] + aa0 := a*cs + b*sn + bb0 := -a*sn + b*cs + cc0 := c*cs + d*sn + dd0 := -c*sn + d*cs + // [a b; c d] = [cs sn; -sn cs]·[aa bb; cc dd] + a = aa0*cs + cc0*sn + b = bb0*cs + dd0*sn + c = -aa0*sn + cc0*cs + d = -bb0*sn + dd0*cs + temp = 0.5 * (a + d) + a, d = temp, temp + if c != 0 && (b == 0 || math.Signbit(b) == math.Signbit(c)) { + // Real eigenvalues: reduce to upper triangular form. (For + // b = 0 this is LAPACK's separate swap of b and c: sab = 0 + // makes the rotation a quarter turn.) + sab := math.Sqrt(math.Abs(b)) + sac := math.Sqrt(math.Abs(c)) + p = math.Copysign(sab*sac, c) + tau = 1 / math.Sqrt(math.Abs(b+c)) + a = temp + p + d = temp - p + b -= c + c = 0 + cs1, sn1 := sab*tau, sac*tau + cs, sn = cs*cs1-sn*sn1, cs*sn1+sn*cs1 + } + } + rt1r, rt2r = a, d + if c != 0 { + rt1i = math.Sqrt(math.Abs(b)) * math.Sqrt(math.Abs(c)) + rt2i = -rt1i + } + return a, b, c, d, rt1r, rt1i, rt2r, rt2i, cs, sn +} + +// Shift strategy constants shared by the two QR iterations (LAPACK): after +// kexsh iterations without a deflation, an exceptional shift from the top +// of the active block; after 2·kexsh, one from the bottom. kexsh is a +// variable so tests can reach them. +var kexsh = 10 + +const ( + dat1 = 3.0 / 4 + dat2 = -0.4375 +) + +// lahqr is the double-shift QR algorithm (LAPACK dlahqr) on the real upper +// Hessenberg n×n h, rows and columns ilo..ihi: eigenvalues into wr, wi; with +// wantt, h becomes the real Schur form T; with zt (the transpose of Z, n×n) +// not nil, Z is postmultiplied by the Schur vectors. It reports false if an +// eigenvalue did not converge. +func lahqr(wantt bool, n, ilo, ihi int, h, wr, wi, zt []float64) bool { + if ilo == ihi { + wr[ilo], wi[ilo] = h[ilo*n+ilo], 0 + return true + } + H := func(i, j int) float64 { return h[i*n+j] } + nh := ihi - ilo + 1 + smlnum := safmin * (float64(nh) / ulp) + i1, i2 := 0, n-1 + itmax := iterFactor * max(10, nh) + kdefl := 0 + var v [3]float64 + for i := ihi; i >= ilo; { + l := ilo + converged := false + for its := 0; its <= itmax; its++ { + // Look for a single small subdiagonal element. + k := i + for ; k > l; k-- { + if math.Abs(H(k, k-1)) <= smlnum { + break + } + tst := math.Abs(H(k-1, k-1)) + math.Abs(H(k, k)) + if tst == 0 { + if k-2 >= ilo { + tst += math.Abs(H(k-1, k-2)) + } + if k+1 <= ihi { + tst += math.Abs(H(k+1, k)) + } + } + // Ahues & Tisseur's conservative deflation criterion. + if math.Abs(H(k, k-1)) <= ulp*tst { + ab := math.Max(math.Abs(H(k, k-1)), math.Abs(H(k-1, k))) + ba := math.Min(math.Abs(H(k, k-1)), math.Abs(H(k-1, k))) + aa := math.Max(math.Abs(H(k, k)), math.Abs(H(k-1, k-1)-H(k, k))) + bb := math.Min(math.Abs(H(k, k)), math.Abs(H(k-1, k-1)-H(k, k))) + s := aa + ab + if ba*(ab/s) <= math.Max(smlnum, ulp*(bb*(aa/s))) { + break + } + } + } + l = k + if l > ilo { + h[l*n+l-1] = 0 + } + if l >= i-1 { + converged = true + break + } + kdefl++ + if !wantt { + i1, i2 = l, i + } + var h11, h12, h21, h22 float64 + switch { + case kdefl%(2*kexsh) == 0: + s := math.Abs(H(i, i-1)) + math.Abs(H(i-1, i-2)) + h11 = dat1*s + H(i, i) + h12, h21, h22 = dat2*s, s, h11 + case kdefl%kexsh == 0: + s := math.Abs(H(l+1, l)) + math.Abs(H(l+2, l+1)) + h11 = dat1*s + H(l, l) + h12, h21, h22 = dat2*s, s, h11 + default: + h11, h21, h12, h22 = H(i-1, i-1), H(i, i-1), H(i-1, i), H(i, i) + } + var rt1r, rt1i, rt2r, rt2i float64 + if s := math.Abs(h11) + math.Abs(h12) + math.Abs(h21) + math.Abs(h22); s != 0 { + h11 /= s + h21 /= s + h12 /= s + h22 /= s + tr := (h11 + h22) / 2 + det := (h11-tr)*(h22-tr) - h12*h21 + rtdisc := math.Sqrt(math.Abs(det)) + if det >= 0 { + // A complex conjugate pair of shifts. + rt1r, rt1i = tr*s, rtdisc*s + rt2r, rt2i = rt1r, -rt1i + } else { + // Two real shifts: use only the one closer to h22. + rt1r, rt2r = tr+rtdisc, tr-rtdisc + if math.Abs(rt1r-h22) <= math.Abs(rt2r-h22) { + rt1r *= s + rt2r = rt1r + } else { + rt2r *= s + rt1r = rt2r + } + } + } + // Look for two consecutive small subdiagonal elements. + m := i - 2 + for ; ; m-- { + h21s := H(m+1, m) + s := math.Abs(H(m, m)-rt2r) + math.Abs(rt2i) + math.Abs(h21s) + h21s = H(m+1, m) / s + v[0] = h21s*H(m, m+1) + (H(m, m)-rt1r)*((H(m, m)-rt2r)/s) - rt1i*(rt2i/s) + v[1] = h21s * (H(m, m) + H(m+1, m+1) - rt1r - rt2r) + v[2] = h21s * H(m+2, m+1) + s = math.Abs(v[0]) + math.Abs(v[1]) + math.Abs(v[2]) + v[0] /= s + v[1] /= s + v[2] /= s + if m == l { + break + } + h00 := math.Abs(H(m, m-1)) * (math.Abs(v[1]) + math.Abs(v[2])) + h01 := math.Abs(v[0]) * (math.Abs(H(m-1, m-1)) + math.Abs(H(m, m)) + math.Abs(H(m+1, m+1))) + if h00 <= ulp*h01 { + break + } + } + // The double-shift QR step. + for k := m; k < i; k++ { + nr := min(3, i-k+1) + if k > m { + for r := 0; r < nr; r++ { + v[r] = H(k+r, k-1) + } + } + var t1 float64 + v[0], t1 = larfg(nr, v[0], v[1:], 1) + if k > m { + h[k*n+k-1] = v[0] + h[(k+1)*n+k-1] = 0 + if k < i-1 { + h[(k+2)*n+k-1] = 0 + } + } else if m > l { + // Not h(k,k-1) = -h(k,k-1): that misbehaves when v[1] + // and v[2] underflow. + h[k*n+k-1] *= 1 - t1 + } + v2 := v[1] + t2 := t1 * v2 + if nr == 3 { + v3 := v[2] + t3 := t1 * v3 + refl3(h[k*n+k:k*n+i2+1], h[(k+1)*n+k:], h[(k+2)*n+k:], v2, v3, t1, t2, t3) + for j := i1; j <= min(k+3, i); j++ { + row := h[j*n : j*n+n] + sum := row[k] + v2*row[k+1] + v3*row[k+2] + row[k] -= sum * t1 + row[k+1] -= sum * t2 + row[k+2] -= sum * t3 + } + if zt != nil { + refl3(zt[k*n:k*n+n], zt[(k+1)*n:], zt[(k+2)*n:], v2, v3, t1, t2, t3) + } + continue + } + refl2(h[k*n+k:k*n+i2+1], h[(k+1)*n+k:], v2, t1, t2) + for j := i1; j <= i; j++ { + row := h[j*n : j*n+n] + sum := row[k] + v2*row[k+1] + row[k] -= sum * t1 + row[k+1] -= sum * t2 + } + if zt != nil { + refl2(zt[k*n:k*n+n], zt[(k+1)*n:], v2, t1, t2) + } + } + } + if !converged { + return false + } + if l == i { + // One eigenvalue converged. + wr[i], wi[i] = H(i, i), 0 + } else { + // A pair converged: standardize the 2×2 block. + a, b, c, d, rt1r, rt1i, rt2r, rt2i, cs, sn := lanv2(H(i-1, i-1), H(i-1, i), H(i, i-1), H(i, i)) + h[(i-1)*n+i-1], h[(i-1)*n+i], h[i*n+i-1], h[i*n+i] = a, b, c, d + wr[i-1], wi[i-1], wr[i], wi[i] = rt1r, rt1i, rt2r, rt2i + if wantt { + if i2 > i { + rot(h[(i-1)*n+i+1:(i-1)*n+i2+1], h[i*n+i+1:i*n+i2+1], cs, sn) + } + for r := i1; r < i-1; r++ { + x, y := h[r*n+i-1], h[r*n+i] + h[r*n+i-1], h[r*n+i] = cs*x+sn*y, cs*y-sn*x + } + } + if zt != nil { + rot(zt[(i-1)*n:i*n], zt[i*n:(i+1)*n], cs, sn) + } + } + kdefl = 0 + i = l - 1 + } + return true +} + +// refl3 applies the reflector I - t·[1 v2 v3]ᵀ[1 v2 v3] (t1 = t, +// t2 = t·v2, t3 = t·v3) to the rows x, y, z: the inner loop of the +// double-shift sweep. +func refl3(x, y, z []float64, v2, v3, t1, t2, t3 float64) { + y, z = y[:len(x)], z[:len(x)] + for j, xv := range x { + sum := xv + v2*y[j] + v3*z[j] + x[j] = xv - sum*t1 + y[j] -= sum * t2 + z[j] -= sum * t3 + } +} + +// refl2 is refl3 for a 2-element reflector. +func refl2(x, y []float64, v2, t1, t2 float64) { + y = y[:len(x)] + for j, xv := range x { + sum := xv + v2*y[j] + x[j] = xv - sum*t1 + y[j] -= sum * t2 + } +} + +// zlahqr is the single-shift complex QR algorithm (LAPACK zlahqr) on the +// complex upper Hessenberg n×n h, rows and columns ilo..ihi: eigenvalues +// into w; with wantt, h becomes the triangular Schur form; zt (the +// transpose of Z) is postmultiplied by the Schur vectors when not nil. +func zlahqr(wantt bool, n, ilo, ihi int, h, w, zt []complex128) bool { + if ilo == ihi { + w[ilo] = h[ilo*n+ilo] + return true + } + H := func(i, j int) complex128 { return h[i*n+j] } + scaleRow := func(r, c0, c1 int, s complex128) { // h[r, c0..c1-1] *= s + for c := c0; c < c1; c++ { + h[r*n+c] *= s + } + } + scaleCol := func(c, r0, r1 int, s complex128) { // h[r0..r1-1, c] *= s + for r := r0; r < r1; r++ { + h[r*n+c] *= s + } + } + scaleZ := func(c int, s complex128) { + if zt != nil { + row := zt[c*n : c*n+n] + for j := range row { + row[j] *= s + } + } + } + // The subdiagonal is real: gehrd's reflectors make it so (zlahqr's + // first loop, which scales it real, has nothing to do here). + nh := ihi - ilo + 1 + smlnum := safmin * (float64(nh) / ulp) + i1, i2 := 0, n-1 + itmax := iterFactor * max(10, nh) + kdefl := 0 + var v [2]complex128 + for i := ihi; i >= ilo; { + l := ilo + converged := false + for its := 0; its <= itmax; its++ { + k := i + for ; k > l; k-- { + if abs1(H(k, k-1)) <= smlnum { + break + } + tst := abs1(H(k-1, k-1)) + abs1(H(k, k)) + if tst == 0 { + if k-2 >= ilo { + tst += math.Abs(real(H(k-1, k-2))) + } + if k+1 <= ihi { + tst += math.Abs(real(H(k+1, k))) + } + } + if math.Abs(real(H(k, k-1))) <= ulp*tst { + ab := math.Max(abs1(H(k, k-1)), abs1(H(k-1, k))) + ba := math.Min(abs1(H(k, k-1)), abs1(H(k-1, k))) + aa := math.Max(abs1(H(k, k)), abs1(H(k-1, k-1)-H(k, k))) + bb := math.Min(abs1(H(k, k)), abs1(H(k-1, k-1)-H(k, k))) + s := aa + ab + if ba*(ab/s) <= math.Max(smlnum, ulp*(bb*(aa/s))) { + break + } + } + } + l = k + if l > ilo { + h[l*n+l-1] = 0 + } + if l >= i { + converged = true + break + } + kdefl++ + if !wantt { + i1, i2 = l, i + } + var t complex128 + switch { + case kdefl%(2*kexsh) == 0: + t = complex(dat1*math.Abs(real(H(i, i-1))), 0) + H(i, i) + case kdefl%kexsh == 0: + t = complex(dat1*math.Abs(real(H(l+1, l))), 0) + H(l, l) + default: + // Wilkinson's shift. + t = H(i, i) + u := cmplx.Sqrt(H(i-1, i)) * cmplx.Sqrt(H(i, i-1)) + if s := abs1(u); s != 0 { + x := 0.5 * (H(i-1, i-1) - t) + sx := abs1(x) + s = math.Max(s, sx) + cs := complex(s, 0) + y := cs * cmplx.Sqrt((x/cs)*(x/cs)+(u/cs)*(u/cs)) + if sx > 0 { + xs := x / complex(sx, 0) + if real(xs)*real(y)+imag(xs)*imag(y) < 0 { + y = -y + } + } + t -= u * (u / (x + y)) + } + } + // Look for two consecutive small subdiagonal elements. + m := i - 1 + for ; m > l; m-- { + h11, h22 := H(m, m), H(m+1, m+1) + h11s := h11 - t + h21 := real(H(m+1, m)) + s := abs1(h11s) + math.Abs(h21) + h11s /= complex(s, 0) + h21 /= s + v[0], v[1] = h11s, complex(h21, 0) + h10 := real(H(m, m-1)) + if math.Abs(h10)*math.Abs(h21) <= ulp*(abs1(h11s)*(abs1(h11)+abs1(h22))) { + break + } + } + if m == l { + h11s := H(l, l) - t + h21 := real(H(l+1, l)) + s := abs1(h11s) + math.Abs(h21) + v[0], v[1] = h11s/complex(s, 0), complex(h21/s, 0) + } + // The single-shift QR step. + for k := m; k < i; k++ { + if k > m { + v[0], v[1] = H(k, k-1), H(k+1, k-1) + } + var t1 complex128 + v[0], t1 = larfg(2, v[0], v[1:], 1) + if k > m { + h[k*n+k-1], h[(k+1)*n+k-1] = v[0], 0 + } + v2 := v[1] + t2 := complex(real(t1*v2), 0) + r0, r1 := h[k*n:k*n+n], h[(k+1)*n:(k+1)*n+n] + for j := k; j <= i2; j++ { + sum := cmplx.Conj(t1)*r0[j] + t2*r1[j] + r0[j] -= sum + r1[j] -= sum * v2 + } + for j := i1; j <= min(k+2, i); j++ { + row := h[j*n : j*n+n] + sum := t1*row[k] + t2*row[k+1] + row[k] -= sum + row[k+1] -= sum * cmplx.Conj(v2) + } + if zt != nil { + z0, z1 := zt[k*n:k*n+n], zt[(k+1)*n:(k+1)*n+n] + for j := range z0 { + sum := t1*z0[j] + t2*z1[j] + z0[j] -= sum + z1[j] -= sum * cmplx.Conj(v2) + } + } + if k == m && m > l { + // The step started below a small subdiagonal: rescale + // so that h(m, m-1) stays real. + temp := 1 - t1 + temp /= complex(cmplx.Abs(temp), 0) + h[(m+1)*n+m] *= cmplx.Conj(temp) + if m+2 <= i { + h[(m+2)*n+m+1] *= temp + } + for j := m; j <= i; j++ { + if j == m+1 { + continue + } + if i2 > j { + scaleRow(j, j+1, i2+1, temp) + } + scaleCol(j, i1, j, cmplx.Conj(temp)) + scaleZ(j, cmplx.Conj(temp)) + } + } + } + // Make h(i, i-1) real. + if temp := H(i, i-1); imag(temp) != 0 { + rtemp := cmplx.Abs(temp) + h[i*n+i-1] = complex(rtemp, 0) + temp /= complex(rtemp, 0) + if i2 > i { + scaleRow(i, i+1, i2+1, cmplx.Conj(temp)) + } + scaleCol(i, i1, i, temp) + scaleZ(i, temp) + } + } + if !converged { + return false + } + w[i] = H(i, i) + kdefl = 0 + i = l - 1 + } + return true +} + +// solve2 solves the 2×2 complex system [a b; c d]·y = r by Gaussian +// elimination with complete pivoting, perturbing pivots below smin to smin +// (as LAPACK dlaln2 does). The returned scale (<= 1) must also be applied +// to the rest of the vector: the solution is of scale·r, which keeps |y| +// below bignum. +func solve2(a, b, c, d, r1, r2 complex128, smin, bignum float64) (y1, y2 complex128, scale float64) { + m := [4]complex128{a, b, c, d} + p := 0 + for i := 1; i < 4; i++ { + if abs1(m[i]) > abs1(m[p]) { + p = i + } + } + pr, pc := p/2, p%2 // pivot row and column + or, oc := 1-pr, 1-pc + rhs := [2]complex128{r1, r2} + u11 := m[pr*2+pc] + if abs1(u11) < smin { + u11 = complex(smin, 0) + } + l21 := m[or*2+pc] / u11 + u12 := m[pr*2+oc] + u22 := m[or*2+oc] - l21*u12 + if abs1(u22) < smin { + u22 = complex(smin, 0) + } + b1 := rhs[pr] + b2 := rhs[or] - l21*b1 + scale = 1 + if bn := math.Max(abs1(b1), abs1(b2)); bn > 1 && abs1(u22) < 1 && bn > bignum*abs1(u22) { + scale = 1 / bn + b1 *= complex(scale, 0) + b2 *= complex(scale, 0) + } + var y [2]complex128 + y[oc] = b2 / u22 + y[pc] = (b1 - u12*y[oc]) / u11 + return y[0], y[1], scale +} + +// eigvec solves (T(0:k,0:k) - lambda)·x = x(0:k) in place for the upper +// (quasi-)triangular n×n t, from row k-1 up: in a real t a nonzero +// subdiagonal entry marks a 2×2 block. x is rescaled as a whole when its +// entries would overflow (LAPACK trevc's scheme). cnorm holds the 1-norms +// of t's strictly upper columns. +func eigvec[T scalar](n int, t []T, k int, lambda complex128, x []complex128, cnorm []float64, smin, bignum float64) { + tc := func(i, j int) complex128 { + v := t[i*n+j] + return complex(re(v), im(v)) + } + rescale := func(s float64) { + for i := range x { + x[i] *= complex(s, 0) + } + } + for j := k - 1; j >= 0; { + if j > 0 && t[j*n+j-1] != 0 { + y1, y2, s := solve2(tc(j-1, j-1)-lambda, tc(j-1, j), tc(j, j-1), tc(j, j)-lambda, x[j-1], x[j], smin, bignum) + if s != 1 { + rescale(s) + } + x[j-1], x[j] = y1, y2 + if xm := math.Max(abs1(y1), abs1(y2)); xm > 1 && math.Max(cnorm[j-1], cnorm[j]) > bignum/xm { + rescale(1 / xm) + } + for r := 0; r < j-1; r++ { + x[r] -= x[j-1]*tc(r, j-1) + x[j]*tc(r, j) + } + j -= 2 + continue + } + den := tc(j, j) - lambda + if abs1(den) < smin { + den = complex(smin, 0) + } + if xj := abs1(x[j]); xj > 1 && abs1(den) < 1 && xj > bignum*abs1(den) { + rescale(1 / xj) + } + x[j] /= den + if xm := abs1(x[j]); xm > 1 && cnorm[j] > bignum/xm { + rescale(1 / xm) + } + for r := 0; r < j; r++ { + x[r] -= x[j] * tc(r, j) + } + j-- + } +} + +// eigReal is dgeev for one real n×n matrix a (destroyed): the eigenvalues +// and, when wantV, the unit eigenvectors as the columns of an n×n complex +// matrix, normalized as dgeev normalizes them (Euclidean norm 1, largest +// component real). +func eigReal(n int, a []float64, wantV bool) ([]complex128, []complex128, bool) { + ilo, ihi, perm, scale := gebal(n, a) + q := gehrd(n, ilo, ihi, a, wantV) + wr, wi := make([]float64, n), make([]float64, n) + for i := 0; i < n; i++ { + if i < ilo || i > ihi { + wr[i] = a[i*n+i] + } + } + var zt []float64 + if wantV { + zt = transposed(n, n, q) + } + if n > 0 && !lahqr(wantV, n, ilo, ihi, a, wr, wi, zt) { + return nil, nil, false + } + w := make([]complex128, n) + for i := range w { + w[i] = complex(wr[i], wi[i]) + } + if !wantV { + return w, nil, true + } + // Eigenvectors of the Schur form T, as dtrevc stores them: a real one + // in its column, a complex pair as its real and imaginary parts in two. + cnorm := make([]float64, n) + for j := 0; j < n; j++ { + for i := 0; i < j; i++ { + cnorm[j] += math.Abs(a[i*n+j]) + } + } + smlnum := safmin * (float64(n) / ulp) + bignum := (1 - ulp) / smlnum + xs := make([]float64, n*n) + x := make([]complex128, n) + for ki := n - 1; ki >= 0; ki-- { + clear(x) + smin := math.Max(ulp*(math.Abs(wr[ki])+math.Abs(wi[ki])), smlnum) + if wi[ki] == 0 { + x[ki] = 1 + for k := 0; k < ki; k++ { + x[k] = complex(-a[k*n+ki], 0) + } + eigvec(n, a, ki, complex(wr[ki], 0), x, cnorm, smin, bignum) + normalizeInf(x[:ki+1]) + for k := 0; k <= ki; k++ { + xs[k*n+ki] = real(x[k]) + } + continue + } + // A complex pair at ki-1, ki: the vector of wr + i·|wi|. + wpos := math.Abs(wi[ki]) + smin = math.Max(ulp*(math.Abs(wr[ki])+wpos), smlnum) + if math.Abs(a[(ki-1)*n+ki]) >= math.Abs(a[ki*n+ki-1]) { + x[ki-1] = 1 + x[ki] = complex(0, wpos/a[(ki-1)*n+ki]) + } else { + x[ki-1] = complex(-wpos/a[ki*n+ki-1], 0) + x[ki] = complex(0, 1) + } + for k := 0; k < ki-1; k++ { + x[k] = -x[ki-1]*complex(a[k*n+ki-1], 0) - x[ki]*complex(a[k*n+ki], 0) + } + eigvec(n, a, ki-1, complex(wr[ki], wpos), x, cnorm, smin, bignum) + normalizeInf(x[:ki+1]) + for k := 0; k <= ki; k++ { + xs[k*n+ki-1], xs[k*n+ki] = real(x[k]), imag(x[k]) + } + ki-- + } + vr := make([]float64, n*n) + gemm(n, n, n, 1, rm(zt, 0, n).t(), rm(xs, 0, n), 0, vr, n) + gebak(n, ilo, ihi, perm, scale, vr) + // Normalize as dgeev does, and assemble the complex vectors. + vt := transposed(n, n, vr) // rows are the eigenvectors + out := make([]complex128, n*n) + for i := 0; i < n; i++ { + ri := vt[i*n : i*n+n] + if wi[i] == 0 { + s := 1 / nrm2(n, ri, 1) + for r, v := range ri { + out[r*n+i] = complex(v*s, 0) + } + continue + } + rj := vt[(i+1)*n : (i+1)*n+n] + s := 1 / lapy2(nrm2(n, ri, 1), nrm2(n, rj, 1)) + k, best := 0, -1.0 + for r := range ri { + ri[r] *= s + rj[r] *= s + if v := ri[r]*ri[r] + rj[r]*rj[r]; v > best { + k, best = r, v + } + } + cs, sn, _ := lartg(ri[k], rj[k]) + rot(ri, rj, cs, sn) + rj[k] = 0 + // wi[i] > 0: the pair is (re + i·im) for w[i], its conjugate for w[i+1]. + sg := math.Copysign(1, wi[i]) + for r := range ri { + out[r*n+i] = complex(ri[r], sg*rj[r]) + out[r*n+i+1] = complex(ri[r], -sg*rj[r]) + } + i++ + } + return w, out, true +} + +// normalizeInf scales x so its largest |re|+|im| is 1 (a zero x is left). +func normalizeInf(x []complex128) { + m := 0.0 + for _, v := range x { + m = math.Max(m, abs1(v)) + } + if m > 0 { + for i := range x { + x[i] /= complex(m, 0) + } + } +} + +// eigCplx is zgeev for one complex n×n matrix a (destroyed). +func eigCplx(n int, a []complex128, wantV bool) ([]complex128, []complex128, bool) { + ilo, ihi, perm, scale := gebal(n, a) + q := gehrd(n, ilo, ihi, a, wantV) + w := make([]complex128, n) + for i := 0; i < n; i++ { + if i < ilo || i > ihi { + w[i] = a[i*n+i] + } + } + var zt []complex128 + if wantV { + zt = transposed(n, n, q) + } + if n > 0 && !zlahqr(wantV, n, ilo, ihi, a, w, zt) { + return nil, nil, false + } + if !wantV { + return w, nil, true + } + cnorm := make([]float64, n) + for j := 0; j < n; j++ { + for i := 0; i < j; i++ { + cnorm[j] += abs1(a[i*n+j]) + } + } + smlnum := safmin * (float64(n) / ulp) + bignum := (1 - ulp) / smlnum + xs := make([]complex128, n*n) + x := make([]complex128, n) + for ki := n - 1; ki >= 0; ki-- { + clear(x) + lam := a[ki*n+ki] + smin := math.Max(ulp*abs1(lam), smlnum) + x[ki] = 1 + for k := 0; k < ki; k++ { + x[k] = -a[k*n+ki] + } + eigvec(n, a, ki, lam, x, cnorm, smin, bignum) + normalizeInf(x[:ki+1]) + for k := 0; k <= ki; k++ { + xs[k*n+ki] = x[k] + } + } + vr := make([]complex128, n*n) + gemm(n, n, n, 1, rm(zt, 0, n).t(), rm(xs, 0, n), 0, vr, n) + gebak(n, ilo, ihi, perm, scale, vr) + vt := transposed(n, n, vr) + for i := 0; i < n; i++ { + ri := vt[i*n : i*n+n] + s := complex(1/nrm2(n, ri, 1), 0) + k, best := 0, -1.0 + for r := range ri { + ri[r] *= s + if v := absSq(ri[r]); v > best { + k, best = r, v + } + } + ph := cmplx.Conj(ri[k]) / complex(math.Sqrt(best), 0) + for r := range ri { + ri[r] *= ph + } + ri[k] = complex(real(ri[k]), 0) + } + return w, transposed(n, n, vt), true +} + +// Eig is numpy.linalg.eig: the eigenvalues w (..., M) and right +// eigenvectors v (..., M, M), as columns, of each square matrix of a. As in +// numpy 2, both are always complex (complex64 for single-precision inputs). +// Each vector has Euclidean norm 1 and its largest component real; for a +// real matrix, conjugate eigenvalues have conjugate vectors. The algorithm +// is LAPACK geev's: balancing, Hessenberg reduction, Francis double-shift QR +// (single-shift for complex input), back-substitution. An input with Inf or +// NaN gives an error wrapping ErrLinAlg, as in numpy. +func Eig(a *ndarray.Array) (w, v *ndarray.Array, err error) { + return eigT(a, true) +} + +// Eigvals is numpy.linalg.eigvals: the eigenvalues of Eig only. +func Eigvals(a *ndarray.Array) (*ndarray.Array, error) { + w, _, err := eigT(a, false) + return w, err +} + +func eigT(a *ndarray.Array, wantV bool) (*ndarray.Array, *ndarray.Array, error) { + o, err := square(a, "Eig") + if err != nil { + return nil, nil, err + } + n, cnt := o.n, o.count() + var data []complex128 + var rdata []float64 + if o.cplx { + data = load[complex128](o.a) + } else { + rdata = load[float64](o.a) + } + if !finite(data) || !finite(rdata) { + return nil, nil, errNonFinite + } + ws := make([]complex128, cnt*n) + var vs []complex128 + if wantV { + vs = make([]complex128, cnt*n*n) + } + var fe firstErr + parallelFor(cnt, 10*n*n*n, func(i int) { + var w, v []complex128 + var ok bool + if o.cplx { + w, v, ok = eigCplx(n, data[i*n*n:(i+1)*n*n], wantV) + } else { + w, v, ok = eigReal(n, rdata[i*n*n:(i+1)*n*n], wantV) + } + if !ok { + fe.set(i, ErrNoConvergence) + return + } + copy(ws[i*n:], w) + if wantV { + copy(vs[i*n*n:], v) + } + }) + if fe.err != nil { + return nil, nil, fe.err + } + w := wrap(ws, shapeOf(o.batch, n), o.single) + if !wantV { + return w, nil, nil + } + return w, wrap(vs, shapeOf(o.batch, n, n), o.single), nil +} + +// finite reports whether every element of x is finite. +func finite[T scalar](x []T) bool { + for _, v := range x { + if r, i := re(v), im(v); math.IsInf(r, 0) || math.IsNaN(r) || math.IsInf(i, 0) || math.IsNaN(i) { + return false + } + } + return true +} diff --git a/linalg/eigh.go b/linalg/eigh.go new file mode 100644 index 0000000..5fdde8d --- /dev/null +++ b/linalg/eigh.go @@ -0,0 +1,595 @@ +package linalg + +import ( + "math" + + "github.com/go-ndarray/ndarray/internal/kernels" + + "github.com/go-ndarray/ndarray" +) + +// hermitianFrom completes the n×n matrix a into a Hermitian one from the +// triangle uplo selects, dropping the imaginary part of the diagonal as +// LAPACK does. +func hermitianFrom[T scalar](n int, a []T, uplo UPLO) { + for r := 0; r < n; r++ { + a[r*n+r] = fromReal[T](re(a[r*n+r])) + for c := r + 1; c < n; c++ { + if uplo == Lower { + a[r*n+c] = conj(a[c*n+r]) + } else { + a[c*n+r] = conj(a[r*n+c]) + } + } + } +} + +// hetrdBlock is the panel width of the blocked tridiagonalization, and +// hetrdMin the order below which it is not blocked. +var hetrdBlock, hetrdMin = 32, 96 + +// hetrd reduces the full Hermitian n×n matrix a to real symmetric +// tridiagonal form T = Qᴴ·A·Q by Householder reflections (LAPACK hetrd, on +// the lower triangle). Panels of hetrdBlock columns are reduced as in +// latrd, accumulating W so the trailing matrix is updated once per panel by +// A -= V·Wᴴ + W·Vᴴ through the GEMM; the rest is hetd2. The matrix is kept +// in full storage, so the matrix-vector products run along contiguous rows. +// It returns the diagonal d, the off-diagonal e and, when wantQ, Q as an +// n×n row-major matrix. a is destroyed. +func hetrd[T scalar](n int, a []T, wantQ bool) (d, e []float64, q []T) { + d = make([]float64, n) + e = make([]float64, max(n-1, 0)) + var refl []T // the reflectors, in geqrf's layout on the trailing (n-1)×(n-1) block + taus := make([]T, max(n-1, 0)) + if wantQ && n > 1 { + refl = make([]T, (n-1)*(n-1)) + } + store := func(i int, v []T, tau T) { + taus[i] = tau + if refl != nil { + for r := 1; r < len(v); r++ { + refl[(i+r)*(n-1)+i] = v[r] + } + } + } + i := 0 + if n >= hetrdMin { + nb := hetrdBlock + for ; i+nb < n-nb; i += nb { + latrd(n, i, nb, a, d, e, store) + } + } + v := make([]T, n) + p := make([]T, n) + for ; i < n-1; i++ { + s := n - i - 1 // order of the trailing block A22 = a[i+1:, i+1:] + var x []T + if s > 1 { + x = a[(i+2)*n+i:] + } + beta, tau := larfg(s, a[(i+1)*n+i], x, n) + e[i] = re(beta) + v[0] = 1 + for r := 1; r < s; r++ { + v[r] = a[(i+1+r)*n+i] + } + if tau != 0 { + // p = tau·A22·v; w = p - (tau/2)(pᴴv)·v; A22 -= v·wᴴ + w·vᴴ. + matvec(s, s, a[(i+1)*n+i+1:], n, v, p) + var pv T + for r := 0; r < s; r++ { + p[r] *= tau + pv += conj(p[r]) * v[r] + } + alpha := -0.5 * tau * pv + for r := 0; r < s; r++ { + p[r] += alpha * v[r] + } + for r := 0; r < s; r++ { + row := a[(i+1+r)*n+i+1 : (i+1+r)*n+n] + vr, wr := v[r], p[r] + for c := range row { + row[c] -= vr*conj(p[c]) + wr*conj(v[c]) + } + } + } + d[i] = re(a[i*n+i]) + store(i, v[:s], tau) + } + if n > 0 { + d[n-1] = re(a[(n-1)*n+n-1]) + } + if !wantQ { + return d, e, nil + } + q = make([]T, n*n) + if n > 0 { + q[0] = 1 + } + if n > 1 { + qs := orgqr(n-1, n-1, n-1, refl, n-1, taus) + for r := 0; r < n-1; r++ { + copy(q[(r+1)*n+1:(r+2)*n], qs[r*(n-1):(r+1)*(n-1)]) + } + } + return d, e, q +} + +// latrd reduces the nb columns i0..i0+nb-1 of the full Hermitian n×n a +// (LAPACK latrd, lower), then updates the trailing matrix +// a[i0+nb:, i0+nb:] -= V·Wᴴ + W·Vᴴ with the panel's reflectors V and the +// accumulated W. V and W are kept transposed (one reflector per row), so +// the corrections are dot products and axpys along contiguous rows. +func latrd[T scalar](n, i0, nb int, a []T, d, e []float64, store func(int, []T, T)) { + m := n - i0 // rows of the panel, indexed from i0 + vt := make([]T, nb*m) + wt := make([]T, nb*m) + vc := make([]T, m) + tmp := make([]T, m) + t1, t2 := make([]T, nb), make([]T, nb) + for j := 0; j < nb; j++ { + c := i0 + j + // Bring column c up to date with the panel's earlier reflectors: + // a[c:, c] -= V·conj(W[j, :j]) + W·conj(V[j, :j]). + if j > 0 { + acc := tmp[:m-j] + clear(acc) + for p := 0; p < j; p++ { + vp, wp := vt[p*m:(p+1)*m], wt[p*m:(p+1)*m] + axpy(acc, vp[j:], conj(wp[j])) + axpy(acc, wp[j:], conj(vp[j])) + } + for r, x := range acc { + a[(c+r)*n+c] -= x + } + } + d[c] = re(a[c*n+c]) + s := n - c - 1 + var x []T + if s > 1 { + x = a[(c+2)*n+c:] + } + beta, tau := larfg(s, a[(c+1)*n+c], x, n) + e[c] = re(beta) + v := vc[:s] // the reflector's vector + v[0] = 1 + for r := 1; r < s; r++ { + v[r] = a[(c+1+r)*n+c] + } + copy(vt[j*m+j+1:], v) + store(c, v, tau) + // w = tau·(A22 - V·Wᴴ - W·Vᴴ)·v, then w -= (tau/2)(wᴴv)·v. + w := wt[j*m+j+1 : (j+1)*m] + matvec(s, s, a[(c+1)*n+c+1:], n, v, w) + for p := 0; p < j; p++ { + t1[p] = dotc(wt[p*m+j+1:(p+1)*m], v) + t2[p] = dotc(vt[p*m+j+1:(p+1)*m], v) + } + for p := 0; p < j; p++ { + axpy(w, vt[p*m+j+1:(p+1)*m], -t1[p]) + axpy(w, wt[p*m+j+1:(p+1)*m], -t2[p]) + } + var pv T + for r := range w { + w[r] *= tau + pv += conj(w[r]) * v[r] + } + axpy(w, v, -0.5*tau*pv) + } + // The trailing update, on the full block. + t := m - nb + c0 := (i0+nb)*n + i0 + nb + vv, ww := rm(vt, nb, m), rm(wt, nb, m) // nb×t: Vᵀ and Wᵀ + gemm(t, t, nb, -1, vv.t(), ww.c(), 1, a[c0:], n) + gemm(t, t, nb, -1, ww.t(), vv.c(), 1, a[c0:], n) +} + +// axpy is y += alpha·x over len(x) elements. +func axpy[T scalar](y, x []T, alpha T) { + y = y[:len(x)] + for i, v := range x { + y[i] += alpha * v + } +} + +// dotc is xᴴ·y. +func dotc[T scalar](x, y []T) T { + if xf, ok := any(x).([]float64); ok { + return any(kernels.Dot1DP(xf, any(y).([]float64)[:len(xf)])).(T) + } + var s T + for i, v := range x { + s += conj(v) * y[i] + } + return s +} + +// matvecPar is the size (m·k) from which matvec spreads its rows over the +// processors: below it, waking the workers costs more than the work. +var matvecPar = 1 << 18 + +// matvec computes y = A·x for the m×k row-major A (leading dimension lda), +// by contiguous row dot products — SIMD and spread over the processors for +// float64. +func matvec[T scalar](m, k int, a []T, lda int, x, y []T) { + if af, ok := any(a).([]float64); ok { + if m*k < matvecPar { + yf, xf := any(y).([]float64), any(x).([]float64) + for r := 0; r < m; r++ { + yf[r] = kernels.Dot1DP(af[r*lda:r*lda+k], xf[:k]) + } + return + } + kernels.MatVecStridedP(any(y).([]float64), af, lda, any(x).([]float64)[:k], m, k) + return + } + for r := 0; r < m; r++ { + row := a[r*lda : r*lda+k] + var s T + for c, v := range row { + s += v * x[c] + } + y[r] = s + } +} + +// laev2 is the eigendecomposition of the symmetric 2×2 [a b; b c] (LAPACK +// dlaev2): rt1 has the larger absolute value, and (cs, sn) is the unit right +// eigenvector for rt1. +func laev2(a, b, c float64) (rt1, rt2, cs, sn float64) { + sm, df := a+c, a-c + adf, tb := math.Abs(df), b+b + ab := math.Abs(tb) + acmx, acmn := a, c + if math.Abs(a) <= math.Abs(c) { + acmx, acmn = c, a + } + var rt float64 + switch { + case adf > ab: + rt = adf * math.Sqrt(1+(ab/adf)*(ab/adf)) + case adf < ab: + rt = ab * math.Sqrt(1+(adf/ab)*(adf/ab)) + default: + rt = ab * math.Sqrt2 + } + var sgn1 float64 + switch { + case sm < 0: + rt1, sgn1 = 0.5*(sm-rt), -1 + rt2 = (acmx/rt1)*acmn - (b/rt1)*b + case sm > 0: + rt1, sgn1 = 0.5*(sm+rt), 1 + rt2 = (acmx/rt1)*acmn - (b/rt1)*b + default: + rt1, rt2, sgn1 = 0.5*rt, -0.5*rt, 1 + } + var cs0, sgn2 float64 + if df >= 0 { + cs0, sgn2 = df+rt, 1 + } else { + cs0, sgn2 = df-rt, -1 + } + if math.Abs(cs0) > ab { + ct := -tb / cs0 + sn = 1 / math.Sqrt(1+ct*ct) + cs = ct * sn + } else if ab == 0 { + cs, sn = 1, 0 + } else { + tn := -cs0 / tb + cs = 1 / math.Sqrt(1+tn*tn) + sn = tn * cs + } + if sgn1 == sgn2 { + cs, sn = -sn, cs + } + return rt1, rt2, cs, sn +} + +// steqr computes the eigenvalues of the symmetric tridiagonal matrix (d, e) +// by the implicit QL or QR method with Wilkinson shifts (LAPACK dsteqr), +// sorted ascending into d. When z is not nil, its n rows of length ldz are +// the columns of a matrix Z (z holds Zᵀ) and receive the same rotations, +// so Z·(eigenvectors of T) comes out — row k of z is the eigenvector of +// d[k]. It reports false if some eigenvalue did not converge in 30·n +// iterations. +func steqr[T scalar](n int, d, e []float64, z []T, ldz int) bool { + if n <= 1 { + return true + } + row := func(i int) []T { return z[i*ldz : i*ldz+ldz] } + rotate := func(i, j int, c, s float64) { + if z != nil { + rot(row(i), row(j), c, s) + } + } + const eps2 = eps * eps + ssfmax := math.Sqrt(1/safmin) / 3 + ssfmin := math.Sqrt(safmin) / eps2 + nmaxit := n * iterFactor + jtot := 0 + for l1 := 0; l1 < n && jtot < nmaxit; { + if l1 > 0 { + e[l1-1] = 0 + } + m := n - 1 + for mm := l1; mm < n-1; mm++ { + tst := math.Abs(e[mm]) + if tst == 0 { + m = mm + break + } + if tst <= math.Sqrt(math.Abs(d[mm]))*math.Sqrt(math.Abs(d[mm+1]))*eps { + e[mm] = 0 + m = mm + break + } + } + l, lend := l1, m + lsv, lendsv := l, lend + l1 = m + 1 + if lend == l { + continue + } + // Scale the submatrix in rows and columns l to lend. + anorm := 0.0 + for i := l; i <= lend; i++ { + anorm = math.Max(anorm, math.Abs(d[i])) + if i < lend { + anorm = math.Max(anorm, math.Abs(e[i])) + } + } + if math.IsNaN(anorm) { + return false + } + // anorm > 0: the block holds a nonzero off-diagonal entry. + scale := 1.0 + switch { + case anorm > ssfmax: + scale = ssfmax / anorm + case anorm < ssfmin: + scale = ssfmin / anorm + } + scaleTri(d, e, lsv, lendsv, scale) + // Choose between QL and QR iteration. + if math.Abs(d[lend]) < math.Abs(d[l]) { + l, lend = lendsv, lsv + } + if lend > l { + // QL iteration: look for a small subdiagonal element. + for l <= lend { + m = lend + for mm := l; mm < lend; mm++ { + if tst := e[mm] * e[mm]; tst <= (eps2*math.Abs(d[mm]))*math.Abs(d[mm+1])+safmin { + m = mm + break + } + } + if m < lend { + e[m] = 0 + } + p := d[l] + if m == l { + l++ // eigenvalue found + continue + } + if m == l+1 { + rt1, rt2, c, s := laev2(d[l], e[l], d[l+1]) + rotate(l, l+1, c, s) + d[l], d[l+1], e[l] = rt1, rt2, 0 + l += 2 + continue + } + if jtot >= nmaxit { + break + } + jtot++ + g := (d[l+1] - p) / (2 * e[l]) + r := lapy2(g, 1) + g = d[m] - p + (e[l] / (g + math.Copysign(r, g))) + s, c := 1.0, 1.0 + p = 0 + for i := m - 1; i >= l; i-- { + f, b := s*e[i], c*e[i] + c, s, r = lartg(g, f) + if i != m-1 { + e[i+1] = r + } + g = d[i+1] - p + r = (d[i]-g)*s + 2*c*b + p = s * r + d[i+1] = g + p + g = c*r - b + rotate(i, i+1, c, -s) + } + d[l] -= p + e[l] = g + } + } else { + // QR iteration: look for a small superdiagonal element. + for l >= lend { + m = lend + for mm := l; mm > lend; mm-- { + if tst := e[mm-1] * e[mm-1]; tst <= (eps2*math.Abs(d[mm]))*math.Abs(d[mm-1])+safmin { + m = mm + break + } + } + if m > lend { + e[m-1] = 0 + } + p := d[l] + if m == l { + l-- + continue + } + if m == l-1 { + rt1, rt2, c, s := laev2(d[l-1], e[l-1], d[l]) + rotate(l-1, l, c, s) + d[l-1], d[l], e[l-1] = rt1, rt2, 0 + l -= 2 + continue + } + if jtot >= nmaxit { + break + } + jtot++ + g := (d[l-1] - p) / (2 * e[l-1]) + r := lapy2(g, 1) + g = d[m] - p + (e[l-1] / (g + math.Copysign(r, g))) + s, c := 1.0, 1.0 + p = 0 + for i := m; i < l; i++ { + f, b := s*e[i], c*e[i] + c, s, r = lartg(g, f) + if i != m { + e[i-1] = r + } + g = d[i] - p + r = (d[i+1]-g)*s + 2*c*b + p = s * r + d[i] = g + p + g = c*r - b + rotate(i, i+1, c, s) + } + d[l] -= p + e[l-1] = g + } + } + scaleTri(d, e, lsv, lendsv, 1/scale) + } + // Every converged block zeroes its off-diagonal entries: one left + // means the iteration limit stopped the loop first. + for _, v := range e { + if v != 0 { + return false + } + } + // Sort the eigenvalues increasing, with their vectors (selection sort: + // at most n-1 row swaps). + for i := 0; i < n-1; i++ { + k := i + for j := i + 1; j < n; j++ { + if d[j] < d[k] { + k = j + } + } + if k != i { + d[i], d[k] = d[k], d[i] + if z != nil { + ri, rk := row(i), row(k) + for c := range ri { + ri[c], rk[c] = rk[c], ri[c] + } + } + } + } + return true +} + +// scaleTri multiplies d[lo..hi] and e[lo..hi-1] by s. +func scaleTri(d, e []float64, lo, hi int, s float64) { + if s == 1 { + return + } + for i := lo; i <= hi; i++ { + d[i] *= s + if i < hi { + e[i] *= s + } + } +} + +// Eigh is numpy.linalg.eigh: the eigenvalues, ascending, and the orthonormal +// eigenvectors (as columns) of each Hermitian matrix of a (..., M, M), read +// from the triangle uplo selects. w is (..., M) and real (float32 for +// single-precision inputs); v is (..., M, M) in a's precision. Eigenvectors +// are unique only up to a unit factor (a sign, or a phase), and within an +// eigenspace of a repeated eigenvalue only up to a unitary mix: compare +// A·v = λ·v, not the vectors themselves. +func Eigh(a *ndarray.Array, uplo UPLO) (w, v *ndarray.Array, err error) { + o, err := square(a, "Eigh") + if err != nil { + return nil, nil, err + } + if o.cplx { + return eighT[complex128](o, uplo, true) + } + return eighT[float64](o, uplo, true) +} + +// Eigvalsh is numpy.linalg.eigvalsh: the eigenvalues of Eigh only, which +// costs a fraction of the vectors. +func Eigvalsh(a *ndarray.Array, uplo UPLO) (*ndarray.Array, error) { + o, err := square(a, "Eigvalsh") + if err != nil { + return nil, err + } + var w *ndarray.Array + if o.cplx { + w, _, err = eighT[complex128](o, uplo, false) + } else { + w, _, err = eighT[float64](o, uplo, false) + } + return w, err +} + +func eighT[T scalar](o operand, uplo UPLO, wantV bool) (*ndarray.Array, *ndarray.Array, error) { + n, cnt := o.n, o.count() + data := load[T](o.a) + ws := make([]float64, cnt*n) + var fe firstErr + parallelFor(cnt, 4*n*n*n, func(i int) { + x := data[i*n*n : (i+1)*n*n] + vals, vecs, ok := eighOne(n, x, uplo, wantV) + if !ok { + fe.set(i, ErrNoConvergence) + return + } + copy(ws[i*n:], vals) + copy(x, vecs) + }) + if fe.err != nil { + return nil, nil, fe.err + } + w := wrap(ws, shapeOf(o.batch, n), o.single) + if !wantV { + return w, nil, nil + } + return w, wrap(data, shapeOf(o.batch, n, n), o.single), nil +} + +// eighOne is the eigendecomposition of one Hermitian matrix: ascending +// eigenvalues and, when wantV, the eigenvectors as the columns of an n×n +// row-major matrix. +func eighOne[T scalar](n int, x []T, uplo UPLO, wantV bool) ([]float64, []T, bool) { + hermitianFrom(n, x, uplo) + d, e, q := hetrd(n, x, wantV) + if !wantV { + return d, nil, steqr[T](n, d, e, nil, 0) + } + if n > dcLeaf { + zt, ok := stedc(n, d, e) + if !ok { + return nil, nil, false + } + return d, mulMixed(n, n, n, rm(q, 0, n), rm(zt, 0, n).t()), true + } + zt := transposed(n, n, q) // rows of zt are the columns of Q + if !steqr(n, d, e, zt, n) { + return nil, nil, false + } + return d, transposed(n, n, zt), true +} + +// transposed returns the n×m transpose of the m×n row-major matrix x. +func transposed[T scalar](m, n int, x []T) []T { + out := make([]T, m*n) + for r := 0; r < m; r++ { + for c := 0; c < n; c++ { + out[c*m+r] = x[r*n+c] + } + } + return out +} diff --git a/linalg/fixture_test.go b/linalg/fixture_test.go new file mode 100644 index 0000000..7ff409a --- /dev/null +++ b/linalg/fixture_test.go @@ -0,0 +1,763 @@ +package linalg + +import ( + _ "embed" + "encoding/json" + "errors" + "fmt" + "math" + "math/cmplx" + "strconv" + "testing" + + "github.com/go-ndarray/ndarray" +) + +// cases.json holds numpy.linalg's results on seeded inputs — every +// function, the five computation dtypes and integers, stacked, empty, +// ill-conditioned, rank-deficient, singular, defective and non-finite +// matrices, and every error numpy raises on them — generated by +// testdata/cases.py. It is embedded because the emulated CI architectures +// run the test binary from another directory. +// +//go:embed testdata/cases.json +var casesJSON []byte + +type fxArray struct { + DType string `json:"dtype"` + Shape []int `json:"shape"` + Data []json.RawMessage `json:"data"` +} + +type fxCase struct { + Fn string `json:"fn"` + Tag string `json:"tag"` + Args map[string]any `json:"args"` + Inputs []fxArray `json:"inputs"` + Outputs []fxArray `json:"outputs"` + Error string `json:"error"` + Message string `json:"message"` + Kappa float64 `json:"kappa"` + Etol float64 `json:"etol"` +} + +func (c fxCase) String() string { + shapes := make([][]int, len(c.Inputs)) + for i, in := range c.Inputs { + shapes[i] = in.Shape + } + return fmt.Sprintf("%s[%s] %v %v", c.Fn, c.Tag, c.Args, shapes) +} + +func loadCases(t *testing.T) []fxCase { + t.Helper() + var fx struct { + NumPy string `json:"numpy"` + Cases []fxCase `json:"cases"` + } + if err := json.Unmarshal(casesJSON, &fx); err != nil { + t.Fatal(err) + } + return fx.Cases +} + +func parseNum(t *testing.T, s string) float64 { + t.Helper() + f, err := strconv.ParseFloat(s, 64) + if err != nil { + t.Fatal(err) + } + return f +} + +func fxValues(t *testing.T, raw []json.RawMessage) []complex128 { + t.Helper() + out := make([]complex128, len(raw)) + for i, r := range raw { + var pair []string + if json.Unmarshal(r, &pair) == nil { + out[i] = complex(parseNum(t, pair[0]), parseNum(t, pair[1])) + continue + } + var s string + if err := json.Unmarshal(r, &s); err != nil { + t.Fatal(err) + } + out[i] = complex(parseNum(t, s), 0) + } + return out +} + +func fxArrayOf(t *testing.T, a fxArray) *ndarray.Array { + t.Helper() + dt, err := ndarray.ParseDType(a.DType) + if err != nil { + t.Fatal(err) + } + x, err := ndarray.FromSlice(fxValues(t, a.Data), a.Shape...) + if err != nil { + t.Fatal(err) + } + return x.AsType(dt) +} + +// vals returns an array's elements as complex128. +func vals(a *ndarray.Array) []complex128 { + d, _ := ndarray.Data[complex128](a.AsType(ndarray.Complex128)) + return d +} + +func argFloat(args map[string]any, k string) (float64, bool) { + v, ok := args[k] + if !ok || v == nil { + return 0, false + } + return v.(float64), true +} + +func argOrd(v any) Ord { + switch o := v.(type) { + case nil: + return Ord{} + case float64: + return P(o) + case string: + switch o { + case "fro": + return Fro + case "nuc": + return Nuc + case "inf": + return P(math.Inf(1)) + case "-inf": + return P(math.Inf(-1)) + } + } + panic(fmt.Sprintf("bad ord %v", v)) +} + +func argAxes(v any) []int { + switch a := v.(type) { + case nil: + return nil + case float64: + return []int{int(a)} + } + var out []int + for _, x := range v.([]any) { + out = append(out, int(x.(float64))) + } + return out +} + +func uploOf(args map[string]any) UPLO { + if args["uplo"] == "U" { + return Upper + } + return Lower +} + +func one(a *ndarray.Array, err error) ([]*ndarray.Array, error) { + if err != nil { + return nil, err + } + return []*ndarray.Array{a}, nil +} + +func many(err error, a ...*ndarray.Array) ([]*ndarray.Array, error) { + if err != nil { + return nil, err + } + return a, nil +} + +var qrModes = map[string]QRMode{"reduced": ModeReduced, "complete": ModeComplete, "r": ModeR, "raw": ModeRaw} + +// runCase calls the Go function a case names. +func runCase(c fxCase, in []*ndarray.Array) ([]*ndarray.Array, error) { + a := c.Args + switch c.Fn { + case "solve": + return one(Solve(in[0], in[1])) + case "inv": + return one(Inv(in[0])) + case "det": + return one(Det(in[0])) + case "slogdet": + s, l, err := Slogdet(in[0]) + return many(err, s, l) + case "cholesky": + uplo := Lower + if a["upper"] == true { + uplo = Upper + } + return one(Cholesky(in[0], uplo)) + case "qr": + mode := qrModes[a["mode"].(string)] + q, r, err := QR(in[0], mode) + if mode == ModeR { + return many(err, r) + } + return many(err, q, r) + case "svd": + u, s, vh, err := SVD(in[0], a["full"].(bool)) + return many(err, u, s, vh) + case "svdvals": + return one(SVDVals(in[0])) + case "eigh": + w, v, err := Eigh(in[0], uploOf(a)) + return many(err, w, v) + case "eigvalsh": + return one(Eigvalsh(in[0], uploOf(a))) + case "eig": + w, v, err := Eig(in[0]) + return many(err, w, v) + case "eigvals": + return one(Eigvals(in[0])) + case "lstsq": + var r LstsqResult + var err error + if rc, ok := argFloat(a, "rcond"); ok { + r, err = LstsqRcond(in[0], in[1], rc) + } else { + r, err = Lstsq(in[0], in[1]) + } + if err != nil { + return nil, err + } + rank, _ := ndarray.FromSlice([]int64{int64(r.Rank)}) + return many(nil, r.X, r.Residuals, rank, r.S) + case "pinv": + if rc, ok := argFloat(a, "rcond"); ok { + return one(PinvRcond(in[0], rc)) + } + return one(Pinv(in[0])) + case "matrix_rank": + if tol, ok := argFloat(a, "tol"); ok { + return one(MatrixRankTol(in[0], tol)) + } + return one(MatrixRank(in[0])) + case "norm": + keep, _ := a["keepdims"].(bool) + return one(Norm(in[0], argOrd(a["ord"]), argAxes(a["axis"]), keep)) + case "cond": + return one(Cond(in[0], argOrd(a["p"]))) + case "matrix_power": + return one(MatrixPower(in[0], int(a["n"].(float64)))) + case "multi_dot": + return one(MultiDot(in...)) + } + panic("unknown function " + c.Fn) +} + +var errKinds = map[string]error{ + "singular": ErrSingular, + "posdef": ErrNotPosDef, + "converge": ErrNoConvergence, + "nonfinite": ErrLinAlg, + "shape": ErrShape, + "args": ErrArgs, +} + +// knownDifferences are the cases where this package deliberately departs +// from numpy, with the reason. +var knownDifferences = map[string]string{ + // numpy takes the nuclear norm of the NaN-filled "inverse" of a singular + // matrix, whose SVD then fails; a singular matrix has condition number + // +Inf in every norm here. + "cond[singular] map[p:nuc]": "+Inf instead of LinAlgError", + // numpy sums the squares unscaled and overflows; the 2-norm here is + // rescaled (TestNormRescales checks the value). + "norm[overflow] map[]": "1.414e200 instead of +Inf", +} + +func knownDifference(c fxCase) (string, bool) { + r, ok := knownDifferences[fmt.Sprintf("%s[%s] %v", c.Fn, c.Tag, c.Args)] + return r, ok +} + +// TestMatchesNumPy runs every fixture case: errors must be the same kind as +// numpy's; results must have numpy's dtype and shape; values are compared +// directly where they are unique, and through the defining identities where +// they are not (see checkCase). +func TestMatchesNumPy(t *testing.T) { + cases := loadCases(t) + checked := 0 + for i, c := range cases { + in := make([]*ndarray.Array, len(c.Inputs)) + for j, x := range c.Inputs { + in[j] = fxArrayOf(t, x) + } + got, err := runCase(c, in) + if reason, ok := knownDifference(c); ok { + t.Logf("case %d %v: known difference: %s", i, c, reason) + continue + } + if c.Error != "" { + if err == nil { + t.Errorf("case %d %v: no error, numpy raises %s", i, c, c.Message) + } else if !errors.Is(err, errKinds[c.Error]) { + t.Errorf("case %d %v: error %v, numpy raises %s", i, c, err, c.Message) + } + continue + } + if err != nil { + t.Errorf("case %d %v: %v", i, c, err) + continue + } + if len(got) != len(c.Outputs) { + t.Fatalf("case %d %v: %d results, numpy %d", i, c, len(got), len(c.Outputs)) + } + ok := true + for j, w := range c.Outputs { + if got[j].DType().String() != w.DType || !sameShape(got[j].Shape(), w.Shape) { + t.Errorf("case %d %v: result %d is %v %v, numpy %s %v", i, c, j, got[j].DType(), got[j].Shape(), w.DType, w.Shape) + ok = false + } + } + if !ok { + continue + } + if msg := checkCase(t, c, in, got); msg != "" { + t.Errorf("case %d %v: %s", i, c, msg) + continue + } + checked++ + } + t.Logf("%d cases, %d with values checked", len(cases), checked) +} + +func sameShape(a, b []int) bool { + if len(a) != len(b) { + return false + } + for i := range a { + if a[i] != b[i] { + return false + } + } + return true +} + +// Tolerances. A value computed by a backward-stable algorithm in double +// precision is within c·n·eps·κ of the exact one, relative to the size of +// the result, where κ is the condition number of the problem: 1 for +// singular values, Hermitian eigenvalues and norms, cond(A) for solves, +// inverses and determinants. numpy's value carries the same kind of error, +// so the two may differ by twice that. c = 32 leaves room for the +// constants of the error bounds (which grow like n, not n²) without +// letting a wrong result through: an error in an algorithm shows as O(1) +// relative differences, or as residuals far above eps. A single-precision +// result is a double one rounded, so it may differ by one rounding more: +// eps32. +const tolC = 32 + +func isSingle(dt string) bool { return dt == "float32" || dt == "complex64" } + +// valueTol is the absolute tolerance on the values of output w of a case +// with matrices of order n. +func valueTol(c fxCase, w []complex128, dt string, n int, kappa float64) float64 { + scale := 0.0 + for _, v := range w { + if !cmplx.IsInf(v) && !cmplx.IsNaN(v) { + scale = math.Max(scale, cmplx.Abs(v)) + } + } + tol := scale * tolC * float64(max(n, 1)) * eps64 * math.Min(kappa, 1/eps64) + if isSingle(dt) { + tol += 2 * scale * eps32 + } + return tol +} + +// order is the largest matrix dimension of a case's first input. +func order(c fxCase) int { + s := c.Inputs[0].Shape + n := 1 + for _, d := range s[max(len(s)-2, 0):] { + n = max(n, d) + } + return n +} + +// compareValues compares got with numpy's values elementwise. +func compareValues(t *testing.T, got *ndarray.Array, want fxArray, tol float64) string { + g, w := vals(got), fxValues(t, want.Data) + for i := range w { + if sameSpecial(g[i], w[i]) { + continue + } + if d := cmplx.Abs(g[i] - w[i]); !(d <= tol) { + return fmt.Sprintf("value %d = %v, numpy %v (|diff| %.3g > tol %.3g)", i, g[i], w[i], d, tol) + } + } + return "" +} + +// sameSpecial reports whether g and w are the same NaN or infinity. +func sameSpecial(g, w complex128) bool { + same := func(a, b float64) bool { + return (math.IsNaN(a) && math.IsNaN(b)) || (math.IsInf(a, 0) && a == b) || a == b + } + return (cmplx.IsNaN(w) || cmplx.IsInf(w)) && same(real(g), real(w)) && same(imag(g), imag(w)) +} + +// checkCase checks a case's values; it returns a description of the first +// failure, or "". +func checkCase(t *testing.T, c fxCase, in, got []*ndarray.Array) string { + n := order(c) + switch c.Fn { + case "qr": + if c.Args["mode"] == "raw" { + return compareAll(t, c, got, n, c.Kappa) + } + return checkQR(c, in[0], got) + case "svd": + if msg := compareOne(t, c, got, 1, n, 1); msg != "" { + return msg + } + return checkSVD(c, in[0], got) + case "eigh": + if msg := compareOne(t, c, got, 0, n, 1); msg != "" { + return msg + } + return checkEigh(c, in[0], got) + case "eig", "eigvals": + return checkEig(t, c, in[0], got) + case "svdvals", "eigvalsh", "norm": + return compareAll(t, c, got, n, 1) + case "lstsq": + k := c.Kappa + for j, kj := range []float64{k, k * k, 1, 1} { + if msg := compareOne(t, c, got, j, n, kj); msg != "" { + return msg + } + } + return "" + case "matrix_power": + p := math.Abs(c.Args["n"].(float64)) + k := 1.0 + if c.Args["n"].(float64) < 0 { + k = c.Kappa + } + return compareAll(t, c, got, n*int(p+1), k) + case "cond": + return compareAll(t, c, got, n, c.Kappa) + } + // solve, inv, det, slogdet, cholesky, pinv, matrix_rank, multi_dot + return compareAll(t, c, got, n, c.Kappa) +} + +func compareAll(t *testing.T, c fxCase, got []*ndarray.Array, n int, kappa float64) string { + for j := range got { + if msg := compareOne(t, c, got, j, n, kappa); msg != "" { + return msg + } + } + return "" +} + +func compareOne(t *testing.T, c fxCase, got []*ndarray.Array, j, n int, kappa float64) string { + w := c.Outputs[j] + tol := valueTol(c, fxValues(t, w.Data), w.DType, n, kappa) + if msg := compareValues(t, got[j], w, tol); msg != "" { + return fmt.Sprintf("result %d: %s", j, msg) + } + return "" +} + +// The identities. Matrices are taken out of the stacks as complex128 m×n +// row-major slices, and products are formed here with plain loops, apart +// from the package's own GEMM. + +type cmat struct { + m, n int + d []complex128 +} + +func (a cmat) at(i, j int) complex128 { return a.d[i*a.n+j] } + +// stack splits an array (..., m, n) into its matrices. +func stack(a *ndarray.Array) []cmat { + s := a.Shape() + m, n := s[len(s)-2], s[len(s)-1] + d := vals(a) + cnt := 1 + for _, v := range s[:len(s)-2] { + cnt *= v + } + out := make([]cmat, cnt) + for i := range out { + out[i] = cmat{m, n, d[i*m*n : (i+1)*m*n]} + } + return out +} + +// vecs splits an array (..., k) into its vectors. +func vecs(a *ndarray.Array) [][]complex128 { + s := a.Shape() + k := s[len(s)-1] + d := vals(a) + out := make([][]complex128, 0) + for i := 0; i+k <= len(d) && k > 0; i += k { + out = append(out, d[i:i+k]) + } + if k == 0 { + cnt := 1 + for _, v := range s[:len(s)-1] { + cnt *= v + } + out = make([][]complex128, cnt) + } + return out +} + +func mul(a, b cmat) cmat { + c := cmat{a.m, b.n, make([]complex128, a.m*b.n)} + for i := 0; i < a.m; i++ { + for p := 0; p < a.n; p++ { + x := a.at(i, p) + for j := 0; j < b.n; j++ { + c.d[i*b.n+j] += x * b.at(p, j) + } + } + } + return c +} + +func adj(a cmat) cmat { + c := cmat{a.n, a.m, make([]complex128, len(a.d))} + for i := 0; i < a.m; i++ { + for j := 0; j < a.n; j++ { + c.d[j*a.m+i] = cmplx.Conj(a.at(i, j)) + } + } + return c +} + +// fro is the Frobenius norm. +func fro(a cmat) float64 { + s := 0.0 + for _, v := range a.d { + s += real(v)*real(v) + imag(v)*imag(v) + } + return math.Sqrt(s) +} + +// diffFro is ‖a - b‖_F. +func diffFro(a, b cmat) float64 { + s := 0.0 + for i := range a.d { + d := a.d[i] - b.d[i] + s += real(d)*real(d) + imag(d)*imag(d) + } + return math.Sqrt(s) +} + +// orthErr is ‖QᴴQ - I‖_F. +func orthErr(q cmat) float64 { + g := mul(adj(q), q) + for i := 0; i < g.m; i++ { + g.d[i*g.n+i]-- + } + return fro(g) +} + +// idTol is the tolerance on an identity of factors of order n with output +// dtype dt, relative to the norm of the matrix. +func idTol(n int, dt string) float64 { + e := eps64 + if isSingle(dt) { + e = eps32 + } + return tolC * float64(max(n, 1)) * e +} + +func checkQR(c fxCase, a *ndarray.Array, got []*ndarray.Array) string { + dt := c.Outputs[0].DType + n := order(c) + rs := stack(got[len(got)-1]) + as := stack(a) + for i, r := range rs { + for p := 0; p < r.m; p++ { + for q := 0; q < min(p, r.n); q++ { + if r.at(p, q) != 0 { + return fmt.Sprintf("R[%d] not upper triangular at (%d, %d)", i, p, q) + } + } + } + if len(got) == 1 { + continue + } + q := stack(got[0])[i] + if e := orthErr(q); e > idTol(n, dt) { + return fmt.Sprintf("‖QᴴQ - I‖ = %.3g", e) + } + if e := diffFro(mul(q, r), as[i]); e > idTol(n, dt)*fro(as[i]) { + return fmt.Sprintf("‖A - QR‖ = %.3g, ‖A‖ = %.3g", e, fro(as[i])) + } + } + return "" +} + +func checkSVD(c fxCase, a *ndarray.Array, got []*ndarray.Array) string { + dt := c.Outputs[0].DType + n := order(c) + us, ss, vs := stack(got[0]), vecs(got[1]), stack(got[2]) + for i, x := range stack(a) { + u, vh := us[i], vs[i] + if e := orthErr(u); e > idTol(n, dt) { + return fmt.Sprintf("‖UᴴU - I‖ = %.3g", e) + } + if e := orthErr(adj(vh)); e > idTol(n, dt) { + return fmt.Sprintf("‖VhVhᴴ - I‖ = %.3g", e) + } + k := min(x.m, x.n) + us := cmat{x.m, k, make([]complex128, x.m*k)} + for r := 0; r < x.m; r++ { + for j := 0; j < k; j++ { + us.d[r*k+j] = u.at(r, j) * ss[i][j] + } + } + if e := diffFro(mul(us, cmat{k, x.n, vh.d[:k*x.n]}), x); e > idTol(n, dt)*fro(x) { + return fmt.Sprintf("‖A - USVh‖ = %.3g, ‖A‖ = %.3g", e, fro(x)) + } + } + return "" +} + +// hermitianOf completes x from the triangle uplo selects, as the +// eigensolver reads it. +func hermitianOf(x cmat, upper bool) cmat { + h := cmat{x.m, x.n, make([]complex128, len(x.d))} + for i := 0; i < x.n; i++ { + for j := 0; j < x.n; j++ { + src := x.at(i, j) + if (j > i) != upper && i != j { + src = cmplx.Conj(x.at(j, i)) + } + if i == j { + src = complex(real(src), 0) + } + h.d[i*x.n+j] = src + } + } + return h +} + +func checkEigh(c fxCase, a *ndarray.Array, got []*ndarray.Array) string { + dt := c.Outputs[1].DType + n := order(c) + ws, vs := vecs(got[0]), stack(got[1]) + for i, x := range stack(a) { + h := hermitianOf(x, c.Args["uplo"] == "U") + v := vs[i] + if e := orthErr(v); e > idTol(n, dt) { + return fmt.Sprintf("‖VᴴV - I‖ = %.3g", e) + } + vw := cmat{v.m, v.n, make([]complex128, len(v.d))} + for r := 0; r < v.m; r++ { + for j := 0; j < v.n; j++ { + vw.d[r*v.n+j] = v.at(r, j) * ws[i][j] + } + } + if e := diffFro(mul(h, v), vw); e > idTol(n, dt)*math.Max(fro(h), math.SmallestNonzeroFloat64) { + return fmt.Sprintf("‖AV - VW‖ = %.3g, ‖A‖ = %.3g", e, fro(h)) + } + for j := 1; j < len(ws[i]); j++ { + if real(ws[i][j]) < real(ws[i][j-1]) { + return "eigenvalues not ascending" + } + } + } + return "" +} + +func checkEig(t *testing.T, c fxCase, a *ndarray.Array, got []*ndarray.Array) string { + dt := c.Outputs[0].DType + n := order(c) + want := vecs(fxArrayOf(t, c.Outputs[0])) + ws := vecs(got[0]) + realInput := !isComplexDType(c.Inputs[0].DType) + for i, x := range stack(a) { + // Eigenvalues: each of numpy's matched to the nearest unused one. + tol := math.Max(c.Etol, idTol(n, "float64")*fro(x)) + if isSingle(dt) { + tol += 2 * eps32 * maxAbs(want[i]) + } + used := make([]bool, len(ws[i])) + for _, w := range want[i] { + best, bd := -1, math.Inf(1) + for j, g := range ws[i] { + if d := cmplx.Abs(g - w); !used[j] && d < bd { + best, bd = j, d + } + } + if !(bd <= tol) { + return fmt.Sprintf("eigenvalue %v of numpy: nearest is %.3g away (tol %.3g)", w, bd, tol) + } + used[best] = true + } + if realInput { + for j := 0; j < len(ws[i]); j++ { + if imag(ws[i][j]) > 0 { + if j+1 == len(ws[i]) || ws[i][j+1] != cmplx.Conj(ws[i][j]) { + return fmt.Sprintf("eigenvalue %d %v not followed by its conjugate", j, ws[i][j]) + } + j++ + } + } + } + if len(got) == 1 { + continue + } + v := stack(got[1])[i] + av := mul(x, v) + for j := 0; j < v.n; j++ { + nrm, res, big := 0.0, 0.0, 0.0 + for r := 0; r < v.m; r++ { + z := v.at(r, j) + nrm += real(z)*real(z) + imag(z)*imag(z) + big = math.Max(big, cmplx.Abs(z)) + d := av.at(r, j) - z*ws[i][j] + res += real(d)*real(d) + imag(d)*imag(d) + } + if math.Abs(math.Sqrt(nrm)-1) > idTol(n, dt) { + return fmt.Sprintf("eigenvector %d has norm %v", j, math.Sqrt(nrm)) + } + // The residual of a computed eigenpair is backward-stable: + // small relative to ‖A‖ even for ill-conditioned eigenvalues. + if r := math.Sqrt(res); r > idTol(n, dt)*fro(x) { + return fmt.Sprintf("‖Av - λv‖ = %.3g for eigenvector %d, ‖A‖ = %.3g", r, j, fro(x)) + } + hasReal := false + for r := 0; r < v.m; r++ { + z := v.at(r, j) + if imag(z) == 0 && cmplx.Abs(z) >= big*(1-1e-6) { + hasReal = true + } + } + if !hasReal { + return fmt.Sprintf("eigenvector %d: its largest component is not real", j) + } + if realInput && imag(ws[i][j]) > 0 { + for r := 0; r < v.m; r++ { + if v.at(r, j+1) != cmplx.Conj(v.at(r, j)) { + return fmt.Sprintf("eigenvectors %d and %d are not conjugate", j, j+1) + } + } + } + } + } + return "" +} + +func isComplexDType(dt string) bool { return dt == "complex64" || dt == "complex128" } + +func maxAbs(x []complex128) float64 { + m := 0.0 + for _, v := range x { + m = math.Max(m, cmplx.Abs(v)) + } + return m +} diff --git a/linalg/kernels_test.go b/linalg/kernels_test.go new file mode 100644 index 0000000..1f3e215 --- /dev/null +++ b/linalg/kernels_test.go @@ -0,0 +1,298 @@ +package linalg + +import ( + "errors" + "math" + "math/cmplx" + "math/rand/v2" + "testing" + + "github.com/go-ndarray/ndarray" +) + +// Unit tests of the scalar kernels, on the inputs that reach their special +// cases, each checked through its defining property. + +func close(a, b, scale float64) bool { return math.Abs(a-b) <= 1e-14*scale } + +func TestLaev2(t *testing.T) { + r := rand.New(rand.NewPCG(1, 2)) + cases := [][3]float64{{1, 0, 1}, {1, 1, -1}, {2, 0, 1}, {-3, 1, 2}, {1, 2, 1}} + for i := 0; i < 20; i++ { + cases = append(cases, [3]float64{r.NormFloat64(), r.NormFloat64(), r.NormFloat64()}) + } + for _, c := range cases { + a, b, cc := c[0], c[1], c[2] + rt1, rt2, cs, sn := laev2(a, b, cc) + sc := math.Abs(a) + math.Abs(b) + math.Abs(cc) + if !close(a*cs+b*sn, rt1*cs, sc) || !close(b*cs+cc*sn, rt1*sn, sc) || !close(rt1+rt2, a+cc, sc) || + math.Abs(rt1) < math.Abs(rt2) || !close(cs*cs+sn*sn, 1, 1) { + t.Errorf("laev2%v = %v %v %v %v", c, rt1, rt2, cs, sn) + } + } +} + +// sv2 is the singular values of [f g; 0 h], from the eigenvalues of AᵀA. +func sv2(f, g, h float64) (float64, float64) { + tr := f*f + g*g + h*h + det := math.Abs(f * h) + big := math.Sqrt((tr + math.Sqrt(math.Max(tr*tr-4*det*det, 0))) / 2) + if big == 0 { + return 0, 0 + } + return det / big, big +} + +func TestLas2Lasv2(t *testing.T) { + r := rand.New(rand.NewPCG(3, 4)) + cases := [][3]float64{ + {0, 0, 0}, {0, 3, 0}, {0, 3, 4}, {4, 3, 0}, {1, 1e20, 1}, {1e-200, 1e200, 1e-200}, + {1, 1e-300, 1}, {2, 1e-300, 1}, {1, 0, 2}, {2, 0, 1}, {1e-17, 1, 2}, {2, 1, 1e-17}, + {3, 1e17, 2}, {0.5, 1e17, 0.25}, {-2, 5, 3}, {2, -5, -3}, + } + for i := 0; i < 30; i++ { + cases = append(cases, [3]float64{r.NormFloat64(), r.NormFloat64() * 10, r.NormFloat64()}) + } + for _, c := range cases { + f, g, h := c[0], c[1], c[2] + lo, hi := sv2(f, g, h) + smin, smax := las2(f, g, h) + if !close(smin, lo, hi) || !close(smax, hi, hi) { + t.Errorf("las2%v = %v %v, want %v %v", c, smin, smax, lo, hi) + } + ssmin, ssmax, snr, csr, snl, csl := lasv2(f, g, h) + // [csl snl; -snl csl]·[f g; 0 h]·[csr -snr; snr csr] = diag(ssmax, ssmin) + a11 := csl*(f*csr+g*snr) + snl*(h*snr) + a12 := csl*(-f*snr+g*csr) + snl*(h*csr) + a21 := -snl*(f*csr+g*snr) + csl*(h*snr) + a22 := -snl*(-f*snr+g*csr) + csl*(h*csr) + if !close(a11, ssmax, hi) || !close(a22, ssmin, hi) || !close(a12, 0, hi) || !close(a21, 0, hi) || + !close(math.Abs(ssmax), hi, hi) || !close(math.Abs(ssmin), lo, hi) { + t.Errorf("lasv2%v = %v %v: %v %v %v %v", c, ssmin, ssmax, a11, a12, a21, a22) + } + } +} + +func TestLanv2(t *testing.T) { + r := rand.New(rand.NewPCG(5, 6)) + cases := [][4]float64{ + {1, 2, 0, 3}, {1, 0, 2, 3}, {1, 2, -3, 1}, {1, 1, 1e-17, 1}, {1, 1e-17, 1, 1}, + {1e200, 3e200, -2e200, 1.0000001e200}, {1e-200, 3e-200, -2e-200, 1e-200}, {1e-200, 3e-200, -2e-200, 1.5e-200}, {1, 0, -1e-17, 1}, + {1, 1, 1, 1}, {2, -1, 1, 2}, {0, 1, -1, 0}, {1, 2, 2, 1}, + } + for i := 0; i < 50; i++ { + cases = append(cases, [4]float64{r.NormFloat64(), r.NormFloat64(), r.NormFloat64(), r.NormFloat64()}) + } + for _, c := range cases { + a, b, cc, d := c[0], c[1], c[2], c[3] + aa, bb, c2, dd, rt1r, rt1i, rt2r, rt2i, cs, sn := lanv2(a, b, cc, d) + sc := math.Abs(a) + math.Abs(b) + math.Abs(cc) + math.Abs(d) + // [a b; c d] = [cs -sn; sn cs]·[aa bb; cc dd]·[cs sn; -sn cs] + m11, m12 := cs*aa-sn*c2, cs*bb-sn*dd + m21, m22 := sn*aa+cs*c2, sn*bb+cs*dd + r11, r12 := m11*cs-m12*sn, m11*sn+m12*cs + r21, r22 := m21*cs-m22*sn, m21*sn+m22*cs + std := c2 == 0 || (aa == dd && math.Signbit(bb) != math.Signbit(c2)) + eig := rt1r == aa && rt2r == dd && (c2 == 0) == (rt1i == 0) && rt2i == -rt1i + if !close(r11, a, sc) || !close(r12, b, sc) || !close(r21, cc, sc) || !close(r22, d, sc) || !std || !eig { + t.Errorf("lanv2%v = %v %v %v %v", c, aa, bb, c2, dd) + } + } +} + +func TestLartgExtremes(t *testing.T) { + for _, c := range [][2]float64{{1e300, 1e300}, {1e-300, 1e-300}, {3, 4}, {0, -2}, {5, 0}, {-1e300, 1e-300}} { + f, g := c[0], c[1] + cs, sn, r := lartg(f, g) + sc := math.Max(math.Abs(f), math.Abs(g)) + if !close(cs*f+sn*g, r, sc) || !close(-sn*f+cs*g, 0, sc) || !close(cs*cs+sn*sn, 1, 1) { + t.Errorf("lartg%v = %v %v %v", c, cs, sn, r) + } + } +} + +func TestNrm2Lapy3(t *testing.T) { + if lapy3(0, 0, 0) != 0 || lapy3(3, 0, 4) != 5 { + t.Error("lapy3") + } + for _, c := range []struct { + x []float64 + want float64 + }{ + {[]float64{1e200, 1e200}, 1e200 * math.Sqrt2}, + {[]float64{1e-200, 1e-200}, 1e-200 * math.Sqrt2}, + {[]float64{0, 0}, 0}, + {[]float64{3, 4}, 5}, + {[]float64{math.Inf(1), 1}, math.Inf(1)}, + } { + if got := nrm2(len(c.x), c.x, 1); !close(got, c.want, c.want) && got != c.want { + t.Errorf("nrm2%v = %v", c.x, got) + } + } + if !math.IsNaN(nrm2(2, []float64{math.NaN(), 1}, 1)) { + t.Error("nrm2 of NaN") + } + if got := nrm2(1, []complex128{3e200 + 4e200i}, 1); !close(got, 5e200, 5e200) { + t.Errorf("complex nrm2 = %v", got) + } +} + +func TestQuadRoot(t *testing.T) { + // c = 0: one root of the linear equation. + if x := quadRoot(0, 1, 1, 0, 2, 0, 2); !close(x, 1, 1) { + t.Errorf("linear root %v", x) + } + // No real root. + if x := quadRoot(1, 1, -1, 0, 1, 0, 1); !math.IsNaN(x) { + t.Errorf("complex roots gave %v", x) + } + // The root in the bracket, from either formula. + for _, c := range [][5]float64{{1, 1, 1, 0, 3}, {-1, 1, 1, 0, 3}} { + x := quadRoot(c[0], c[1], c[2], c[3], c[4], c[3], c[4]) + if f := c[0] + c[1]/(c[3]-x) + c[2]/(c[4]-x); !(x > c[3] && x < c[4]) || math.Abs(f) > 1e-12 { + t.Errorf("quadRoot%v = %v (f = %v)", c, x, f) + } + } +} + +func TestSolve2(t *testing.T) { + r := rand.New(rand.NewPCG(7, 8)) + for i := 0; i < 20; i++ { + var m [4]complex128 + for j := range m { + m[j] = complex(r.NormFloat64(), r.NormFloat64()) + } + r1, r2 := complex(r.NormFloat64(), 0), complex(0, r.NormFloat64()) + y1, y2, s := solve2(m[0], m[1], m[2], m[3], r1, r2, 1e-300, 1e300) + if s != 1 || cmplx.Abs(m[0]*y1+m[1]*y2-r1) > 1e-12 || cmplx.Abs(m[2]*y1+m[3]*y2-r2) > 1e-12 { + t.Errorf("solve2: %v %v", y1, y2) + } + } + // A matrix below smin is perturbed to smin·I, and a huge solution scaled. + y1, y2, s := solve2(1e-30, 0, 0, 1e-30, 1e3, 1e3, 1e-10, 1e5) + if y1 != y2 || s >= 1 || cmplx.Abs(y1) > 1e10 { + t.Errorf("perturbed solve2: %v %v %v", y1, y2, s) + } +} + +// TestBdsqrGraded runs the bidiagonal QR on graded matrices, where it +// chases the bulge both ways with zero shifts, and checks B = U·S·Vᵀ to +// within rounding of ‖B‖. +func TestBdsqrGraded(t *testing.T) { + type bidi struct{ d, e []float64 } + var cases []bidi + for _, dir := range []float64{1, -1} { + n := 12 + d, e := make([]float64, n), make([]float64, n-1) + for i := range d { + d[i] = math.Pow(10, dir*float64(i-n/2)*2) + } + for i := range e { + e[i] = math.Pow(10, dir*float64(i-n/2)*2) * 0.5 + } + cases = append(cases, bidi{d, e}) + } + // Found by a search: a zero-shift sweep from the bottom deflates at the top. + cases = append(cases, bidi{ + []float64{1.087, 14326.298, 1.6734395e+07, 6.8290222558999234e+13, 6.1163180038118536e+16, 5.214581644332932e+21, 1.2978735465153189e+25}, + []float64{0.156, 833.053, 704284.894, 8.620413772825534e+21, 1.631282114430647e+12, 4.406843216880124e+22}, + }) + for _, c := range cases { + n := len(c.d) + d, e := append([]float64(nil), c.d...), append([]float64(nil), c.e...) + vt, ut := make([]float64, n*n), make([]float64, n*n) + for i := 0; i < n; i++ { + vt[i*n+i], ut[i*n+i] = 1, 1 + } + if !bdsqr(n, d, e, vt, n, ut, n) { + t.Fatal("bdsqr did not converge") + } + for i := 0; i < n; i++ { + for j := 0; j < n; j++ { + s := 0.0 + for k := 0; k < n; k++ { + s += ut[k*n+i] * d[k] * vt[k*n+j] + } + want := 0.0 + if i == j { + want = c.d[i] + } else if j == i+1 { + want = c.e[i] + } + if math.Abs(s-want) > 1e-13*d[0] { + t.Errorf("B[%d,%d] = %v, want %v", i, j, s, want) + } + } + } + } +} + +func TestNegligibleShift(t *testing.T) { + if negligibleShift(1e-9, 1) != 0 || negligibleShift(1e-7, 1) != 1e-7 || negligibleShift(1, 0) != 1 { + t.Error("negligibleShift") + } +} + +func TestSmallAPI(t *testing.T) { + for o, s := range map[Ord]string{{}: "None", Fro: "fro", Nuc: "nuc", P(2): "2"} { + if o.String() != s { + t.Errorf("Ord %q", o.String()) + } + } + a, _ := ndarray.Ones(3, 3) + if _, _, err := QR(a, ModeRaw+1); !errors.Is(err, ErrArgs) { + t.Errorf("QR bad mode: %v", err) + } + v, _ := ndarray.Ones(3) + if _, err := SVDVals(v); !errors.Is(err, ErrShape) { + t.Errorf("SVDVals 1-D: %v", err) + } +} + +// TestBdsdcZeroDiagonal: zero diagonal entries give the halves zero +// singular values, which the merge must keep apart from the arrow's own +// zero pole. +func TestBdsdcZeroDiagonal(t *testing.T) { + r := rand.New(rand.NewPCG(9, 10)) + n := 60 + for _, zeros := range [][]int{{5}, {5, 17}, {29, 30, 45}, {0, 59}} { + d, e := make([]float64, n), make([]float64, n-1) + for i := range d { + d[i] = r.NormFloat64() + } + for i := range e { + e[i] = r.NormFloat64() + } + for _, z := range zeros { + d[z] = 0 + } + d0, e0 := append([]float64(nil), d...), append([]float64(nil), e...) + u, vt, ok := bdsdc(n, d, e) + if !ok { + t.Fatal("bdsdc did not converge") + } + worst := 0.0 + for i := 0; i < n; i++ { + for j := 0; j < n; j++ { + s := 0.0 + for k := 0; k < n; k++ { + s += u[i*n+k] * d[k] * vt[k*n+j] + } + want := 0.0 + if i == j { + want = d0[i] + } else if j == i+1 { + want = e0[i] + } + worst = math.Max(worst, math.Abs(s-want)) + } + } + uc, vc := cmat{n, n, make([]complex128, n*n)}, cmat{n, n, make([]complex128, n*n)} + for i := range u { + uc.d[i], vc.d[i] = complex(u[i], 0), complex(vt[i], 0) + } + if worst > 1e-13*d[0] || orthErr(uc) > 1e-12 || orthErr(adj(vc)) > 1e-12 { + t.Errorf("zeros %v: ‖B - USVᵀ‖∞ = %.3g, orth %.3g %.3g", zeros, worst, orthErr(uc), orthErr(adj(vc))) + } + } +} diff --git a/linalg/linalg.go b/linalg/linalg.go new file mode 100644 index 0000000..836e03f --- /dev/null +++ b/linalg/linalg.go @@ -0,0 +1,244 @@ +// Package linalg is numpy.linalg for ndarray arrays, in pure Go: solving, +// inverting, determinants, the Cholesky, QR, singular-value and eigenvalue +// decompositions, least squares, pseudo-inverses, ranks, norms and condition +// numbers. +// +// Every function takes arrays of any dtype and treats the last two axes as +// the matrix, so a (..., M, N) input is a stack of matrices processed one by +// one (in parallel), as numpy's gufuncs do. As in numpy, the computation is +// always in double precision: float64 and complex128 natively, integers and +// bools after promotion to float64, float32 and complex64 after widening — +// their results are rounded back to float32 or complex64, as numpy's are. +// +// Decompositions follow LAPACK's algorithms (the ones numpy calls) and its +// sign and ordering conventions: LU with partial pivoting, Householder QR, +// Cholesky, Householder tridiagonalisation with implicit QL/QR for Hermitian +// eigenproblems, Golub–Kahan bidiagonalisation with implicit-shift QR for the +// SVD, and balancing, Hessenberg reduction, Francis QR and back-substitution +// for general eigenproblems. Large blocks are updated with the ndarray +// package's packed SIMD GEMM. docs/linalg.md describes the algorithms, their +// accuracy and their speed against numpy. +// +// Errors: numerical failures — a singular matrix, a matrix that is not +// positive definite, an iteration that does not converge — wrap ErrLinAlg, +// numpy's LinAlgError. Operands of the wrong rank or shape give ErrShape, and +// invalid options ErrArgs. +package linalg + +import ( + "errors" + "fmt" + "runtime" + "sync" + "sync/atomic" + + "github.com/go-ndarray/ndarray" +) + +// ErrLinAlg is numpy's LinAlgError: the computation failed numerically. The +// more specific ErrSingular, ErrNotPosDef and ErrNoConvergence wrap it. +var ErrLinAlg = errors.New("linalg: LinAlgError") + +// The numerical failures, each wrapping ErrLinAlg. +var ( + ErrSingular = fmt.Errorf("%w: singular matrix", ErrLinAlg) + ErrNotPosDef = fmt.Errorf("%w: matrix is not positive definite", ErrLinAlg) + ErrNoConvergence = fmt.Errorf("%w: did not converge", ErrLinAlg) +) + +// ErrShape is returned for operands whose rank or shape the operation cannot +// take: fewer than two dimensions, a non-square matrix where a square one is +// needed, mismatched or non-broadcastable dimensions, or an empty matrix +// where numpy refuses one. (numpy raises LinAlgError for some of these and +// ValueError for others.) +var ErrShape = errors.New("linalg: invalid shape") + +// ErrArgs is returned for invalid options: an unknown norm order, repeated +// or out-of-range axes, too few arrays. +var ErrArgs = errors.New("linalg: invalid argument") + +// iterFactor scales the iteration limits of the QR iterations (LAPACK's 30 +// sweeps per eigenvalue); tests lower it to exercise their failure paths. +var iterFactor = 30 + +// UPLO selects which triangle of a Hermitian matrix is read. +type UPLO uint8 + +// The two triangles. Lower is numpy's default. +const ( + Lower UPLO = iota + Upper +) + +// operand is an input whose last two axes are the matrix. +type operand struct { + a *ndarray.Array + batch []int // the leading dimensions + m, n int + cplx bool // computed in complex128 (else float64) + single bool // results are float32 / complex64 +} + +// count is the number of matrices in the stack. +func (o operand) count() int { return prod(o.batch) } + +func prod(s []int) int { + p := 1 + for _, v := range s { + p *= v + } + return p +} + +// precision reports how numpy computes and returns a group of operands (its +// _commonType): complex if any is complex; single precision only if every +// one is float32 or complex64 — integers and bools count as double. +func precision(arrays ...*ndarray.Array) (cplx, single bool) { + single = true + for _, a := range arrays { + dt := a.DType() + cplx = cplx || dt.IsComplex() + if dt != ndarray.Float32 && dt != ndarray.Complex64 { + single = false + } + } + return cplx, single +} + +// stacked checks that a has at least two dimensions and describes it. +func stacked(a *ndarray.Array, fn string) (operand, error) { + s := a.Shape() + if len(s) < 2 { + return operand{}, fmt.Errorf("%w: %s needs an array of at least two dimensions, got %d-D", ErrShape, fn, len(s)) + } + cplx, single := precision(a) + return operand{a: a, batch: s[:len(s)-2], m: s[len(s)-2], n: s[len(s)-1], cplx: cplx, single: single}, nil +} + +// square is stacked for operations on square matrices. +func square(a *ndarray.Array, fn string) (operand, error) { + o, err := stacked(a, fn) + if err == nil && o.m != o.n { + err = fmt.Errorf("%w: %s needs square matrices, got %d×%d", ErrShape, fn, o.m, o.n) + } + return o, err +} + +// load returns a contiguous copy of a's elements converted to T. +func load[T scalar](a *ndarray.Array) []T { + if isCplx[T]() { + d, _ := ndarray.Data[complex128](a.AsType(ndarray.Complex128)) + return any(d).([]T) + } + d, _ := ndarray.Data[float64](a.AsType(ndarray.Float64)) + return any(d).([]T) +} + +// wrap makes an array of the given shape from computed values, rounded to +// single precision when single is set. +func wrap[T scalar](d []T, shape []int, single bool) *ndarray.Array { + if z, ok := any(d).([]complex128); ok { + r, _ := ndarray.FromSlice(z, shape...) + if single { + r = r.AsType(ndarray.Complex64) + } + return r + } + r, _ := ndarray.FromSlice(any(d).([]float64), shape...) + if single { + r = r.AsType(ndarray.Float32) + } + return r +} + +// shapeOf joins a batch shape and trailing dimensions. +func shapeOf(batch []int, tail ...int) []int { + return append(append([]int(nil), batch...), tail...) +} + +// parallelFor runs f(0..count-1), spreading the calls over the processors +// when there are several and each is worth a goroutine (work is the rough +// cost of one call in flops). +func parallelFor(count, work int, f func(i int)) { + w := min(runtime.GOMAXPROCS(0), count) + if w <= 1 || count*work < 1<<15 { + for i := 0; i < count; i++ { + f(i) + } + return + } + var next atomic.Int64 + var wg sync.WaitGroup + for g := 0; g < w; g++ { + wg.Add(1) + go func() { + defer wg.Done() + for { + i := int(next.Add(1)) - 1 + if i >= count { + return + } + f(i) + } + }() + } + wg.Wait() +} + +// firstErr collects the error of the lowest-indexed failing matrix of a +// parallel loop, so the error reported does not depend on scheduling. +type firstErr struct { + mu sync.Mutex + idx int + err error +} + +func (f *firstErr) set(i int, err error) { + f.mu.Lock() + if f.err == nil || i < f.idx { + f.idx, f.err = i, err + } + f.mu.Unlock() +} + +// broadcast returns the broadcast of two batch shapes, numpy's rules. +func broadcast(x, y []int) ([]int, bool) { + n := max(len(x), len(y)) + out := make([]int, n) + for i := range out { + a, b := dimAt(x, i-n+len(x)), dimAt(y, i-n+len(y)) + switch { + case a == b || b == 1: + out[i] = a + case a == 1: + out[i] = b + default: + return nil, false + } + } + return out, true +} + +func dimAt(s []int, i int) int { + if i < 0 { + return 1 + } + return s[i] +} + +// batchIndex maps the linear index of an element of the broadcast batch +// shape out to the linear index in the batch shape in that broadcasts to it. +func batchIndex(out, in []int, i int) int { + idx, stride := 0, 1 + for ax := len(out) - 1; ax >= 0; ax-- { + c := i % out[ax] + i /= out[ax] + if j := ax - len(out) + len(in); j >= 0 { + if in[j] != 1 { + idx += c * stride + } + stride *= in[j] + } + } + return idx +} diff --git a/linalg/linalg.test b/linalg/linalg.test new file mode 100755 index 0000000..4a5edad Binary files /dev/null and b/linalg/linalg.test differ diff --git a/linalg/lstsq.go b/linalg/lstsq.go new file mode 100644 index 0000000..3702afe --- /dev/null +++ b/linalg/lstsq.go @@ -0,0 +1,347 @@ +package linalg + +import ( + "fmt" + "math" + + "github.com/go-ndarray/ndarray" +) + +// eps64 and eps32 are numpy's finfo(float64).eps and finfo(float32).eps. +const ( + eps64 = 0x1p-52 + eps32 = 0x1p-23 +) + +// LstsqResult is numpy.linalg.lstsq's result. +type LstsqResult struct { + // X is the least-squares solution, (N,) for a vector b and (N, K) for a + // matrix b; the minimum-norm one when a is rank deficient. + X *ndarray.Array + // Residuals holds the squared Euclidean norm of each column of b - a·X, + // (K,) — (1,) for a vector b, as in numpy — when the rank is N and + // M > N; otherwise it is empty, (0,). + Residuals *ndarray.Array + // Rank is the effective rank of a. + Rank int + // S holds a's singular values, decreasing, (min(M, N),). + S *ndarray.Array +} + +// Lstsq is numpy.linalg.lstsq(a, b) with numpy 2's default cutoff: the X +// minimizing ‖b - a·X‖ for an (M, N) a and a b of shape (M,) or (M, K). +// Singular values at most eps·max(M, N) times the largest are treated as +// zero (eps = 2⁻⁵²). +func Lstsq(a, b *ndarray.Array) (LstsqResult, error) { + return lstsq(a, b, math.NaN()) +} + +// LstsqRcond is Lstsq with numpy's rcond: singular values at most rcond +// times the largest are treated as zero. As in LAPACK's gelsd, which numpy +// calls, an rcond outside (0, 1) means the machine precision 2⁻⁵³. +func LstsqRcond(a, b *ndarray.Array, rcond float64) (LstsqResult, error) { + return lstsq(a, b, rcond) +} + +func lstsq(a, b *ndarray.Array, rcond float64) (LstsqResult, error) { + as, bs := a.Shape(), b.Shape() + vec := len(bs) == 1 + if len(as) != 2 || (len(bs) != 2 && !vec) { + return LstsqResult{}, fmt.Errorf("%w: Lstsq needs a 2-D a and a 1-D or 2-D b, got %d-D and %d-D", ErrShape, len(as), len(bs)) + } + m, n := as[0], as[1] + k := 1 + if !vec { + k = bs[1] + } + if bs[0] != m { + return LstsqResult{}, fmt.Errorf("%w: Lstsq a has %d rows, b %d", ErrShape, m, bs[0]) + } + switch { + case math.IsNaN(rcond): + rcond = eps64 * float64(max(m, n)) + case rcond <= 0 || rcond >= 1: + rcond = eps + } + cplx, single := precision(a, b) + if cplx { + return lstsqT[complex128](a, b, m, n, k, vec, rcond, single) + } + return lstsqT[float64](a, b, m, n, k, vec, rcond, single) +} + +func lstsqT[T scalar](a, b *ndarray.Array, m, n, k int, vec bool, rcond float64, single bool) (LstsqResult, error) { + ad, bd := load[T](a), load[T](b) + p := min(m, n) + s, u, vh, ok := svdOne(m, n, append([]T(nil), ad...), true, false) + if !ok { + return LstsqResult{}, fmt.Errorf("%w in linear least squares", ErrNoConvergence) + } + tol := 0.0 + if p > 0 { + tol = rcond * s[0] + } + rank := 0 + for _, v := range s { + if v > tol { + rank++ + } + } + // X = V·S⁺·Uᴴ·b over the rank largest singular values. + c := make([]T, p*k) + gemm(p, k, m, 1, rm(u, 0, p).h(), rm(bd, 0, k), 0, c, k) + for i := 0; i < p; i++ { + row := c[i*k : i*k+k] + for j := range row { + if i < rank { + row[j] = rscale(row[j], 1/s[i]) + } else { + row[j] = 0 + } + } + } + x := make([]T, n*k) + gemm(n, k, p, 1, rm(vh, 0, n).h(), rm(c, 0, k), 0, x, k) + var resid []float64 + if rank == n && m > n { + // From the QR factorization: the residual is the part of Qᴴb below + // row n, free of the cancellation in b - a·x. + tau := make([]T, n) + geqrf(m, n, ad, n, tau) + v := make([]T, m) + w := make([]T, k) + for i := 0; i < n; i++ { + v[0] = 1 + for r := i + 1; r < m; r++ { + v[r-i] = ad[r*n+i] + } + larfL(m-i, k, v[:m-i], conj(tau[i]), bd[i*k:], k, w) + } + resid = make([]float64, k) + for j := range resid { + resid[j] = math.Pow(nrm2(m-n, bd[n*k+j:], k), 2) + } + } + xs := []int{n, k} + if vec { + xs = xs[:1] + } + return LstsqResult{ + X: wrap(x, xs, single), + Residuals: wrap(resid, []int{len(resid)}, single), + Rank: rank, + S: wrap(s, []int{p}, single), + }, nil +} + +// Pinv is numpy.linalg.pinv(a) with numpy's default cutoff: the +// Moore–Penrose pseudo-inverse (..., N, M) of each matrix of a (..., M, N), +// from its SVD, singular values at most 1e-15 times the largest being +// treated as zero. +func Pinv(a *ndarray.Array) (*ndarray.Array, error) { return PinvRcond(a, 1e-15) } + +// PinvRcond is Pinv with numpy's rcond: singular values at most rcond times +// the largest are treated as zero. numpy's rtol=None is +// PinvRcond(a, max(M, N)·eps). +func PinvRcond(a *ndarray.Array, rcond float64) (*ndarray.Array, error) { + o, err := stacked(a, "Pinv") + if err != nil { + return nil, err + } + if o.m*o.n == 0 { + // numpy returns an empty array of a's own dtype. + return ndarray.ZerosOf(a.DType(), shapeOf(o.batch, o.n, o.m)...) + } + if o.cplx { + return pinvT[complex128](o, rcond) + } + return pinvT[float64](o, rcond) +} + +func pinvT[T scalar](o operand, rcond float64) (*ndarray.Array, error) { + m, n, cnt := o.m, o.n, o.count() + p := min(m, n) + data := load[T](o.a) + out := make([]T, cnt*n*m) + var fe firstErr + parallelFor(cnt, 8*m*n*p, func(i int) { + s, u, vh, ok := svdOne(m, n, data[i*m*n:(i+1)*m*n], true, false) + if !ok { + fe.set(i, ErrNoConvergence) + return + } + cut := rcond * s[0] + // S⁺·Uᴴ, p×m. + su := make([]T, p*m) + for r := 0; r < p; r++ { + if !(s[r] > cut) { + continue + } + inv := 1 / s[r] + for c := 0; c < m; c++ { + su[r*m+c] = rscale(conj(u[c*p+r]), inv) + } + } + gemm(n, m, p, 1, rm(vh, 0, n).h(), rm(su, 0, m), 0, out[i*n*m:], m) + }) + if fe.err != nil { + return nil, fe.err + } + return wrap(out, shapeOf(o.batch, n, m), o.single), nil +} + +// singularValues returns the singular values of every matrix of o (k per +// matrix, decreasing), rounded to float32 when o is single precision. +func singularValues(o operand) ([]float64, error) { + var s *ndarray.Array + var err error + if o.cplx { + _, s, _, err = svdT[complex128](o, false, false) + } else { + _, s, _, err = svdT[float64](o, false, false) + } + if err != nil { + return nil, err + } + d, _ := ndarray.Data[float64](s.AsType(ndarray.Float64)) + return d, nil +} + +// MatrixRank is numpy.linalg.matrix_rank(A) with its default tolerance: the +// number of singular values of each matrix of a greater than +// S.max()·max(M, N)·eps, eps being the machine epsilon of the singular +// values' dtype (float32's for single-precision inputs). The result is +// Int64, shaped like a's batch dimensions. As in numpy, an array of fewer +// than two dimensions has rank 1 unless all its elements are zero. +func MatrixRank(a *ndarray.Array) (*ndarray.Array, error) { return matrixRank(a, math.NaN()) } + +// MatrixRankTol is numpy.linalg.matrix_rank(A, tol): singular values +// greater than the absolute threshold tol count. +func MatrixRankTol(a *ndarray.Array, tol float64) (*ndarray.Array, error) { + return matrixRank(a, tol) +} + +func matrixRank(a *ndarray.Array, tol float64) (*ndarray.Array, error) { + if len(a.Shape()) < 2 { + nz := int64(0) + if a.Any() { + nz = 1 + } + return ndarray.FromSlice([]int64{nz}) + } + o, _ := stacked(a, "MatrixRank") + s, err := singularValues(o) + if err != nil { + return nil, err + } + k := min(o.m, o.n) + e := eps64 + if o.single { + e = eps32 + } + ranks := make([]int64, o.count()) + for i := range ranks { + si := s[i*k : (i+1)*k] + t := tol + if math.IsNaN(t) { + t = 0 + if k > 0 { + t = si[0] * float64(max(o.m, o.n)) * e + } + } + for _, v := range si { + if v > t { + ranks[i]++ + } + } + } + return ndarray.FromSlice(ranks, o.batch...) +} + +// Cond is numpy.linalg.cond: the condition number of each matrix of x in +// the norm p. The default P2 (or the zero Ord, numpy's None) and its +// negative are the ratio of the extreme singular values and take any +// (..., M, N); the other orders (Fro, Nuc, ±1, ±Inf) are norm(x)·norm(x⁻¹) +// and need square matrices. A singular matrix has condition number +Inf. +func Cond(x *ndarray.Array, p Ord) (*ndarray.Array, error) { + o, err := stacked(x, "Cond") + if err != nil { + return nil, err + } + if o.m*o.n == 0 { + return nil, fmt.Errorf("%w: Cond is not defined on empty matrices", ErrShape) + } + r := make([]float64, o.count()) + if p.kind == ordDefault || (p.kind == ordNum && math.Abs(p.p) == 2) { + s, err := singularValues(o) + if err != nil { + return nil, err + } + k := min(o.m, o.n) + for i := range r { + if p.p == -2 { + r[i] = s[i*k+k-1] / s[i*k] + } else { + r[i] = s[i*k] / s[i*k+k-1] + } + } + } else { + if o.m != o.n { + return nil, fmt.Errorf("%w: Cond in this norm needs square matrices, got %d×%d", ErrShape, o.m, o.n) + } + if o.cplx { + err = condT[complex128](o, p, r) + } else { + err = condT[float64](o, p, r) + } + if err != nil { + return nil, err + } + } + data := load[complex128](x) + n2 := o.m * o.n + for i, v := range r { + // numpy turns a NaN from a singular matrix into +Inf, unless the + // matrix itself holds a NaN. + if math.IsNaN(v) && finiteOrInf(data[i*n2:(i+1)*n2]) { + r[i] = math.Inf(1) + } + } + return wrap(r, o.batch, o.single), nil +} + +// finiteOrInf reports whether x holds no NaN. +func finiteOrInf(x []complex128) bool { + for _, v := range x { + if math.IsNaN(real(v)) || math.IsNaN(imag(v)) { + return false + } + } + return true +} + +func condT[T scalar](o operand, p Ord, r []float64) error { + n := o.n + d := load[T](o.a) + for i := range r { + x := d[i*n*n : (i+1)*n*n] + nx, err := matNorm(n, n, x, p) + if err != nil { + return err + } + lu := append([]T(nil), x...) + piv := make([]int, n) + if getrf(n, lu, n, piv) >= 0 { + r[i] = math.NaN() + continue + } + inv := make([]T, n*n) + for j := 0; j < n; j++ { + inv[j*n+j] = 1 + } + getrs(n, n, lu, piv, inv) + ni, _ := matNorm(n, n, inv, p) + r[i] = nx * ni + } + return nil +} diff --git a/linalg/lu.go b/linalg/lu.go new file mode 100644 index 0000000..15bba02 --- /dev/null +++ b/linalg/lu.go @@ -0,0 +1,296 @@ +package linalg + +import ( + "fmt" + "math" + "math/cmplx" + + "github.com/go-ndarray/ndarray" +) + +// getrf factors the n×n row-major matrix a (leading dimension lda) in place +// as P·L·U with partial pivoting (LAPACK getrf): L unit lower triangular +// below the diagonal, U upper triangular on and above it. piv[j] is the row +// interchanged with row j at step j. It returns the index of the first +// exactly zero pivot, or -1: like LAPACK, the factorization is still +// completed then. +// +// The recursion (LAPACK getrf2) splits the columns in halves, so all but a +// vanishing fraction of the work is the GEMM update of the trailing block. +// Row interchanges swap whole rows at once, which in row-major storage is a +// contiguous copy and is equivalent to LAPACK's deferred laswp calls. +func getrf[T scalar](n int, a []T, lda int, piv []int) int { + return luRec(n, n, a, lda, piv, 0, n) +} + +// luRec factors columns c0..c0+w-1, rows c0..m-1, of the m×ncol matrix a. +func luRec[T scalar](m, ncol int, a []T, lda int, piv []int, c0, w int) int { + if w <= trsmLeaf { + return luLeaf(m, ncol, a, lda, piv, c0, w) + } + w1 := w / 2 + info := luRec(m, ncol, a, lda, piv, c0, w1) + c1 := c0 + w1 + trsmLLU(w1, w-w1, a[c0*lda+c0:], lda, a[c0*lda+c1:], lda) + gemm(m-c1, w-w1, w1, -1, rm(a, c1*lda+c0, lda), rm(a, c0*lda+c1, lda), 1, a[c1*lda+c1:], lda) + if i2 := luRec(m, ncol, a, lda, piv, c1, w-w1); info < 0 { + info = i2 + } + return info +} + +// luLeaf is the unblocked right-looking factorization of a narrow panel. +func luLeaf[T scalar](m, ncol int, a []T, lda int, piv []int, c0, w int) int { + info := -1 + end := c0 + w + for j := c0; j < end; j++ { + // Like LAPACK's i*amax, a NaN in the first place is kept as the + // pivot (so it propagates); later NaNs never compare larger. + p, best := j, abs1(a[j*lda+j]) + for i := j + 1; i < m; i++ { + if v := abs1(a[i*lda+j]); v > best { + p, best = i, v + } + } + piv[j] = p + if a[p*lda+j] == 0 { + if info < 0 { + info = j + } + } else { + if p != j { + rj, rp := a[j*lda:j*lda+ncol], a[p*lda:p*lda+ncol] + for c := range rj { + rj[c], rp[c] = rp[c], rj[c] + } + } + // Divide rather than multiply by the reciprocal: an entry equal + // to the pivot then gives a multiplier of exactly 1, so + // duplicated rows cancel exactly and a singular matrix is + // reported as such (x·(1/x) is not always 1, in complex + // arithmetic especially). + d := a[j*lda+j] + for i := j + 1; i < m; i++ { + a[i*lda+j] /= d + } + } + rowj := a[j*lda+j+1 : j*lda+end] + for i := j + 1; i < m; i++ { + l := a[i*lda+j] + ri := a[i*lda+j+1 : i*lda+end] + for c, u := range rowj { + ri[c] -= l * u + } + } + } + return info +} + +// getrs solves A·X = B with the LU factors of the n×n A, B n×nrhs in place. +func getrs[T scalar](n, nrhs int, lu []T, piv []int, b []T) { + for i := 0; i < n; i++ { + if p := piv[i]; p != i { + ri, rp := b[i*nrhs:i*nrhs+nrhs], b[p*nrhs:p*nrhs+nrhs] + for c := range ri { + ri[c], rp[c] = rp[c], ri[c] + } + } + } + trsmLLU(n, nrhs, lu, n, b, nrhs) + trsmLUN(n, nrhs, lu, n, b, nrhs) +} + +// Solve is numpy.linalg.solve: the solution X of A·X = B for square A. +// +// a is (..., M, M). b is either a vector (M,), solved against every matrix +// of a, giving (..., M); or a stack (..., M, K), whose leading dimensions +// broadcast against a's, giving (..., M, K). As in numpy 2, a b of two or +// more dimensions is always a stack of matrices. An exactly singular matrix +// gives ErrSingular. +func Solve(a, b *ndarray.Array) (*ndarray.Array, error) { + oa, err := square(a, "Solve") + if err != nil { + return nil, err + } + bs := b.Shape() + var bb []int + k := 1 + vec := len(bs) == 1 + switch { + case vec: + if bs[0] != oa.n { + return nil, fmt.Errorf("%w: Solve %v and %v: b has %d rows, a %d", ErrShape, a.Shape(), bs, bs[0], oa.n) + } + case len(bs) < 2: + return nil, fmt.Errorf("%w: Solve needs b of at least one dimension", ErrShape) + default: + if bs[len(bs)-2] != oa.n { + return nil, fmt.Errorf("%w: Solve %v and %v: b has %d rows, a %d", ErrShape, a.Shape(), bs, bs[len(bs)-2], oa.n) + } + bb, k = bs[:len(bs)-2], bs[len(bs)-1] + } + batch, ok := broadcast(oa.batch, bb) + if !ok { + return nil, fmt.Errorf("%w: Solve batch dimensions %v and %v do not broadcast", ErrShape, oa.batch, bb) + } + cplx, single := precision(a, b) + shape := shapeOf(batch, oa.n) + if !vec { + shape = append(shape, k) + } + if cplx { + return solveT[complex128](oa, b, batch, bb, k, shape, single) + } + return solveT[float64](oa, b, batch, bb, k, shape, single) +} + +func solveT[T scalar](oa operand, b *ndarray.Array, batch, bb []int, k int, shape []int, single bool) (*ndarray.Array, error) { + n := oa.n + ad, bd := load[T](oa.a), load[T](b) + cnt := prod(batch) + out := make([]T, cnt*n*k) + var fe firstErr + parallelFor(cnt, n*n*(n+k), func(i int) { + ai := batchIndex(batch, oa.batch, i) + lu := append([]T(nil), ad[ai*n*n:(ai+1)*n*n]...) + piv := make([]int, n) + if getrf(n, lu, n, piv) >= 0 { + fe.set(i, ErrSingular) + return + } + x := out[i*n*k : (i+1)*n*k] + bi := batchIndex(batch, bb, i) + copy(x, bd[bi*n*k:]) + getrs(n, k, lu, piv, x) + }) + if fe.err != nil { + return nil, fe.err + } + return wrap(out, shape, single), nil +} + +// Inv is numpy.linalg.inv: the inverse of each square matrix of a +// (..., M, M). An exactly singular matrix gives ErrSingular. +func Inv(a *ndarray.Array) (*ndarray.Array, error) { + o, err := square(a, "Inv") + if err != nil { + return nil, err + } + if o.cplx { + return invT[complex128](o) + } + return invT[float64](o) +} + +func invT[T scalar](o operand) (*ndarray.Array, error) { + n := o.n + d := load[T](o.a) + out := make([]T, len(d)) + var fe firstErr + parallelFor(o.count(), 2*n*n*n, func(i int) { + lu := d[i*n*n : (i+1)*n*n] + piv := make([]int, n) + if getrf(n, lu, n, piv) >= 0 { + fe.set(i, ErrSingular) + return + } + x := out[i*n*n : (i+1)*n*n] + for j := 0; j < n; j++ { + x[j*n+j] = 1 + } + getrs(n, n, lu, piv, x) + }) + if fe.err != nil { + return nil, fe.err + } + return wrap(out, shapeOf(o.batch, n, n), o.single), nil +} + +// Det is numpy.linalg.det: the determinant of each square matrix of a +// (..., M, M), shaped (...). It is computed from the LU factors, with the +// product of the pivots accumulated as a mantissa and an exponent, so it +// neither overflows nor underflows before the final result does. A singular +// matrix has determinant 0, not an error. +func Det(a *ndarray.Array) (*ndarray.Array, error) { + o, err := square(a, "Det") + if err != nil { + return nil, err + } + if o.cplx { + return detT[complex128](o) + } + return detT[float64](o) +} + +func detT[T scalar](o operand) (*ndarray.Array, error) { + signs, mants, exps, _ := luDets[T](o) + out := make([]T, len(signs)) + for i := range out { + out[i] = rscale(signs[i], math.Ldexp(mants[i], exps[i])) + } + return wrap(out, o.batch, o.single), nil +} + +// Slogdet is numpy.linalg.slogdet: the sign and the natural logarithm of the +// absolute value of the determinant of each matrix of a (..., M, M), so that +// det = sign·exp(logabsdet) without overflow. For a complex input sign is the +// complex unit det/|det|. A singular matrix has sign 0 and logabsdet -Inf. +// sign has a's (promoted) dtype, logabsdet the matching real dtype. +func Slogdet(a *ndarray.Array) (sign, logabsdet *ndarray.Array, err error) { + o, err := square(a, "Slogdet") + if err != nil { + return nil, nil, err + } + if o.cplx { + return slogdetT[complex128](o) + } + return slogdetT[float64](o) +} + +func slogdetT[T scalar](o operand) (*ndarray.Array, *ndarray.Array, error) { + signs, _, _, logs := luDets[T](o) + return wrap(signs, o.batch, o.single), wrap(logs, o.batch, o.single), nil +} + +// luDets factors every matrix of o and returns, for each, the sign (a unit, +// or 0 when singular), the determinant's modulus as mantissa·2^exponent, +// and its natural logarithm. +func luDets[T scalar](o operand) (signs []T, mants []float64, exps []int, logs []float64) { + n, cnt := o.n, o.count() + d := load[T](o.a) + signs, mants, exps, logs = make([]T, cnt), make([]float64, cnt), make([]int, cnt), make([]float64, cnt) + parallelFor(cnt, n*n*n, func(i int) { + lu := d[i*n*n : (i+1)*n*n] + piv := make([]int, n) + getrf(n, lu, n, piv) + sign, mant, exp, logd := T(1), 1.0, 0, 0.0 + for j := 0; j < n; j++ { + if piv[j] != j { + sign = -sign + } + p := lu[j*n+j] + ap := abs(p) + if ap == 0 { + sign, mant, logd = 0, 0, math.Inf(-1) + break + } + sign *= unit(p, ap) + f, e := math.Frexp(ap) + mant, exp = mant*f, exp+e + f, e = math.Frexp(mant) + mant, exp = f, exp+e + logd += math.Log(ap) + } + signs[i], mants[i], exps[i], logs[i] = sign, mant, exp, logd + }) + return signs, mants, exps, logs +} + +// unit is p/|p| (ap = |p| > 0). +func unit[T scalar](p T, ap float64) T { + if z, ok := any(p).(complex128); ok { + u := complex(real(z)/ap, imag(z)/ap) + return any(u / complex(cmplx.Abs(u), 0)).(T) + } + return fromReal[T](math.Copysign(1, any(p).(float64))) +} diff --git a/linalg/norm.go b/linalg/norm.go new file mode 100644 index 0000000..2922b5a --- /dev/null +++ b/linalg/norm.go @@ -0,0 +1,317 @@ +package linalg + +import ( + "fmt" + "math" + + "github.com/go-ndarray/ndarray" +) + +// Ord is a norm order, numpy's ord argument. The zero value is numpy's +// None: the 2-norm of vectors and the Frobenius norm of matrices. Make a +// numeric order with P (P(math.Inf(1)) is numpy's inf); Fro and Nuc are +// numpy's 'fro' and 'nuc'. +type Ord struct { + kind uint8 + p float64 +} + +const ( + ordDefault uint8 = iota + ordNum + ordFro + ordNuc +) + +// The named matrix norms. +var ( + Fro = Ord{kind: ordFro} // the Frobenius norm + Nuc = Ord{kind: ordNuc} // the nuclear norm, the sum of the singular values +) + +// P is the numeric norm order p: for vectors (Σ|x|^p)^(1/p), with 0 the +// count of nonzeros and ±Inf the largest and smallest |x|; for matrices +// ±1, ±2 and ±Inf only. +func P(p float64) Ord { return Ord{kind: ordNum, p: p} } + +func (o Ord) String() string { + switch o.kind { + case ordFro: + return "fro" + case ordNuc: + return "nuc" + case ordNum: + return fmt.Sprint(o.p) + } + return "None" +} + +// Norm is numpy.linalg.norm(x, ord, axis, keepdims). +// +// With axes nil: the default ord is the 2-norm of all of x flattened, +// whatever its rank; another ord takes x as a vector (1-D) or a matrix +// (2-D). With one axis, vector norms are taken along it; with two, matrix +// norms over them (the first indexes rows). keepdims keeps the reduced axes +// with length one. The result is float64, or float32 for float32 and +// complex64 inputs. +// +// The sums are accumulated in double precision, and the 2-norms are +// rescaled when the sum of squares would overflow or underflow, so a +// float32 norm, or a norm of huge or tiny values, can be more accurate than +// numpy's (which overflows to Inf where this returns the norm). +func Norm(x *ndarray.Array, ord Ord, axes []int, keepdims bool) (*ndarray.Array, error) { + if x.DType().IsComplex() { + return normT[complex128](x, ord, axes, keepdims) + } + return normT[float64](x, ord, axes, keepdims) +} + +func normT[T scalar](x *ndarray.Array, ord Ord, axes []int, keepdims bool) (*ndarray.Array, error) { + shape := x.Shape() + nd := len(shape) + _, single := precision(x) + data := load[T](x) + if axes == nil { + if ord.kind == ordDefault || (ord.kind == ordFro && nd == 2) || (ord.kind == ordNum && ord.p == 2 && nd == 1) { + var os []int + if keepdims { + os = ones(nd) + } + return wrap([]float64{nrm2(len(data), data, 1)}, os, single), nil + } + axes = make([]int, nd) + for i := range axes { + axes[i] = i + } + } + ax := make([]int, len(axes)) + for i, a := range axes { + if a < -nd || a >= nd { + return nil, fmt.Errorf("%w: axis %d out of range for %d-D array", ErrArgs, a, nd) + } + ax[i] = (a + nd) % nd + } + var vals []float64 + var err error + switch len(ax) { + case 1: + moved, rest := moveToEnd(data, shape, ax) + l := shape[ax[0]] + vals = make([]float64, prod(rest)) + for i := range vals { + if vals[i], err = vecNorm(moved[i*l:(i+1)*l], ord); err != nil { + return nil, err + } + } + case 2: + if ax[0] == ax[1] { + return nil, fmt.Errorf("%w: duplicate axes %v", ErrArgs, axes) + } + moved, rest := moveToEnd(data, shape, ax) + r, c := shape[ax[0]], shape[ax[1]] + vals = make([]float64, prod(rest)) + if len(vals) == 0 { + // numpy still rejects the order, and an empty reduction axis. + _, err = matNorm(r, c, make([]T, r*c), ord) + } + for i := range vals { + if vals[i], err = matNorm(r, c, moved[i*r*c:(i+1)*r*c], ord); err != nil { + break + } + } + if err != nil { + return nil, err + } + default: + return nil, fmt.Errorf("%w: Norm over %d axes", ErrArgs, len(ax)) + } + out := make([]int, 0, nd) + for i, s := range shape { + if reduced(ax, i) { + if keepdims { + out = append(out, 1) + } + continue + } + out = append(out, s) + } + return wrap(vals, out, single), nil +} + +func ones(n int) []int { + s := make([]int, n) + for i := range s { + s[i] = 1 + } + return s +} + +func reduced(ax []int, i int) bool { + for _, a := range ax { + if a == i { + return true + } + } + return false +} + +// moveToEnd returns the elements of the row-major array (data, shape) +// reordered so the axes ax come last, in their order, and the shape of the +// other axes. +func moveToEnd[T scalar](data []T, shape, ax []int) ([]T, []int) { + var order, rest []int + for i := range shape { + if !reduced(ax, i) { + order = append(order, i) + rest = append(rest, shape[i]) + } + } + order = append(order, ax...) + strides := make([]int, len(shape)) + s := 1 + for i := len(shape) - 1; i >= 0; i-- { + strides[i] = s + s *= shape[i] + } + out := make([]T, len(data)) + idx := make([]int, len(order)) + for lin := range out { + pos := 0 + for k, a := range order { + pos += idx[k] * strides[a] + } + out[lin] = data[pos] + for k := len(order) - 1; k >= 0; k-- { + idx[k]++ + if idx[k] < shape[order[k]] { + break + } + idx[k] = 0 + } + } + return out, rest +} + +// vecNorm is the vector norm of x. +func vecNorm[T scalar](x []T, ord Ord) (float64, error) { + if ord.kind == ordDefault || (ord.kind == ordNum && ord.p == 2) { + return nrm2(len(x), x, 1), nil + } + if ord.kind != ordNum { + return 0, fmt.Errorf("%w: invalid norm order '%v' for vectors", ErrArgs, ord) + } + p := ord.p + switch { + case math.IsInf(p, 1): + m := 0.0 + for _, v := range x { + m = math.Max(m, abs(v)) + } + return m, nil + case math.IsInf(p, -1): + if len(x) == 0 { + return 0, fmt.Errorf("%w: the -Inf norm of an empty vector", ErrShape) + } + m := math.Inf(1) + for _, v := range x { + m = math.Min(m, abs(v)) + } + return m, nil + case p == 0: + c := 0.0 + for _, v := range x { + if v != 0 { + c++ + } + } + return c, nil + case p == 1: + s := 0.0 + for _, v := range x { + s += abs(v) + } + return s, nil + } + s := 0.0 + for _, v := range x { + s += math.Pow(abs(v), p) + } + return math.Pow(s, 1/p), nil +} + +// matNorm is the matrix norm of the r×c row-major x. +func matNorm[T scalar](r, c int, x []T, ord Ord) (float64, error) { + switch ord.kind { + case ordDefault, ordFro: + return nrm2(len(x), x, 1), nil + case ordNuc: + s, err := svals(r, c, x) + total := 0.0 + for _, v := range s { + total += v + } + return total, err + } + switch p := ord.p; p { + case 2, -2: + s, err := svals(r, c, x) + if err != nil || p == 2 { + return first(s), err + } + if len(s) == 0 { + return 0, fmt.Errorf("%w: the -2 norm of an empty matrix", ErrShape) + } + return s[len(s)-1], nil + case 1, -1: + if p == -1 && c == 0 { + return 0, fmt.Errorf("%w: the -1 norm of a matrix without columns", ErrShape) + } + sums := make([]float64, c) + for i := range x { + sums[i%c] += abs(x[i]) + } + return extreme(sums, p > 0), nil + case math.Inf(1), math.Inf(-1): + if p < 0 && r == 0 { + return 0, fmt.Errorf("%w: the -Inf norm of a matrix without rows", ErrShape) + } + sums := make([]float64, r) + for i := range x { + sums[i/c] += abs(x[i]) + } + return extreme(sums, p > 0), nil + } + return 0, fmt.Errorf("%w: invalid norm order '%v' for matrices", ErrArgs, ord) +} + +// svals is the singular values of the r×c matrix x (left intact). +func svals[T scalar](r, c int, x []T) ([]float64, error) { + s, _, _, ok := svdOne(r, c, append([]T(nil), x...), false, false) + if !ok { + return nil, ErrNoConvergence + } + return s, nil +} + +func first(s []float64) float64 { + if len(s) == 0 { + return 0 + } + return s[0] +} + +// extreme is the largest (or smallest) of s, 0 for an empty s. +func extreme(s []float64, largest bool) float64 { + if len(s) == 0 { + return 0 + } + m := s[0] + for _, v := range s[1:] { + if largest { + m = math.Max(m, v) + } else { + m = math.Min(m, v) + } + } + return m +} diff --git a/linalg/power.go b/linalg/power.go new file mode 100644 index 0000000..94db81e --- /dev/null +++ b/linalg/power.go @@ -0,0 +1,189 @@ +package linalg + +import ( + "fmt" + + "github.com/go-ndarray/ndarray" +) + +// matmul multiplies two arrays of the same shape (..., M, M) matrix by +// matrix, in their common dtype (the ndarray package's MatMul: the packed +// SIMD GEMM for float64). +func matmul(x, y *ndarray.Array) *ndarray.Array { + s := x.Shape() + if len(s) == 2 { + r, _ := x.MatMul(y) + return r + } + m := s[len(s)-1] + cnt := prod(s[:len(s)-2]) + xr, _ := x.Reshape(cnt, m, m) + yr, _ := y.Reshape(cnt, m, m) + parts := make([]*ndarray.Array, cnt) + for i := range parts { + xi, _ := xr.Slice(ndarray.A(i), ndarray.All(), ndarray.All()) + yi, _ := yr.Slice(ndarray.A(i), ndarray.All(), ndarray.All()) + parts[i], _ = xi.MatMul(yi) + } + st, _ := ndarray.Stack(parts, 0) + r, _ := st.Reshape(s...) + return r +} + +// MatrixPower is numpy.linalg.matrix_power: each square matrix of a +// (..., M, M) raised to the integer power n, by repeated squaring in +// numpy's order. n = 0 gives identities of a's dtype; a negative n inverts +// first (so the result is floating point, and a singular matrix gives +// ErrSingular). Integer matrices stay integer for n >= 0. +func MatrixPower(a *ndarray.Array, n int) (*ndarray.Array, error) { + o, err := square(a, "MatrixPower") + if err != nil { + return nil, err + } + if n == 0 { + m := o.m + id := make([]float64, o.count()*m*m) + for b := 0; b < o.count(); b++ { + for i := 0; i < m; i++ { + id[b*m*m+i*m+i] = 1 + } + } + r, _ := ndarray.FromSlice(id, a.Shape()...) + return r.AsType(a.DType()), nil + } + if n < 0 { + if a, err = Inv(a); err != nil { + return nil, err + } + n = -n + } + if o.count() == 0 { + return a.Copy(), nil + } + switch n { + case 1: + return a.Copy(), nil + case 2: + return matmul(a, a), nil + case 3: + return matmul(matmul(a, a), a), nil + } + var z, result *ndarray.Array + for n > 0 { + if z == nil { + z = a + } else { + z = matmul(z, z) + } + bit := n % 2 + n /= 2 + if bit == 1 { + if result == nil { + result = z + } else { + result = matmul(result, z) + } + } + } + return result, nil +} + +// MultiDot is numpy.linalg.multi_dot: the product of two or more arrays, +// evaluated in the order of fewest scalar multiplications (the classic +// matrix-chain dynamic programme, as numpy does). The first and last +// arrays may be vectors — a row and a column vector — the others must be +// 2-D. Two vectors give a 0-d result, one gives a vector. +func MultiDot(arrays ...*ndarray.Array) (*ndarray.Array, error) { + n := len(arrays) + if n < 2 { + return nil, fmt.Errorf("%w: MultiDot needs at least two arrays, got %d", ErrArgs, n) + } + if n == 2 { + r, err := arrays[0].Dot(arrays[1]) + if err != nil { + return nil, fmt.Errorf("%w: %v", ErrShape, err) + } + return r, nil + } + arr := append([]*ndarray.Array(nil), arrays...) + firstVec, lastVec := arr[0].Ndim() == 1, arr[n-1].Ndim() == 1 + if firstVec { + arr[0], _ = arr[0].Reshape(1, -1) + } + if lastVec { + arr[n-1], _ = arr[n-1].Reshape(-1, 1) + } + for i, a := range arr { + if a.Ndim() != 2 { + return nil, fmt.Errorf("%w: MultiDot array %d is %d-D, not 2-D", ErrShape, i, a.Ndim()) + } + } + for i := 0; i+1 < n; i++ { + if arr[i].Shape()[1] != arr[i+1].Shape()[0] { + return nil, fmt.Errorf("%w: MultiDot arrays %d and %d: %v times %v", ErrShape, i, i+1, arr[i].Shape(), arr[i+1].Shape()) + } + } + var res *ndarray.Array + if n == 3 { + a0, a1b0 := arr[0].Shape()[0], arr[0].Shape()[1] + b1c0, c1 := arr[2].Shape()[0], arr[2].Shape()[1] + if a0*b1c0*(a1b0+c1) < a1b0*c1*(a0+b1c0) { + res = dot(dot(arr[0], arr[1]), arr[2]) + } else { + res = dot(arr[0], dot(arr[1], arr[2])) + } + } else { + order := chainOrder(arr) + res = chain(arr, order, 0, n-1) + } + switch { + case firstVec && lastVec: + r, _ := res.Reshape() + return r, nil + case firstVec || lastVec: + return res.Ravel(), nil + } + return res, nil +} + +func dot(a, b *ndarray.Array) *ndarray.Array { + r, _ := a.MatMul(b) + return r +} + +// chainOrder is numpy's _multi_dot_matrix_chain_order: s[i][j] is where the +// product of arrays i..j is best split. +func chainOrder(arr []*ndarray.Array) [][]int { + n := len(arr) + p := make([]float64, n+1) + for i, a := range arr { + p[i] = float64(a.Shape()[0]) + } + p[n] = float64(arr[n-1].Shape()[1]) + m := make([][]float64, n) + s := make([][]int, n) + for i := range m { + m[i] = make([]float64, n) + s[i] = make([]int, n) + } + for l := 1; l < n; l++ { + for i := 0; i < n-l; i++ { + j := i + l + m[i][j] = -1 + for k := i; k < j; k++ { + q := m[i][k] + m[k+1][j] + p[i]*p[k+1]*p[j+1] + if m[i][j] < 0 || q < m[i][j] { + m[i][j], s[i][j] = q, k + } + } + } + } + return s +} + +func chain(arr []*ndarray.Array, s [][]int, i, j int) *ndarray.Array { + if i == j { + return arr[i] + } + return dot(chain(arr, s, i, s[i][j]), chain(arr, s, s[i][j]+1, j)) +} diff --git a/linalg/properties_test.go b/linalg/properties_test.go new file mode 100644 index 0000000..d9241b7 --- /dev/null +++ b/linalg/properties_test.go @@ -0,0 +1,533 @@ +package linalg + +import ( + "errors" + "fmt" + "math" + "math/cmplx" + "math/rand/v2" + "runtime" + "testing" + + "github.com/go-ndarray/ndarray" +) + +// These tests check the identities that define each result on inputs the +// fixture keeps small to stay small: orders above the blocking and +// divide-and-conquer thresholds, stacks large enough to run in parallel, +// and matrices built to reach the rarely taken branches (deflation, +// rescaling, exceptional shifts, failure to converge). + +func randC(r *rand.Rand, cplx bool, shape ...int) *ndarray.Array { + n := 1 + for _, s := range shape { + n *= s + } + d := make([]complex128, n) + for i := range d { + d[i] = complex(r.NormFloat64(), 0) + if cplx { + d[i] += complex(0, r.NormFloat64()) + } + } + a, _ := ndarray.FromSlice(d, shape...) + if !cplx { + return a.AsType(ndarray.Float64) + } + return a +} + +func toArray(c cmat, cplx bool) *ndarray.Array { + a, _ := ndarray.FromSlice(c.d, c.m, c.n) + if !cplx { + return a.AsType(ndarray.Float64) + } + return a +} + +func eye(n int) cmat { + c := cmat{n, n, make([]complex128, n*n)} + for i := 0; i < n; i++ { + c.d[i*n+i] = 1 + } + return c +} + +// within fails the test if got exceeds tol. +func within(t *testing.T, what string, got, tol float64) { + t.Helper() + if !(got <= tol) { + t.Errorf("%s = %.3g, above %.3g", what, got, tol) + } +} + +func TestLargeIdentities(t *testing.T) { + r := rand.New(rand.NewPCG(11, 12)) + for _, cplx := range []bool{false, true} { + for _, n := range []int{100, 131} { + a := randC(r, cplx, n, n) + x := stack(a)[0] + tol := idTol(n, "float64") + // Inverse, solve, determinant. + inv, err := Inv(a) + if err != nil { + t.Fatal(err) + } + within(t, "‖A·A⁻¹ - I‖", diffFro(mul(x, stack(inv)[0]), eye(n)), tol*1e3) + // Cholesky of A·Aᴴ + nI. + s := mul(x, adj(x)) + for i := 0; i < n; i++ { + s.d[i*n+i] += complex(float64(n), 0) + } + l, err := Cholesky(toArray(s, cplx), Lower) + if err != nil { + t.Fatal(err) + } + lc := stack(l)[0] + within(t, "‖LLᴴ - S‖/‖S‖", diffFro(mul(lc, adj(lc)), s)/fro(s), tol) + // QR, tall and wide. + for _, sh := range [][2]int{{n, n - 30}, {n - 30, n}} { + b := randC(r, cplx, sh[0], sh[1]) + q, rr, err := QR(b, ModeComplete) + if err != nil { + t.Fatal(err) + } + qc, rc, bc := stack(q)[0], stack(rr)[0], stack(b)[0] + within(t, "‖QᴴQ - I‖", orthErr(qc), tol) + within(t, "‖A - QR‖/‖A‖", diffFro(mul(qc, rc), bc)/fro(bc), tol) + u, sv, vh, err := SVD(b, true) + if err != nil { + t.Fatal(err) + } + c := fxCase{Outputs: []fxArray{{DType: "float64"}}, Inputs: []fxArray{{Shape: []int{sh[0], sh[1]}}}} + if msg := checkSVD(c, b, []*ndarray.Array{u, sv, vh}); msg != "" { + t.Errorf("SVD %v: %s", sh, msg) + } + } + // Hermitian eigenproblem. + h := toArray(hermitianOf(x, false), cplx) + w, v, err := Eigh(h, Lower) + if err != nil { + t.Fatal(err) + } + c := fxCase{Args: map[string]any{"uplo": "L"}, Outputs: []fxArray{{}, {DType: "float64"}}, Inputs: []fxArray{{Shape: []int{n, n}}}} + if msg := checkEigh(c, h, []*ndarray.Array{w, v}); msg != "" { + t.Errorf("Eigh n=%d: %s", n, msg) + } + wv, _ := Eigvalsh(h, Lower) + if d := maxDiff(vals(wv), vals(w)); d > tol*fro(x) { + t.Errorf("Eigvalsh differs from Eigh by %.3g", d) + } + checkEigResidual(t, a) + } + } +} + +func maxDiff(a, b []complex128) float64 { + m := 0.0 + for i := range a { + m = math.Max(m, cmplx.Abs(a[i]-b[i])) + } + return m +} + +// checkEigResidual checks ‖Av - λv‖ ≤ tol·‖A‖ for every eigenpair, and +// that Eigvals agrees with Eig. +func checkEigResidual(t *testing.T, a *ndarray.Array) { + t.Helper() + w, v, err := Eig(a) + if err != nil { + t.Fatal(err) + } + x := stack(a)[0] + n := x.n + vc := stack(v)[0] + av := mul(x, vc) + ws := vals(w) + for j := 0; j < n; j++ { + res, nrm := 0.0, 0.0 + for i := 0; i < n; i++ { + d := av.at(i, j) - vc.at(i, j)*ws[j] + res += real(d)*real(d) + imag(d)*imag(d) + nrm += real(vc.at(i, j))*real(vc.at(i, j)) + imag(vc.at(i, j))*imag(vc.at(i, j)) + } + within(t, "eigenvector norm - 1", math.Abs(math.Sqrt(nrm)-1), idTol(n, "float64")) + within(t, "‖Av - λv‖/‖A‖", math.Sqrt(res)/math.Max(fro(x), 1e-300), idTol(n, "float64")) + } + ev, err := Eigvals(a) + if err != nil { + t.Fatal(err) + } + if d := maxDiff(vals(ev), ws); d > idTol(n, "float64")*fro(x) { + t.Errorf("Eigvals differs from Eig by %.3g", d) + } +} + +// TestStackInParallel checks that a stack large enough to be spread over +// the processors gives each matrix's own result. +func TestStackInParallel(t *testing.T) { + defer runtime.GOMAXPROCS(runtime.GOMAXPROCS(4)) // also on a one-CPU runner + r := rand.New(rand.NewPCG(13, 14)) + a := randC(r, false, 16, 40, 40) + inv, err := Inv(a) + if err != nil { + t.Fatal(err) + } + got := vals(inv) + for i, x := range stack(a) { + one, _ := Inv(toArray(x, false)) + if d := maxDiff(got[i*1600:(i+1)*1600], vals(one)); d != 0 { + t.Errorf("matrix %d differs by %g", i, d) + } + } + // The error of the first failing matrix is reported. + z := make([]float64, 16*40*40) + zs, _ := ndarray.FromSlice(z, 16, 40, 40) + if _, err := Inv(zs); !errors.Is(err, ErrSingular) { + t.Errorf("Inv of zeros: %v", err) + } +} + +// TestNoConvergence lowers the iteration limits until every iteration fails +// and checks each function reports it. +func TestNoConvergence(t *testing.T) { + defer func(f int) { iterFactor = f }(iterFactor) + iterFactor = -1 + r := rand.New(rand.NewPCG(15, 16)) + for _, n := range []int{6, 40} { + for _, cplx := range []bool{false, true} { + a := randC(r, cplx, n, n) + x := stack(a)[0] + h := toArray(hermitianOf(x, false), cplx) + b := randC(r, cplx, n) + calls := map[string]func() error{ + "Eigh": func() error { _, _, err := Eigh(h, Lower); return err }, + "Eigvalsh": func() error { _, err := Eigvalsh(h, Lower); return err }, + "SVD": func() error { _, _, _, err := SVD(a, false); return err }, + "SVDVals": func() error { _, err := SVDVals(a); return err }, + "Eig": func() error { _, _, err := Eig(a); return err }, + "Eigvals": func() error { _, err := Eigvals(a); return err }, + "Lstsq": func() error { _, err := Lstsq(a, b); return err }, + "Pinv": func() error { _, err := Pinv(a); return err }, + "MatrixRank": func() error { _, err := MatrixRank(a); return err }, + "Cond": func() error { _, err := Cond(a, Ord{}); return err }, + "Norm2": func() error { _, err := Norm(a, P(2), nil, false); return err }, + } + for name, f := range calls { + if err := f(); !errors.Is(err, ErrNoConvergence) { + t.Errorf("%s n=%d complex=%v: %v, want ErrNoConvergence", name, n, cplx, err) + } + } + } + } +} + +// TestIterationLimit lets the tridiagonal QL/QR iteration start but stop +// short (one sweep per eigenvalue), in both directions. +func TestIterationLimit(t *testing.T) { + defer func(f int) { iterFactor = f }(iterFactor) + iterFactor = 1 + r := rand.New(rand.NewPCG(25, 26)) + for i := 0; i < 8; i++ { + x := stack(randC(r, false, 7, 7))[0] + h := hermitianOf(x, false) + if i%2 == 1 { + for j := range h.d { + h.d[j] = -h.d[j] * complex(float64(1+j%7), 0) + } + h = hermitianOf(h, false) + } + if _, err := Eigvalsh(toArray(h, false), Lower); !errors.Is(err, ErrNoConvergence) { + t.Errorf("matrix %d: %v", i, err) + } + } +} + +// orthogonal is a seeded random n×n orthogonal matrix (the Q of a QR). +func orthogonal(r *rand.Rand, n int) cmat { + q, _, _ := QR(randC(r, false, n, n), ModeReduced) + return stack(q)[0] +} + +func diagMat(d ...float64) cmat { + n := len(d) + c := cmat{n, n, make([]complex128, n*n)} + for i, v := range d { + c.d[i*n+i] = complex(v, 0) + } + return c +} + +// TestDegenerateSpectra reaches the deflations of the divide-and-conquer +// solvers: repeated eigenvalues and singular values (close poles), and a +// zero matrix (zero updating vector). +func TestDegenerateSpectra(t *testing.T) { + r := rand.New(rand.NewPCG(17, 18)) + n := 40 + d := make([]float64, n) + for i := range d { + d[i] = float64(1 + i%3) + } + q := orthogonal(r, n) + a := mul(mul(q, diagMat(d...)), adj(q)) + h := toArray(a, false) + w, v, err := Eigh(h, Lower) + if err != nil { + t.Fatal(err) + } + c := fxCase{Args: map[string]any{"uplo": "L"}, Outputs: []fxArray{{}, {DType: "float64"}}, Inputs: []fxArray{{Shape: []int{n, n}}}} + if msg := checkEigh(c, h, []*ndarray.Array{w, v}); msg != "" { + t.Errorf("Eigh repeated: %s", msg) + } + // Singular values 1, 2, 3 repeated: U·diag·Vᵀ. + p := orthogonal(r, n) + b := toArray(mul(mul(q, diagMat(d...)), p), false) + cs := fxCase{Outputs: []fxArray{{DType: "float64"}}, Inputs: []fxArray{{Shape: []int{n, n}}}} + rd := toArray(mul(stack(randC(r, false, n, 7))[0], stack(randC(r, false, 7, n))[0]), false) + for _, m := range []*ndarray.Array{b, h, rd, zerosArr(n, n), zerosArr(n+3, n)} { + u, s, vh, err := SVD(m, false) + if err != nil { + t.Fatal(err) + } + cs.Inputs[0].Shape = m.Shape() + if msg := checkSVD(cs, m, []*ndarray.Array{u, s, vh}); msg != "" { + t.Errorf("SVD degenerate %v: %s", m.Shape(), msg) + } + } + // A zero matrix: every eigenvalue 0, vectors the identity's. + w, _, err = Eigh(zerosArr(n, n), Lower) + if err != nil || maxAbs(vals(w)) != 0 { + t.Errorf("Eigh of zeros: %v %v", vals(w), err) + } + // Tiny off-diagonal entries split the tridiagonal matrix. + tri := diagMat(1, 2, 3, 4, 5) + for i := 0; i < 4; i++ { + tri.d[i*5+i+1], tri.d[(i+1)*5+i] = 1e-20, 1e-20 + } + w, _, err = Eigh(toArray(tri, false), Lower) + if err != nil || maxDiff(vals(w), []complex128{1, 2, 3, 4, 5}) > 1e-15 { + t.Errorf("Eigh split: %v %v", vals(w), err) + } + // numpy returns NaN eigenvalues for a NaN input; the QL iteration here + // reports it instead. + nan := mul(mul(q, diagMat(d...)), adj(q)) + nan.d[0] = complex(math.NaN(), 0) + if _, _, err := Eigh(toArray(nan, false), Lower); !errors.Is(err, ErrNoConvergence) { + t.Errorf("Eigh of NaN: %v", err) + } +} + +func zerosArr(shape ...int) *ndarray.Array { + a, _ := ndarray.Zeros(shape...) + return a +} + +// TestCholeskyFailsInBlocks: the leading minor that is not positive +// definite lies in the first, then the second half of the recursion. +func TestCholeskyFailsInBlocks(t *testing.T) { + n := 100 + for _, bad := range []int{10, 80} { + d := make([]float64, n) + for i := range d { + d[i] = 1 + } + d[bad] = -1 + if _, err := Cholesky(toArray(diagMat(d...), false), Lower); !errors.Is(err, ErrNotPosDef) { + t.Errorf("minor %d: %v", bad, err) + } + } +} + +// TestParallelMatvec forces the parallel matrix-vector product of the +// tridiagonalization. +func TestParallelMatvec(t *testing.T) { + defer func(v int) { matvecPar = v }(matvecPar) + r := rand.New(rand.NewPCG(19, 20)) + a := randC(r, false, 120, 120) + h := toArray(hermitianOf(stack(a)[0], false), false) + want, _ := Eigvalsh(h, Lower) + matvecPar = 1 + got, _ := Eigvalsh(h, Lower) + if d := maxDiff(vals(got), vals(want)); d > 1e-12 { + t.Errorf("parallel matvec changes the eigenvalues by %g", d) + } +} + +// TestEigBranches runs Eig on matrices that reach its rarer branches and +// checks the eigenpairs. +func TestEigBranches(t *testing.T) { + mk := func(n int, f func(i, j int) complex128) cmat { + c := cmat{n, n, make([]complex128, n*n)} + for i := 0; i < n; i++ { + for j := 0; j < n; j++ { + c.d[i*n+j] = f(i, j) + } + } + return c + } + cases := map[string]cmat{ + // Column 0 isolates an eigenvalue: balancing permutes it left. + "isolated-column": mk(4, func(i, j int) complex128 { + if j == 0 && i > 0 { + return 0 + } + return complex(float64(1+i+2*j), 0) + }), + // Cyclic permutations defeat the standard shifts. + "cyclic": mk(12, func(i, j int) complex128 { + if j == (i+1)%12 { + return 1 + } + return 0 + }), + // A large Jordan block: the back-substitution overflows unless rescaled. + "jordan": mk(60, func(i, j int) complex128 { + switch j - i { + case 0: + return 2 + case 1: + return 1 + } + return 0 + }), + // Two equal rotation blocks coupled: a defective complex pair. + "defective-pair": mk(4, func(i, j int) complex128 { + rot := [2][2]float64{{0, -1}, {1, 0}} + switch { + case i/2 == j/2: + return complex(rot[i%2][j%2], 0) + case j/2 == i/2+1 && i%2 == j%2: + return 1 + } + return 0 + }), + // A zero diagonal. + "zero-diagonal": mk(5, func(i, j int) complex128 { + if i-j == 1 || j-i == 1 { + return 1 + } + return 0 + }), + // Badly scaled: balancing scales by large powers of two. + "scaled": mk(4, func(i, j int) complex128 { + return complex(math.Pow(1e30, float64(i-j))*float64(1+(i*3+j)%5), 0) + }), + } + r := rand.New(rand.NewPCG(21, 22)) + for _, n := range []int{8, 16} { + cases[fmt.Sprintf("orthogonal-%d", n)] = orthogonal(r, n) // eigenvalues on the unit circle + cases[fmt.Sprintf("grcar-%d", n)] = mk(n, func(i, j int) complex128 { + if j == i-1 { + return -1 + } + if j >= i && j <= i+3 { + return 1 + } + return 0 + }) + cases[fmt.Sprintf("jordan-1e10-%d", n)] = mk(n, func(i, j int) complex128 { + switch j - i { + case 0: + return 3 + case 1: + return 1e10 + } + return 0 + }) + // Coupled rotation blocks: a defective complex pair of high order. + cases[fmt.Sprintf("rotations-%d", n)] = mk(2*n, func(i, j int) complex128 { + rot := [2][2]float64{{0, -1}, {1, 0}} + switch { + case i/2 == j/2: + return complex(rot[i%2][j%2], 0) + case j/2 == i/2+1 && i%2 == j%2: + return 1e5 + } + return 0 + }) + } + for _, n := range []int{6, 30} { + cases[fmt.Sprintf("frank-%d", n)] = mk(n, func(i, j int) complex128 { + if j >= i-1 { + return complex(float64(n-max(i, j)), 0) + } + return 0 + }) + cases[fmt.Sprintf("rotations-%d", n)] = mk(2*n, func(i, j int) complex128 { + rot := [2][2]float64{{0, -1}, {1, 0}} + switch { + case i/2 == j/2: + return complex(rot[i%2][j%2], 0) + case j/2 == i/2+1 && i%2 == j%2: + return 1e5 + } + return 0 + }) + cases[fmt.Sprintf("jordan-1e10-%d", n)] = mk(n, func(i, j int) complex128 { + switch j - i { + case 0: + return 3 + case 1: + return 1e10 + } + return 0 + }) + cases[fmt.Sprintf("binary-%d", n)] = mk(n, func(i, j int) complex128 { return complex(float64(r.IntN(2)), 0) }) + } + // Columns 2 and 0 isolate eigenvalues. + cases["isolated-columns"] = mk(5, func(i, j int) complex128 { + if (j == 2 && i != 2) || (j == 0 && i != 0) { + return 0 + } + return complex(float64(1+i+2*j), 0) + }) + // Scaling at the limits of the exponent range. + cases["extreme-scaling"] = cmat{2, 2, []complex128{1, 1e300, 1e-300, 1}} + cases["extreme-scaling-t"] = cmat{2, 2, []complex128{1, 1e-300, 1e300, 1}} + // A chain whose balancing would need scale factors beyond the exponent + // range: balancing stops at its limits. + cases["scaling-chain"] = cmat{3, 3, []complex128{1, 1e300, 0, 1e-300, 1, 1e300, 0, 1e-300, 1}} + cases["scaling-chain-t"] = cmat{3, 3, []complex128{1, 1e-300, 0, 1e300, 1, 1e-300, 0, 1e300, 1}} + // Equal pairs far from the origin: λ - T is below smin. + cases["shifted-pairs"] = mk(8, func(i, j int) complex128 { + rot := [2][2]float64{{1e20, -1}, {1, 1e20}} + switch { + case i/2 == j/2: + return complex(rot[i%2][j%2], 0) + case j/2 == i/2+1 && i%2 == j%2: + return 1 + } + return 0 + }) + for name, c := range cases { + for _, cplx := range []bool{false, true} { + m := c + if cplx { + m = cmat{c.m, c.n, append([]complex128(nil), c.d...)} + m.d[0] += 1i + } + t.Run(name, func(t *testing.T) { checkEigResidual(t, toArray(m, cplx)) }) + } + } +} + +// TestExceptionalShifts makes every iteration use an exceptional shift: +// the QR iterations must then still converge to correct eigenpairs, or +// say they did not. +func TestExceptionalShifts(t *testing.T) { + defer func(k int) { kexsh = k }(kexsh) + kexsh = 1 + r := rand.New(rand.NewPCG(23, 24)) + for _, cplx := range []bool{false, true} { + for _, n := range []int{3, 8} { + a := randC(r, cplx, n, n) + if _, _, err := Eig(a); err != nil { + if !errors.Is(err, ErrNoConvergence) { + t.Errorf("n=%d complex=%v: %v", n, cplx, err) + } + continue + } + checkEigResidual(t, a) + } + } +} diff --git a/linalg/qr.go b/linalg/qr.go new file mode 100644 index 0000000..f79beda --- /dev/null +++ b/linalg/qr.go @@ -0,0 +1,157 @@ +package linalg + +import ( + "fmt" + + "github.com/go-ndarray/ndarray" +) + +// qrBlock is the panel width of the blocked Householder QR: each panel is +// factored with geqr2 and applied to the rest of the matrix as one block +// reflector, through the GEMM. +var qrBlock = 32 + +// geqr2 is the unblocked Householder QR of the m×n matrix a (LAPACK geqr2): +// R overwrites the upper triangle, the reflectors' vectors (with an implicit +// unit first entry) the part below it, and tau their scalars. +func geqr2[T scalar](m, n int, a []T, lda int, tau []T) { + k := min(m, n) + v := make([]T, m) + w := make([]T, n) + for i := 0; i < k; i++ { + var x []T + if i+1 < m { + x = a[(i+1)*lda+i:] + } + beta, t := larfg(m-i, a[i*lda+i], x, lda) + tau[i] = t + if i+1 < n { + v[0] = 1 + for r := i + 1; r < m; r++ { + v[r-i] = a[r*lda+i] + } + larfL(m-i, n-i-1, v[:m-i], conj(t), a[i*lda+i+1:], lda, w) + } + a[i*lda+i] = beta + } +} + +// geqrf is the blocked Householder QR (LAPACK geqrf), with geqr2's output. +func geqrf[T scalar](m, n int, a []T, lda int, tau []T) { + k := min(m, n) + nb := qrBlock + if k <= nb { + geqr2(m, n, a, lda, tau) + return + } + for j := 0; j < k; j += nb { + jb := min(nb, k-j) + geqr2(m-j, jb, a[j*lda+j:], lda, tau[j:j+jb]) + if j+jb < n { + br := newBlockReflector(m-j, jb, a[j*lda+j:], lda, tau[j:j+jb]) + br.applyLeft(n-j-jb, a[j*lda+j+jb:], lda, true) + } + } +} + +// orgqr forms the first q columns of Q = H1·H2···Hk from geqrf's output +// (LAPACK orgqr), as an m×q row-major matrix. The reflectors are applied +// backwards a block at a time. +func orgqr[T scalar](m, q, k int, a []T, lda int, tau []T) []T { + out := make([]T, m*q) + for i := 0; i < min(m, q); i++ { + out[i*q+i] = 1 + } + nb := qrBlock + for j := (k - 1) / nb * nb; k > 0 && j >= 0; j -= nb { + jb := min(nb, k-j) + br := newBlockReflector(m-j, jb, a[j*lda+j:], lda, tau[j:j+jb]) + br.applyLeft(q-j, out[j*q+j:], q, false) + } + return out +} + +// QRMode selects what QR returns, as numpy.linalg.qr's mode does. +type QRMode uint8 + +// The modes. With M×N matrices and K = min(M, N): +const ( + // ModeReduced (numpy 'reduced', the default) returns Q (..., M, K) and + // R (..., K, N). + ModeReduced QRMode = iota + // ModeComplete returns Q (..., M, M) and R (..., M, N). + ModeComplete + // ModeR returns only R (..., K, N); q is nil. + ModeR + // ModeRaw returns LAPACK's compact form, as numpy's 'raw': h + // (..., N, M), the transpose of the factored matrix with R in its upper + // part and the Householder vectors below, and tau (..., K). + ModeRaw +) + +// QR is numpy.linalg.qr: the factorization A = Q·R of each matrix of a +// (..., M, N), Q with orthonormal columns and R upper triangular, by +// Householder reflections as LAPACK's geqrf computes it (R's diagonal may be +// negative, as in numpy). See QRMode for what is returned. +func QR(a *ndarray.Array, mode QRMode) (q, r *ndarray.Array, err error) { + o, err := stacked(a, "QR") + if err != nil { + return nil, nil, err + } + if mode > ModeRaw { + return nil, nil, fmt.Errorf("%w: QR mode %d", ErrArgs, mode) + } + if o.cplx { + q, r = qrT[complex128](o, mode) + } else { + q, r = qrT[float64](o, mode) + } + return q, r, nil +} + +func qrT[T scalar](o operand, mode QRMode) (*ndarray.Array, *ndarray.Array) { + m, n, cnt := o.m, o.n, o.count() + k := min(m, n) + mc := k + if mode == ModeComplete { + mc = m + } + d := load[T](o.a) + taus := make([]T, cnt*k) + var qs, rs []T + if mode != ModeRaw { + rs = make([]T, cnt*mc*n) + } + if mode == ModeReduced || mode == ModeComplete { + qs = make([]T, cnt*m*mc) + } + parallelFor(cnt, 2*m*n*k, func(i int) { + x := d[i*m*n : (i+1)*m*n] + tau := taus[i*k : (i+1)*k] + geqrf(m, n, x, n, tau) + if rs != nil { + r := rs[i*mc*n : (i+1)*mc*n] + for row := 0; row < k; row++ { + copy(r[row*n+row:row*n+n], x[row*n+row:row*n+n]) + } + } + if qs != nil { + copy(qs[i*m*mc:], orgqr(m, mc, k, x, n, tau)) + } + }) + switch mode { + case ModeRaw: + h := make([]T, len(d)) + for i := 0; i < cnt; i++ { + for r := 0; r < m; r++ { + for c := 0; c < n; c++ { + h[i*m*n+c*m+r] = d[i*m*n+r*n+c] + } + } + } + return wrap(h, shapeOf(o.batch, n, m), o.single), wrap(taus, shapeOf(o.batch, k), o.single) + case ModeR: + return nil, wrap(rs, shapeOf(o.batch, mc, n), o.single) + } + return wrap(qs, shapeOf(o.batch, m, mc), o.single), wrap(rs, shapeOf(o.batch, mc, n), o.single) +} diff --git a/linalg/svd.go b/linalg/svd.go new file mode 100644 index 0000000..7bc6365 --- /dev/null +++ b/linalg/svd.go @@ -0,0 +1,754 @@ +package linalg + +import ( + "math" + + "github.com/go-ndarray/ndarray" +) + +// gebrdBlock is the panel width of the blocked bidiagonalization, and +// gebrdMin the order below which it is not blocked. +var gebrdBlock, gebrdMin = 32, 96 + +// bidiag is a bidiagonal reduction B = Qᴴ·A·P: B's diagonal d and +// superdiagonal e, the left reflectors (stored in the reduced matrix below +// its diagonal, with scalars tauq) and the right ones (in pv, geqrf's +// layout on an (n-1)×(n-1) block, with scalars taup), Q = H0·H1···H(n-1), +// P = G0·G1···G(n-2). +type bidiag[T scalar] struct { + d, e []float64 + tauq, taup []T + pv []T +} + +// gebrd reduces the m×n matrix a (m >= n) to real upper bidiagonal form by +// Householder reflections from the left and the right (LAPACK gebrd): +// panels of gebrdBlock rows and columns as labrd, the trailing matrix +// updated once per panel through the GEMM, the rest as gebd2. The right +// reflectors are kept only when wantP. +func gebrd[T scalar](m, n int, a []T, wantP bool) bidiag[T] { + b := bidiag[T]{ + d: make([]float64, n), e: make([]float64, max(n-1, 0)), + tauq: make([]T, n), taup: make([]T, max(n-1, 0)), + } + if wantP && n > 1 { + b.pv = make([]T, (n-1)*(n-1)) + } + i0 := 0 + if n >= gebrdMin { + nb := gebrdBlock + for ; i0+nb < n-nb; i0 += nb { + labrd(m, n, i0, nb, a, &b) + } + } + gebd2(m, n, i0, a, &b) + return b +} + +// gebd2 is the unblocked bidiagonalization (LAPACK gebd2) of rows and +// columns i0 onwards. +func gebd2[T scalar](m, n, i0 int, a []T, b *bidiag[T]) { + v := make([]T, m) + w := make([]T, n) + for i := i0; i < n; i++ { + var x []T + if i+1 < m { + x = a[(i+1)*n+i:] + } + beta, tq := larfg(m-i, a[i*n+i], x, n) + b.d[i], b.tauq[i] = re(beta), tq + if i+1 < n { + v[0] = 1 + for r := i + 1; r < m; r++ { + v[r-i] = a[r*n+i] + } + larfL(m-i, n-i-1, v[:m-i], conj(tq), a[i*n+i+1:], n, w) + } + a[i*n+i] = beta + if i+1 >= n { + continue + } + row := a[i*n+i+1 : i*n+n] + for c := range row { + row[c] = conj(row[c]) + } + var y []T + if i+2 < n { + y = row[1:] + } + beta, tp := larfg(n-i-1, row[0], y, 1) + b.e[i], b.taup[i] = re(beta), tp + row[0] = 1 + larfR(m-i-1, n-i-1, row, tp, a[(i+1)*n+i+1:], n) + b.storeP(n, i, row) + for c := range row { + row[c] = conj(row[c]) + } + row[0] = beta + } +} + +// storeP records the right reflector u (u[0] = 1, at column i+1) of step i. +func (b *bidiag[T]) storeP(n, i int, u []T) { + if b.pv != nil { + for c := 1; c < len(u); c++ { + b.pv[(i+c)*(n-1)+i] = u[c] + } + } +} + +// labrd reduces the nb rows and columns i0..i0+nb-1 of the m×n a (LAPACK +// labrd), accumulating X and Y so that the trailing matrix becomes +// A - V·Yᴴ - X·Uᴴ (V and U the left and right reflectors' vectors), which +// is then applied by two GEMMs. All four are kept transposed — one vector +// per row — so the corrections are dot products and axpys along contiguous +// rows; A's own products are matrix-vector products along rows, of A for X +// and of a transposed copy of the trailing block for Y. +func labrd[T scalar](m, n, i0, nb int, a []T, b *bidiag[T]) { + mr, nc := m-i0, n-i0 + vt, xt := make([]T, nb*mr), make([]T, nb*mr) + yt, ut := make([]T, nb*nc), make([]T, nb*nc) + at := make([]T, nc*mr) // the trailing block, transposed and conjugated + for r := 0; r < mr; r++ { + row := a[(i0+r)*n+i0 : (i0+r)*n+n] + for c, x := range row { + at[c*mr+r] = conj(x) + } + } + acc := make([]T, max(mr, nc)) + t1, t2 := make([]T, nb), make([]T, nb) + for j := 0; j < nb; j++ { + i := i0 + j + // Column i of the current matrix, rows i onwards. + col := acc[:mr-j] + clear(col) + for p := 0; p < j; p++ { + axpy(col, vt[p*mr+j:(p+1)*mr], conj(yt[p*nc+j])) + axpy(col, xt[p*mr+j:(p+1)*mr], conj(ut[p*nc+j])) + } + for r, x := range col { + a[(i+r)*n+i] -= x + } + var x []T + if i+1 < m { + x = a[(i+1)*n+i:] + } + beta, tq := larfg(mr-j, a[i*n+i], x, n) + b.d[i], b.tauq[i] = re(beta), tq + a[i*n+i] = beta + v := vt[j*mr : (j+1)*mr] + v[j] = 1 + for r := j + 1; r < mr; r++ { + v[r] = a[(i0+r)*n+i] + } + // y = tauq·(Aᴴv - Y·(Vᴴv) - U·(Xᴴv)), entries j+1 onwards. + y := yt[j*nc+j+1 : (j+1)*nc] + vj := v[j:] + matvec(nc-j-1, mr-j, at[(j+1)*mr+j:], mr, vj, y) + for p := 0; p < j; p++ { + t1[p] = dotc(vt[p*mr+j:(p+1)*mr], vj) + t2[p] = dotc(xt[p*mr+j:(p+1)*mr], vj) + } + for p := 0; p < j; p++ { + axpy(y, yt[p*nc+j+1:(p+1)*nc], -t1[p]) + axpy(y, ut[p*nc+j+1:(p+1)*nc], -t2[p]) + } + for c := range y { + y[c] *= tq + } + // Row i of the matrix after the left reflector, columns i+1 onwards. + row := a[i*n+i+1 : i*n+n] + rc := acc[:nc-j-1] + clear(rc) + for p := 0; p <= j; p++ { + axpy(rc, yt[p*nc+j+1:(p+1)*nc], conj(vt[p*mr+j])) + } + for p := 0; p < j; p++ { + axpy(rc, ut[p*nc+j+1:(p+1)*nc], conj(xt[p*mr+j])) + } + for c := range row { + row[c] = conj(row[c] - conj(rc[c])) + } + var yy []T + if i+2 < n { + yy = row[1:] + } + beta, tp := larfg(nc-j-1, row[0], yy, 1) + b.e[i], b.taup[i] = re(beta), tp + row[0] = 1 + u := ut[j*nc+j+1 : (j+1)*nc] + copy(u, row) + b.storeP(n, i, row) + for c := range row { + row[c] = conj(row[c]) + } + row[0] = beta + // x = taup·(A·u - V·(Yᴴu) - X·(Uᴴu)), entries j+1 onwards. + xr := xt[j*mr+j+1 : (j+1)*mr] + matvec(mr-j-1, nc-j-1, a[(i+1)*n+i+1:], n, u, xr) + for p := 0; p <= j; p++ { + t1[p] = dotc(yt[p*nc+j+1:(p+1)*nc], u) + } + for p := 0; p < j; p++ { + t2[p] = dotc(ut[p*nc+j+1:(p+1)*nc], u) + } + for p := 0; p <= j; p++ { + axpy(xr, vt[p*mr+j+1:(p+1)*mr], -t1[p]) + } + for p := 0; p < j; p++ { + axpy(xr, xt[p*mr+j+1:(p+1)*mr], -t2[p]) + } + for r := range xr { + xr[r] *= tp + } + } + // The trailing update. + tm, tn := mr-nb, nc-nb + c0 := (i0+nb)*n + i0 + nb + vv, xx := rm(vt, nb, mr), rm(xt, nb, mr) // nb×tm: Vᵀ and Xᵀ + yy, uu := rm(yt, nb, nc), rm(ut, nb, nc) // nb×tn: Yᵀ and Uᵀ + gemm(tm, tn, nb, -1, vv.t(), yy.c(), 1, a[c0:], n) + gemm(tm, tn, nb, -1, xx.t(), uu.c(), 1, a[c0:], n) +} + +// las2 returns the singular values of the 2×2 upper triangular [f g; 0 h] +// (LAPACK dlas2). +func las2(f, g, h float64) (ssmin, ssmax float64) { + fa, ga, ha := math.Abs(f), math.Abs(g), math.Abs(h) + fhmn, fhmx := math.Min(fa, ha), math.Max(fa, ha) + switch { + case fhmn == 0: + if fhmx == 0 { + return 0, ga + } + mx, mn := math.Max(fhmx, ga), math.Min(fhmx, ga) + return 0, mx * math.Sqrt(1+(mn/mx)*(mn/mx)) + case ga < fhmx: + as := 1 + fhmn/fhmx + at := (fhmx - fhmn) / fhmx + au := (ga / fhmx) * (ga / fhmx) + c := 2 / (math.Sqrt(as*as+au) + math.Sqrt(at*at+au)) + return fhmn * c, fhmx / c + } + au := fhmx / ga + if au == 0 { + // Avoid underflow in the computation of ssmin. + return (fhmn * fhmx) / ga, ga + } + as := 1 + fhmn/fhmx + at := (fhmx - fhmn) / fhmx + c := 1 / (math.Sqrt(1+(as*au)*(as*au)) + math.Sqrt(1+(at*au)*(at*au))) + return 2 * (fhmn * c) * au, ga / (c + c) +} + +// lasv2 is the SVD of the 2×2 upper triangular [f g; 0 h] (LAPACK dlasv2): +// [csl snl; -snl csl]·[f g; 0 h]·[csr -snr; snr csr] = [ssmax 0; 0 ssmin], +// with |ssmax| >= |ssmin|. +func lasv2(f, g, h float64) (ssmin, ssmax, snr, csr, snl, csl float64) { + ft, fa, ht, ha := f, math.Abs(f), h, math.Abs(h) + pmax := 1 + swap := ha > fa + if swap { + pmax = 3 + ft, ht = ht, ft + fa, ha = ha, fa + } + gt, ga := g, math.Abs(g) + var clt, crt, slt, srt float64 + if ga == 0 { + ssmin, ssmax = ha, fa + clt, crt, slt, srt = 1, 1, 0, 0 + } else { + gasmal := true + if ga > fa { + pmax = 2 + if fa/ga < eps { + gasmal = false + ssmax = ga + if ha > 1 { + ssmin = fa / (ga / ha) + } else { + ssmin = (fa / ga) * ha + } + clt, slt = 1, ht/gt + srt, crt = 1, ft/gt + } + } + if gasmal { + dd := fa - ha + l := 1.0 + if dd != fa { + l = dd / fa + } + m := gt / ft + t := 2 - l + mm, tt := m*m, t*t + s := math.Sqrt(tt + mm) + r := math.Abs(m) + if l != 0 { + r = math.Sqrt(l*l + mm) + } + a := 0.5 * (s + r) + ssmin, ssmax = ha/a, fa*a + switch { + case mm == 0 && l == 0: + t = math.Copysign(2, ft) * math.Copysign(1, gt) + case mm == 0: + t = gt/math.Copysign(dd, ft) + m/t + default: + t = (m/(s+t) + m/(r+l)) * (1 + a) + } + l = math.Sqrt(t*t + 4) + crt, srt = 2/l, t/l + clt = (crt + srt*m) / a + slt = (ht / ft) * srt / a + } + } + if swap { + csl, snl, csr, snr = srt, crt, slt, clt + } else { + csl, snl, csr, snr = clt, slt, crt, srt + } + var tsign float64 + switch pmax { + case 1: + tsign = math.Copysign(1, csr) * math.Copysign(1, csl) * math.Copysign(1, f) + case 2: + tsign = math.Copysign(1, snr) * math.Copysign(1, csl) * math.Copysign(1, g) + default: + tsign = math.Copysign(1, snr) * math.Copysign(1, snl) * math.Copysign(1, h) + } + ssmax = math.Copysign(ssmax, tsign) + ssmin = math.Copysign(ssmin, tsign*math.Copysign(1, f)*math.Copysign(1, h)) + return ssmin, ssmax, snr, csr, snl, csl +} + +// bdsqr computes the SVD of the n×n real upper bidiagonal (d, e) by implicit +// zero-shift and shifted QR (LAPACK dbdsqr, Demmel–Kahan): singular values +// to high relative accuracy, sorted decreasing into d. vt (n rows of length +// ldvt) is premultiplied by the right singular vectors' transpose; ut holds +// the transpose of U (its rows are U's columns, ldu long, at least n of +// them) and is postmultiplied. Either may be nil. It reports false if the +// iteration did not converge. +func bdsqr[T scalar](n int, d, e []float64, vt []T, ldvt int, ut []T, ldu int) bool { + rotV := func(i, j int, c, s float64) { + if vt != nil { + rot(vt[i*ldvt:i*ldvt+ldvt], vt[j*ldvt:j*ldvt+ldvt], c, s) + } + } + rotU := func(i, j int, c, s float64) { + if ut != nil { + rot(ut[i*ldu:i*ldu+ldu], ut[j*ldu:j*ldu+ldu], c, s) + } + } + const maxitr = 6 + const hndrth = 0.01 + if n > 1 { + tolmul := math.Max(10, math.Min(100, math.Pow(eps, -0.125))) + tol := tolmul * eps + // Approximate the largest and the smallest singular values. + smax := 0.0 + for i := 0; i < n; i++ { + smax = math.Max(smax, math.Abs(d[i])) + } + for i := 0; i < n-1; i++ { + smax = math.Max(smax, math.Abs(e[i])) + } + if math.IsNaN(smax) { + return false + } + sminoa := math.Abs(d[0]) + if sminoa != 0 { + mu := sminoa + for i := 1; i < n && sminoa != 0; i++ { + mu = math.Abs(d[i]) * (mu / (mu + math.Abs(e[i-1]))) + sminoa = math.Min(sminoa, mu) + } + } + sminoa /= math.Sqrt(float64(n)) + thresh := math.Max(tol*sminoa, maxitr*(float64(n)*(float64(n)*safmin))) + maxit := maxitr * n * n * iterFactor / 30 + iter := 0 + oldll, oldm := -1, -1 + idir := 0 + m := n - 1 // the last element of the unconverged part + for m > 0 { + if iter > maxit { + return false + } + // Find the diagonal block of the matrix to work on. + smax = math.Abs(d[m]) + ll := -1 + for k := m - 1; k >= 0; k-- { + abss, abse := math.Abs(d[k]), math.Abs(e[k]) + if abse <= thresh { + ll = k + break + } + smax = math.Max(smax, math.Max(abss, abse)) + } + if ll >= 0 { + e[ll] = 0 + if ll == m-1 { + m-- // the bottom singular value converged + continue + } + } + ll++ + // e[ll..m-1] are nonzero, e[ll-1] is zero. + if ll == m-1 { + // A 2×2 block: handle it separately. + sigmn, sigmx, sinr, cosr, sinl, cosl := lasv2(d[m-1], e[m-1], d[m]) + d[m-1], e[m-1], d[m] = sigmx, 0, sigmn + rotV(m-1, m, cosr, sinr) + rotU(m-1, m, cosl, sinl) + m -= 2 + continue + } + // On a new submatrix, choose the direction of the bulge chase + // from the end with the larger diagonal entry. + if ll > oldm || m < oldll { + if math.Abs(d[ll]) >= math.Abs(d[m]) { + idir = 1 // top to bottom + } else { + idir = 2 // bottom to top + } + } + // Convergence tests. + var sminl float64 + converged := false + if idir == 1 { + if math.Abs(e[m-1]) <= tol*math.Abs(d[m]) { + e[m-1] = 0 + continue + } + mu := math.Abs(d[ll]) + sminl = mu + for k := ll; k < m; k++ { + if math.Abs(e[k]) <= tol*mu { + e[k] = 0 + converged = true + break + } + mu = math.Abs(d[k+1]) * (mu / (mu + math.Abs(e[k]))) + sminl = math.Min(sminl, mu) + } + } else { + if math.Abs(e[ll]) <= tol*math.Abs(d[ll]) { + e[ll] = 0 + continue + } + mu := math.Abs(d[m]) + sminl = mu + for k := m - 1; k >= ll; k-- { + if math.Abs(e[k]) <= tol*mu { + e[k] = 0 + converged = true + break + } + mu = math.Abs(d[k]) * (mu / (mu + math.Abs(e[k]))) + sminl = math.Min(sminl, mu) + } + } + if converged { + continue + } + oldll, oldm = ll, m + // The shift: zero if it would ruin relative accuracy. + shift := 0.0 + if float64(n)*tol*(sminl/smax) > math.Max(eps, hndrth*tol) { + var sll float64 + if idir == 1 { + sll = math.Abs(d[ll]) + shift, _ = las2(d[m-1], e[m-1], d[m]) + } else { + sll = math.Abs(d[m]) + shift, _ = las2(d[ll], e[ll], d[ll+1]) + } + shift = negligibleShift(shift, sll) + } + iter += m - ll + switch { + case shift == 0 && idir == 1: + cs, oldcs := 1.0, 1.0 + var sn, oldsn, r float64 + for i := ll; i < m; i++ { + cs, sn, r = lartg(d[i]*cs, e[i]) + if i > ll { + e[i-1] = oldsn * r + } + oldcs, oldsn, d[i] = lartg(oldcs*r, d[i+1]*sn) + rotV(i, i+1, cs, sn) + rotU(i, i+1, oldcs, oldsn) + } + h := d[m] * cs + d[m], e[m-1] = h*oldcs, h*oldsn + if math.Abs(e[m-1]) <= thresh { + e[m-1] = 0 + } + case shift == 0: + cs, oldcs := 1.0, 1.0 + var sn, oldsn, r float64 + for i := m; i > ll; i-- { + cs, sn, r = lartg(d[i]*cs, e[i-1]) + if i < m { + e[i] = oldsn * r + } + oldcs, oldsn, d[i] = lartg(oldcs*r, d[i-1]*sn) + rotV(i-1, i, oldcs, -oldsn) + rotU(i-1, i, cs, -sn) + } + h := d[ll] * cs + d[ll], e[ll] = h*oldcs, h*oldsn + if math.Abs(e[ll]) <= thresh { + e[ll] = 0 + } + case idir == 1: + f := (math.Abs(d[ll]) - shift) * (math.Copysign(1, d[ll]) + shift/d[ll]) + g := e[ll] + for i := ll; i < m; i++ { + cosr, sinr, r := lartg(f, g) + if i > ll { + e[i-1] = r + } + f = cosr*d[i] + sinr*e[i] + e[i] = cosr*e[i] - sinr*d[i] + g = sinr * d[i+1] + d[i+1] *= cosr + cosl, sinl, r := lartg(f, g) + d[i] = r + f = cosl*e[i] + sinl*d[i+1] + d[i+1] = cosl*d[i+1] - sinl*e[i] + if i < m-1 { + g = sinl * e[i+1] + e[i+1] *= cosl + } + rotV(i, i+1, cosr, sinr) + rotU(i, i+1, cosl, sinl) + } + e[m-1] = f + if math.Abs(e[m-1]) <= thresh { + e[m-1] = 0 + } + default: + f := (math.Abs(d[m]) - shift) * (math.Copysign(1, d[m]) + shift/d[m]) + g := e[m-1] + for i := m; i > ll; i-- { + cosr, sinr, r := lartg(f, g) + if i < m { + e[i] = r + } + f = cosr*d[i] + sinr*e[i-1] + e[i-1] = cosr*e[i-1] - sinr*d[i] + g = sinr * d[i-1] + d[i-1] *= cosr + cosl, sinl, r := lartg(f, g) + d[i] = r + f = cosl*e[i-1] + sinl*d[i-1] + d[i-1] = cosl*d[i-1] - sinl*e[i-1] + if i > ll+1 { + g = sinl * e[i-2] + e[i-2] *= cosl + } + rotV(i-1, i, cosl, -sinl) + rotU(i-1, i, cosr, -sinr) + } + e[ll] = f + if math.Abs(e[ll]) <= thresh { + e[ll] = 0 + } + } + } + } + // Make the singular values positive, then sort them decreasing + // (selection sort: one swap per position). + for i := 0; i < n; i++ { + if d[i] < 0 { + d[i] = -d[i] + if vt != nil { + row := vt[i*ldvt : i*ldvt+ldvt] + for c := range row { + row[c] = -row[c] + } + } + } + } + for i := 0; i < n-1; i++ { + k := i + for j := i + 1; j < n; j++ { + if d[j] > d[k] { + k = j + } + } + if k != i { + d[i], d[k] = d[k], d[i] + swapRows(vt, ldvt, i, k) + swapRows(ut, ldu, i, k) + } + } + return true +} + +// negligibleShift is 0 when the shift is negligible against sll, the +// diagonal entry at the end the bulge chase starts from (dbdsqr's test), +// and the shift otherwise. +func negligibleShift(shift, sll float64) float64 { + if sll > 0 && (shift/sll)*(shift/sll) < eps { + return 0 + } + return shift +} + +// swapRows swaps rows i and k of the row-major x (no-op for a nil x). +func swapRows[T scalar](x []T, ld, i, k int) { + if x == nil { + return + } + ri, rk := x[i*ld:i*ld+ld], x[k*ld:k*ld+ld] + for c := range ri { + ri[c], rk[c] = rk[c], ri[c] + } +} + +// svdOne is the SVD of one m×n matrix x (destroyed): the singular values, +// decreasing, and when wantUV, U (m×m if full, else m×k) and Vᴴ (n×n if +// full, else k×n), row-major. +func svdOne[T scalar](m, n int, x []T, wantUV, full bool) (s []float64, u, vh []T, ok bool) { + if m < n { + // Factor Aᴴ = U'·S·V'ᴴ, so A = V'·S·U'ᴴ. + xh := transposed(m, n, x) + for i := range xh { + xh[i] = conj(xh[i]) + } + s, u2, vh2, ok := svdOne(n, m, xh, wantUV, full) + if !wantUV || !ok { + return s, nil, nil, ok + } + ku := len(u2) / n // columns of U' + return s, adjoint(m, m, vh2), adjoint(n, ku, u2), true + } + bd := gebrd(m, n, x, wantUV) + d, e, tauq, pv, taup := bd.d, bd.e, bd.tauq, bd.pv, bd.taup + if !wantUV { + return d, nil, nil, bdsqr[T](n, d, e, nil, 0, nil, 0) + } + ncu := n + if full { + ncu = m + } + q := orgqr(m, ncu, n, x, n, tauq) + ph := make([]T, n*n) // Pᴴ + if n > 0 { + ph[0] = 1 + } + if n > 1 { + p := orgqr(n-1, n-1, n-1, pv, n-1, taup) + for r := 0; r < n-1; r++ { + for c := 0; c < n-1; c++ { + ph[(c+1)*n+r+1] = conj(p[r*(n-1)+c]) + } + } + } + if n > dcLeaf { + // Divide and conquer on the bidiagonal, then U = Q·U_B, Vᴴ = V_Bᵀ·Pᴴ. + ub, vtb, ok := bdsdc(n, d, e) + if !ok { + return nil, nil, nil, false + } + u := mulMixed(m, n, n, rm(q, 0, ncu), rm(ub, 0, n)) + if ncu > n { + full := make([]T, m*ncu) + for r := 0; r < m; r++ { + copy(full[r*ncu:r*ncu+n], u[r*n:(r+1)*n]) + copy(full[r*ncu+n:(r+1)*ncu], q[r*ncu+n:(r+1)*ncu]) + } + u = full + } + return d, u, mulMixedLeft(n, n, n, rm(vtb, 0, n), rm(ph, 0, n)), true + } + ut := transposed(m, ncu, q) + if !bdsqr(n, d, e, ph, n, ut, m) { + return nil, nil, nil, false + } + return d, transposed(ncu, m, ut), ph, true +} + +// adjoint returns the n×m conjugate transpose of the m×n matrix x. +func adjoint[T scalar](m, n int, x []T) []T { + out := transposed(m, n, x) + for i := range out { + out[i] = conj(out[i]) + } + return out +} + +// SVD is numpy.linalg.svd: A = U·diag(S)·Vh for each matrix of a +// (..., M, N), the singular values S (..., K), K = min(M, N), real and +// decreasing. With fullMatrices U is (..., M, M) and Vh (..., N, N), as +// numpy's default; otherwise (..., M, K) and (..., K, N). Singular vectors +// are unique only up to a unit factor per pair (and a unitary mix within +// repeated singular values). It is Golub–Kahan bidiagonalisation followed by +// implicit-shift QR (LAPACK gesvd's algorithm); numpy calls gesdd, whose +// divide and conquer agrees to rounding. +func SVD(a *ndarray.Array, fullMatrices bool) (u, s, vh *ndarray.Array, err error) { + o, err := stacked(a, "SVD") + if err != nil { + return nil, nil, nil, err + } + if o.cplx { + return svdT[complex128](o, true, fullMatrices) + } + return svdT[float64](o, true, fullMatrices) +} + +// SVDVals is numpy.linalg.svdvals (svd with compute_uv=False): the singular +// values of each matrix of a, decreasing, without the vectors. +func SVDVals(a *ndarray.Array) (*ndarray.Array, error) { + o, err := stacked(a, "SVDVals") + if err != nil { + return nil, err + } + var s *ndarray.Array + if o.cplx { + _, s, _, err = svdT[complex128](o, false, false) + } else { + _, s, _, err = svdT[float64](o, false, false) + } + return s, err +} + +func svdT[T scalar](o operand, wantUV, full bool) (*ndarray.Array, *ndarray.Array, *ndarray.Array, error) { + m, n, cnt := o.m, o.n, o.count() + k := min(m, n) + mu, nv := k, k + if full { + mu, nv = m, n + } + data := load[T](o.a) + ss := make([]float64, cnt*k) + var us, vs []T + if wantUV { + us, vs = make([]T, cnt*m*mu), make([]T, cnt*nv*n) + } + var fe firstErr + parallelFor(cnt, 8*m*n*k, func(i int) { + s, u, vh, ok := svdOne(m, n, data[i*m*n:(i+1)*m*n], wantUV, full) + if !ok { + fe.set(i, ErrNoConvergence) + return + } + copy(ss[i*k:], s) + if wantUV { + copy(us[i*m*mu:], u) + copy(vs[i*nv*n:], vh) + } + }) + if fe.err != nil { + return nil, nil, nil, fe.err + } + s := wrap(ss, shapeOf(o.batch, k), o.single) + if !wantUV { + return nil, s, nil, nil + } + return wrap(us, shapeOf(o.batch, m, mu), o.single), s, wrap(vs, shapeOf(o.batch, nv, n), o.single), nil +} diff --git a/linalg/testdata/cases.json b/linalg/testdata/cases.json new file mode 100644 index 0000000..b877d78 --- /dev/null +++ b/linalg/testdata/cases.json @@ -0,0 +1 @@ +{"numpy": "2.5.3", "cases": [{"fn": "solve", 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Array must be two-dimensional"}, {"fn": "multi_dot", "tag": "two-mismatch", "args": {}, "inputs": [{"dtype": "float64", "shape": [3, 4], "data": ["1.8714703004396536", "0.8912439059028768", "1.773222288491882", "-1.0249389277088747", "-2.4533755987750703", "-0.7556526443077655", "-0.036657898427371334", "-0.0719209028725383", "0.2702811358125879", "-1.055928546707357", "-0.02213127117060487", "0.05077353408113492"]}, {"dtype": "float64", "shape": [5, 2], "data": ["1.0069021873645447", "0.5408497796953362", "1.5036550125767312", "0.13450953372898986", "1.4711028282652707", "-1.0220650947700027", "-0.5638707304707449", "-0.6241791951157403", "0.5885579527262468", "-0.07461898855736356"]}], "kappa": 3.395652840508507, "error": "shape", "message": "ValueError: shapes (3,4) and (5,2) not aligned: 4 (dim 1) != 5 (dim 0)"}]} diff --git a/linalg/testdata/cases.py b/linalg/testdata/cases.py new file mode 100644 index 0000000..e2345f5 --- /dev/null +++ b/linalg/testdata/cases.py @@ -0,0 +1,503 @@ +# Generates linalg/testdata/cases.json: +# uv run --with numpy python linalg/testdata/cases.py > linalg/testdata/cases.json +# Inputs come from a seeded generator and are stored with numpy.linalg's +# results; numbers are repr strings, complex numbers [re, im]. "kappa" is the +# 2-norm condition number of the (first) matrix operand, which the Go test +# uses to scale its tolerance on values that depend on it. +import json, math +import numpy as np +import numpy.linalg as la + +rng = np.random.default_rng(20261009) + + +def rand(*shape, dt="float64"): + if np.dtype(dt).kind == "c": + return (rng.standard_normal(shape) + 1j * rng.standard_normal(shape)).astype(dt) + if np.dtype(dt).kind in "iu": + return rng.integers(-5, 6, size=shape).astype(dt) + if np.dtype(dt).kind == "b": + return rng.integers(0, 2, size=shape).astype(bool) + return rng.standard_normal(shape).astype(dt) + + +def spd(n, dt="float64", batch=()): + a = rand(*batch, n, n, dt="complex128" if np.dtype(dt).kind == "c" else "float64") + s = a @ np.conj(np.swapaxes(a, -1, -2)) + n * np.eye(n) + return s.astype(dt) + + +def herm(n, dt="float64", batch=()): + a = rand(*batch, n, n, dt=dt) + return ((a + np.conj(np.swapaxes(a, -1, -2))) / 2).astype(dt) + + +def hilbert(n): + i = np.arange(n) + return 1.0 / (i[:, None] + i[None, :] + 1) + + +def rankdef(m, n, r, dt="float64"): + return (rand(m, r, dt=dt) @ rand(r, n, dt=dt)).astype(dt) + + +def singular(n, dt="float64"): + a = rand(n, n, dt=dt) + a[-1] = a[0] + return a + + +def jordan(n, lam=2.0): + return lam * np.eye(n) + np.eye(n, k=1) + + +def graded(n): + d = 10.0 ** np.arange(-n // 2, n - n // 2) + return d[:, None] * rand(n, n) / d[None, :] + + +def enc(v): + v = np.asarray(v) + if v.dtype.kind == "c": + return [[repr(float(z.real)), repr(float(z.imag))] for z in v.ravel()] + if v.dtype.kind == "b": + return [repr(float(z)) for z in v.ravel()] + return [repr(float(z)) for z in v.ravel()] + + +def arr(v): + v = np.asarray(v) + return {"dtype": str(v.dtype), "shape": list(v.shape), "data": enc(v)} + + +def kappa(a): + a = np.asarray(a) + if a.ndim < 2 or a.size == 0 or a.dtype.kind == "b": + return 1.0 + try: + with np.errstate(all="ignore"): + c = np.max(la.cond(a.astype(np.complex128 if a.dtype.kind == "c" else np.float64))) + except la.LinAlgError: + return 1.0 + return float(c) if np.isfinite(c) else 1e300 + + +def kind(e): + m = str(e) + if isinstance(e, la.LinAlgError): + if "Singular" in m: + return "singular" + if "positive definite" in m: + return "posdef" + if "converge" in m: + return "converge" + if "infs or NaNs" in m: + return "nonfinite" + return "shape" + if "norm order" in m or "Duplicate axes" in m or "Improper number" in m or type(e).__name__ == "AxisError" or "Expecting at least two" in m: + return "args" + return "shape" + + +cases = [] +SHAPE_ONLY = {"qr": [0, 1], "svd": [0, 2], "eigh": [1], "eig": [1]} + + +prng = np.random.default_rng(1) # perturbations, apart from the inputs' stream + + +def matching_move(w, w2): + """Largest distance from an eigenvalue of w to its nearest unused one of w2.""" + w2 = list(w2) + move = 0.0 + for z in w: + j = min(range(len(w2)), key=lambda k: abs(w2[k] - z)) + move = max(move, abs(w2[j] - z)) + w2.pop(j) + return move + + +def eig_tol(a): + """100 times how far the eigenvalues move when each entry is perturbed + by 1e-14 of itself (or of the largest entry, for a zero entry): an + empirical bound on their sensitivity that respects the scaling + balancing exploits.""" + a = np.asarray(a) + if a.ndim < 2 or a.size == 0 or a.shape[-1] != a.shape[-2] or not np.isfinite(a).all(): + return 0.0 + a = a.astype(np.complex128 if a.dtype.kind == "c" else np.float64) + move = 0.0 + for m in a.reshape(-1, a.shape[-2], a.shape[-1]): + size = np.where(m != 0, np.abs(m), np.abs(m).max()) + pert = m + 1e-14 * size * prng.standard_normal(m.shape) + try: + move = max(move, matching_move(la.eigvals(m), la.eigvals(pert))) + except la.LinAlgError: + return float("inf") + return 100 * move + + +def effective_kappa(a, rcond): + """sigma_max / sigma_r over the singular values above the cutoff.""" + a = np.asarray(a) + if a.size == 0 or a.ndim < 2: + return 1.0 + s = la.svd(a.astype(np.complex128 if a.dtype.kind == "c" else np.float64), compute_uv=False) + s = s.reshape(-1, s.shape[-1]) + k = 1.0 + for row in s: + keep = row[row > rcond * row[0]] + if len(keep): + k = max(k, keep[0] / keep[-1]) + return float(k) + + +def add(fn, inputs, args=None, tag="", **check): + args = args or {} + c = {"fn": fn, "tag": tag, "args": args, "inputs": [arr(x) for x in inputs], "kappa": kappa(inputs[0])} + if fn in ("eig", "eigvals"): + c["etol"] = eig_tol(inputs[0]) + if fn == "lstsq": + a = np.asarray(inputs[0]) + if a.ndim == 2: + rc = args.get("rcond") + if rc is None: + rc = np.finfo(np.float64).eps * max(a.shape) + elif rc <= 0 or rc >= 1: + rc = np.finfo(np.float64).eps / 2 + c["kappa"] = effective_kappa(a, rc) + if fn == "pinv": + c["kappa"] = effective_kappa(inputs[0], args.get("rcond", 1e-15)) + c.update(check) + try: + out = call(fn, inputs, args) + c["outputs"] = [arr(o) for o in out] + # Factors that are not unique are checked through invariants; only + # their dtype and shape are kept. + for i in SHAPE_ONLY.get(fn, []) if args.get("mode") not in ("raw", "r") else []: + c["outputs"][i]["data"] = None + except Exception as e: # numpy refused: the Go side must refuse too + c["error"] = kind(e) + c["message"] = f"{type(e).__name__}: {e}" + cases.append(c) + + +def ordarg(o): + if o is None: + return None + if isinstance(o, str): + return o + return float(o) + + +def call(fn, x, a): + if fn == "solve": + return [la.solve(x[0], x[1])] + if fn == "inv": + return [la.inv(x[0])] + if fn == "det": + return [la.det(x[0])] + if fn == "slogdet": + return list(la.slogdet(x[0])) + if fn == "cholesky": + return [la.cholesky(x[0], upper=a.get("upper", False))] + if fn == "qr": + r = la.qr(x[0], mode=a["mode"]) + return [r] if a["mode"] == "r" else list(r) + if fn == "svd": + return list(la.svd(x[0], full_matrices=a["full"])) + if fn == "svdvals": + return [la.svdvals(x[0])] + if fn == "eigh": + return list(la.eigh(x[0], UPLO=a["uplo"])) + if fn == "eigvalsh": + return [la.eigvalsh(x[0], UPLO=a["uplo"])] + if fn == "eig": + return list(la.eig(x[0])) + if fn == "eigvals": + return [la.eigvals(x[0])] + if fn == "lstsq": + rc = a.get("rcond") + xs, res, rank, s = la.lstsq(x[0], x[1], rcond=rc) + return [xs, res, np.int64(rank), s] + if fn == "pinv": + if "rcond" in a: + return [la.pinv(x[0], rcond=a["rcond"])] + return [la.pinv(x[0])] + if fn == "matrix_rank": + return [np.asarray(la.matrix_rank(x[0], tol=a.get("tol")), dtype=np.int64)] + if fn == "norm": + ax = a.get("axis") + if isinstance(ax, list): + ax = tuple(ax) + o = a.get("ord") + if o in ("inf", "-inf"): + o = float(o) + return [np.asarray(la.norm(x[0], ord=o, axis=ax, keepdims=a.get("keepdims", False)))] + if fn == "cond": + o = a.get("p") + if o in ("inf", "-inf"): + o = float(o) + return [np.asarray(la.cond(x[0], o))] + if fn == "matrix_power": + return [la.matrix_power(x[0], a["n"])] + if fn == "multi_dot": + return [np.asarray(la.multi_dot(x))] + raise ValueError(fn) + + +F, F32, C, C64, I = "float64", "float32", "complex128", "complex64", "int32" + +# --- solve +for dt in [F, F32, C, C64, I]: + for n in [1, 3, 8]: + a = rand(n, n, dt=dt) + if dt == I: + a = a + 10 * np.eye(n, dtype=a.dtype) + add("solve", [a, rand(n, dt=dt)], tag=f"vec-{dt}") + add("solve", [a, rand(n, 2, dt=dt)], tag=f"mat-{dt}") +add("solve", [rand(2, 3, 4, 4), rand(3, 4, 2)], tag="broadcast") +add("solve", [rand(3, 4, 4), rand(4)], tag="batched-vec") +add("solve", [rand(2, 1, 4, 4), rand(5, 4, 1)], tag="broadcast-ones") +add("solve", [rand(4, 4), rand(4, 0)], tag="no-rhs") +add("solve", [np.zeros((0, 0)), np.zeros(0)], tag="empty") +add("solve", [hilbert(8), np.ones(8)], tag="hilbert") +add("solve", [rand(4, 4, dt=F), rand(4, 2, dt=C)], tag="mixed") +add("solve", [rand(4, 4, dt=F32), rand(4, dt=I)], tag="f32-int") +add("solve", [singular(5), rand(5)], tag="singular") +add("solve", [np.zeros((3, 3)), rand(3)], tag="zero") +add("solve", [rand(3, 4), rand(3)], tag="nonsquare") +add("solve", [rand(3, 3), rand(4)], tag="badrows") +add("solve", [rand(3, 3), rand(4, 2)], tag="badrows2") +add("solve", [rand(2, 3, 3), rand(3, 3, 1)], tag="nobroadcast") +add("solve", [rand(3), rand(3)], tag="1d") +add("solve", [rand(3, 3), np.float64(1.0)], tag="0d-b") +add("solve", [np.eye(3).copy(), np.array([1.0, np.nan, 2.0])], tag="nan") + +# --- inv, det, slogdet +mats = [(rand(n, n, dt=dt), f"{dt}-{n}") for dt in [F, F32, C, C64] for n in [0, 1, 2, 5, 12]] +mats += [(rand(4, 4, dt=I), "int"), (rand(3, 3, dt="bool") | np.eye(3, dtype=bool), "bool"), + (rand(2, 3, 4, 4), "batched"), (rand(0, 3, 3), "empty-batch"), (hilbert(7), "hilbert"), + (graded(6), "graded"), (rand(40, 40), "n40"), (rand(34, 34, dt=C), "c34")] +for a, tag in mats: + add("inv", [a], tag=tag) + add("det", [a], tag=tag) + add("slogdet", [a], tag=tag) +for a, tag in [(singular(5), "singular"), (np.zeros((3, 3)), "zero"), (np.zeros((2, 2, 2)), "zero-batch"), + (singular(4, dt=C), "singular-c"), (np.array([[1.0, 2.0], [2.0, 4.0]]), "rank1")]: + add("inv", [a], tag=tag) + add("det", [a], tag=tag) + add("slogdet", [a], tag=tag) +scaled = np.diag([1e200, 1e200, 1e-300]) +add("det", [scaled], tag="scaled") +add("slogdet", [scaled], tag="scaled") +add("det", [np.diag([1e200, 1e200, 1e200])], tag="overflow") +add("det", [np.diag([1e-200, 1e-200, 1e-200])], tag="underflow") +add("slogdet", [np.diag([1e-200, 1e-200, 1e-200])], tag="underflow") +for f in ["inv", "det", "slogdet"]: + add(f, [rand(3, 4)], tag="nonsquare") + add(f, [rand(3)], tag="1d") +add("inv", [np.array([[np.nan, 0.0], [0.0, 1.0]])], tag="nan") + +# --- cholesky +for dt in [F, F32, C, C64]: + for n in [0, 1, 2, 5, 12] + ([34] if dt in (F, C) else []): + a = spd(n, dt) + add("cholesky", [a], tag=f"{dt}-{n}") + add("cholesky", [a], {"upper": True}, tag=f"{dt}-{n}-up") +g = spd(6) +gl, gu = g.copy(), g.copy() +gl[np.triu_indices(6, 1)] = 99.0 # garbage numpy must not read +gu[np.tril_indices(6, -1)] = -99.0 +add("cholesky", [gl], tag="lower-only") +add("cholesky", [gu], {"upper": True}, tag="upper-only") +add("cholesky", [spd(4, batch=(2, 3))], tag="batched") +add("cholesky", [spd(3, C, batch=(2,))], {"upper": True}, tag="batched-c-up") +add("cholesky", [np.array([[4, 2], [2, 3]], dtype=np.int64)], tag="int") +add("cholesky", [hilbert(8)], tag="hilbert") +add("cholesky", [np.array([[1.0, 2.0], [2.0, 1.0]])], tag="indefinite") +add("cholesky", [np.array([[0.0, 0.0], [0.0, 1.0]])], tag="semidefinite") +add("cholesky", [np.array([[1.0, 0.0], [0.0, -1.0]])], tag="negative") +add("cholesky", [np.array([[np.nan, 0.0], [0.0, 1.0]])], tag="nan") +add("cholesky", [rand(3, 4)], tag="nonsquare") + +# --- qr +shapes = [(5, 3), (3, 5), (4, 4), (1, 1), (0, 3), (3, 0), (2, 4, 3)] +big = {F: [(36, 20), (20, 36)], C: [(36, 20), (20, 36)]} +for mode in ["reduced", "complete", "r", "raw"]: + for dt in [F, C, F32, C64]: + for s in shapes + big.get(dt, []): + add("qr", [rand(*s, dt=dt)], {"mode": mode}, tag=f"{dt}-{s}") + add("qr", [rankdef(6, 5, 2)], {"mode": mode}, tag="rankdef") + add("qr", [np.zeros((3, 2))], {"mode": mode}, tag="zero") + add("qr", [rand(4, 3, dt=I)], {"mode": mode}, tag="int") + add("qr", [hilbert(8)], {"mode": mode}, tag="hilbert") +add("qr", [rand(3)], {"mode": "reduced"}, tag="1d") + +# --- svd, svdvals +for full in [True, False]: + for dt in [F, C, F32, C64]: + for s in shapes + big.get(dt, []): + add("svd", [rand(*s, dt=dt)], {"full": full}, tag=f"{dt}-{s}") + add("svd", [rankdef(7, 5, 2)], {"full": full}, tag="rankdef") + add("svd", [rankdef(5, 7, 3, dt=C)], {"full": full}, tag="rankdef-c") + add("svd", [np.zeros((4, 3))], {"full": full}, tag="zero") + add("svd", [hilbert(10)], {"full": full}, tag="hilbert") + add("svd", [graded(8)], {"full": full}, tag="graded") + add("svd", [np.diag([3.0, 3.0, 1.0, 1.0])], {"full": full}, tag="repeated") + add("svd", [rand(4, 3, dt=I)], {"full": full}, tag="int") + add("svd", [rand(3)], {"full": full}, tag="1d") +add("svd", [np.array([[np.nan, 0.0], [0.0, 1.0]])], {"full": True}, tag="nan") +for s in shapes + [(9, 9)]: + add("svdvals", [rand(*s)], tag=f"{s}") +add("svdvals", [rand(6, 4, dt=C64)], tag="c64") +add("svdvals", [hilbert(12)], tag="hilbert") + +# --- eigh, eigvalsh +for uplo in ["L", "U"]: + for dt in [F, C, F32, C64]: + for n in [0, 1, 2, 5, 12] + ([34] if dt in (F, C) else []): + a = herm(n, dt) + add("eigh", [a], {"uplo": uplo}, tag=f"{dt}-{n}") + add("eigvalsh", [a], {"uplo": uplo}, tag=f"{dt}-{n}") + g = herm(6, C) + if uplo == "L": + g[np.triu_indices(6, 1)] = 7 + 7j + else: + g[np.tril_indices(6, -1)] = -7 - 7j + add("eigh", [g], {"uplo": uplo}, tag="one-triangle") + add("eigh", [np.eye(5)], {"uplo": uplo}, tag="identity") + add("eigh", [np.diag([2.0, 1.0, 2.0, 1.0, 2.0])], {"uplo": uplo}, tag="repeated") + add("eigh", [herm(4, batch=(2, 3))], {"uplo": uplo}, tag="batched") + add("eigh", [hilbert(10)], {"uplo": uplo}, tag="hilbert") + add("eigh", [np.array([[2, 1], [1, 2]], dtype=np.int64)], {"uplo": uplo}, tag="int") + add("eigh", [np.diag([1e300, 1e-300, 1.0])], {"uplo": uplo}, tag="wide-range") + add("eigh", [1e-310 * herm(4)], {"uplo": uplo}, tag="tiny") + add("eigh", [1e305 * herm(4)], {"uplo": uplo}, tag="huge") + add("eigh", [np.diag(np.ones(7)) + np.diag(1e-9 * np.ones(6), 1) + np.diag(1e-9 * np.ones(6), -1)], {"uplo": uplo}, tag="cluster") +add("eigh", [rand(3, 4)], {"uplo": "L"}, tag="nonsquare") +add("eigvalsh", [rand(3)], {"uplo": "L"}, tag="1d") + +# --- eig, eigvals +emats = [(rand(n, n, dt=dt), f"{dt}-{n}") for dt in [F, C, F32, C64] for n in [0, 1, 2, 5, 12] + ([34] if dt in (F, C) else [])] +emats += [(jordan(4), "jordan"), (np.array([[1.0, 1.0], [0.0, 1.0]]), "defective2"), + (np.roll(np.eye(5), 1, axis=1), "permutation"), (np.triu(rand(6, 6)), "triangular"), + (np.array([[0.0, -1.0], [1.0, 0.0]]), "rotation"), (graded(8), "graded"), + (rand(2, 3, 4, 4), "batched"), (rand(4, 4, dt=I), "int"), (np.zeros((3, 3)), "zero"), + (herm(6), "symmetric"), (np.diag([1.0, 2.0, 3.0]) + np.diag([1e-8, 1e-8], 1), "nearly-diag"), + (np.array([[1.0, 2.0, 0.0], [0.0, 3.0, 0.0], [4.0, 5.0, 6.0]]), "isolated"), + (np.vander(np.arange(1.0, 7.0)), "vander"), (np.ones((5, 5)), "ones"), + (np.array([[1e-300, 1.0], [-1.0, 1e300]]), "extreme")] +for a, tag in emats: + add("eig", [a], tag=tag) + add("eigvals", [a], tag=tag) +for f in ["eig", "eigvals"]: + add(f, [np.array([[np.nan, 0.0], [0.0, 1.0]])], tag="nan") + add(f, [np.array([[np.inf, 0.0], [0.0, 1.0]], dtype=C)], tag="inf") + add(f, [rand(3, 4)], tag="nonsquare") + +# --- lstsq +for dt in [F, C, F32]: + for m, n in [(6, 3), (3, 6), (4, 4), (1, 1)]: + a = rand(m, n, dt=dt) + add("lstsq", [a, rand(m, dt=dt)], tag=f"{dt}-{m}x{n}-vec") + add("lstsq", [a, rand(m, 2, dt=dt)], tag=f"{dt}-{m}x{n}-mat") +for rc in [None, -1.0, 1e-3, 0.5, 2.0]: + add("lstsq", [rankdef(6, 4, 2), rand(6, 2)], {"rcond": rc}, tag=f"rankdef-{rc}") +add("lstsq", [hilbert(8)[:, :6], np.ones(8)], tag="hilbert") +add("lstsq", [rand(5, 3), rand(5, 0)], tag="no-rhs") +add("lstsq", [np.zeros((0, 3)), np.zeros(0)], tag="m0") +add("lstsq", [np.zeros((4, 3)), rand(4)], tag="zero") +add("lstsq", [rand(5, 3, dt=I), rand(5, dt=I)], tag="int") +add("lstsq", [rand(2, 5, 3), rand(2, 5)], tag="3d") +add("lstsq", [rand(5, 3), rand(4)], tag="badrows") +add("lstsq", [rand(5, 3), rand(5, 2, 2)], tag="3d-b") + +# --- pinv +for s in [(5, 3), (3, 5), (4, 4), (2, 3, 4), (1, 1)]: + for dt in [F, C, F32]: + add("pinv", [rand(*s, dt=dt)], tag=f"{dt}-{s}") +add("pinv", [rankdef(6, 5, 2)], tag="rankdef") +add("pinv", [rankdef(6, 5, 2)], {"rcond": 1e-3}, tag="rankdef-rcond") +add("pinv", [hilbert(8)], tag="hilbert") +add("pinv", [np.zeros((3, 2))], tag="zero") +add("pinv", [np.zeros((0, 3), dtype=np.int64)], tag="empty-int") +add("pinv", [rand(4, dt=F)], tag="1d") + +# --- matrix_rank +for a, tag in [(rand(5, 4), "full"), (rankdef(6, 5, 2), "rankdef"), (np.zeros((3, 3)), "zero"), + (hilbert(12), "hilbert"), (rankdef(5, 5, 3, dt=C), "complex"), (rand(2, 3, 4), "batched"), + (rand(5, 5, dt=F32), "f32"), (rankdef(6, 6, 3).astype(F32), "f32-rankdef"), + (np.ones(4), "1d"), (np.zeros(3), "1d-zero"), (np.float64(0.5), "0d"), (np.eye(0), "empty")]: + add("matrix_rank", [a], tag=tag) +add("matrix_rank", [hilbert(12)], {"tol": 1e-8}, tag="tol") + +# --- norm +vec = rand(7) +vecs = [(vec, "f64"), (rand(6, dt=C), "c128"), (rand(5, dt=F32), "f32"), (rand(6, dt=I), "int"), + (np.zeros(0), "empty"), (np.array([0.0, 3.0, -4.0, 0.0]), "zeros")] +vords = [None, "inf", "-inf", 0, 1, -1, 2, -2, 3, 0.5, "fro", "nuc"] +for v, tag in vecs: + for o in vords: + add("norm", [v], {"ord": ordarg(o)}, tag=f"vec-{tag}") +mat = rand(4, 5) +mords = [None, "fro", "nuc", "inf", "-inf", 1, -1, 2, -2, 3, 0] +for m_, tag in [(mat, "f64"), (rand(3, 3, dt=C), "c128"), (rand(4, 2, dt=C64), "c64"), (rand(3, 4, dt=I), "int"), + (np.zeros((0, 3)), "empty-rows"), (np.zeros((3, 0)), "empty-cols")]: + for o in mords: + add("norm", [m_], {"ord": ordarg(o)}, tag=f"mat-{tag}") +t3 = rand(2, 3, 4) +add("norm", [t3], {}, tag="3d-default") +add("norm", [t3], {"ord": 1.0}, tag="3d-ord") +for ax in [0, 1, -1, 2, [0, 2], [2, 0], [-1, -2], [1, 1], [0, 1, 2], 3, -4]: + for o in [None, 1, "inf", 2, "fro", "nuc"]: + add("norm", [t3], {"ord": ordarg(o), "axis": ax}, tag="3d-axis") +for ax in [None, 1, [0, 2]]: + add("norm", [t3], {"axis": ax, "keepdims": True}, tag="keepdims") + add("norm", [t3], {"axis": ax, "keepdims": True, "ord": "inf" if ax == 1 else None}, tag="keepdims-ord") +add("norm", [mat], {"ord": "fro", "keepdims": True}, tag="keepdims-2d") +add("norm", [np.zeros((0, 3, 0))], {"ord": -1.0, "axis": [1, 2]}, tag="empty-batch-bad") +add("norm", [np.zeros((0, 3, 3))], {"ord": -1.0, "axis": [1, 2]}, tag="empty-batch") +add("norm", [np.array([1e200, 1e200])], {}, tag="overflow") +add("norm", [np.float64(-3.0)], {}, tag="0d") + +# --- cond +for p in [None, 2, -2, 1, -1, "inf", "-inf", "fro", "nuc", 3]: + add("cond", [rand(4, 4)], {"p": ordarg(p)}, tag="f64") + add("cond", [rand(3, 3, dt=C)], {"p": ordarg(p)}, tag="c128") + add("cond", [hilbert(6)], {"p": ordarg(p)}, tag="hilbert") + add("cond", [singular(4)], {"p": ordarg(p)}, tag="singular") + add("cond", [rand(2, 3, 3, dt=F32)], {"p": ordarg(p)}, tag="batched-f32") + add("cond", [rand(4, 3)], {"p": ordarg(p)}, tag="nonsquare") +add("cond", [np.zeros((3, 3))], {}, tag="zero") +add("cond", [np.zeros((0, 0))], {}, tag="empty") +add("cond", [np.array([[np.nan, 0.0], [0.0, 1.0]])], {"p": 1.0}, tag="nan") +add("cond", [rand(3)], {}, tag="1d") + +# --- matrix_power +for n in [0, 1, 2, 3, 5, 8, -1, -3]: + add("matrix_power", [rand(4, 4)], {"n": n}, tag="f64") + add("matrix_power", [rand(3, 3, dt=C)], {"n": n}, tag="c128") + add("matrix_power", [rand(2, 3, 3)], {"n": n}, tag="batched") +for n in [0, 1, 2, 3, 6]: + add("matrix_power", [rand(3, 3, dt="int64")], {"n": n}, tag="int") + add("matrix_power", [rand(3, 3, dt=F32)], {"n": n}, tag="f32") +add("matrix_power", [singular(3)], {"n": -1}, tag="singular") +add("matrix_power", [rand(0, 3, 3)], {"n": 4}, tag="empty-batch") +add("matrix_power", [rand(3, 4)], {"n": 2}, tag="nonsquare") + +# --- multi_dot +add("multi_dot", [rand(3, 4), rand(4, 5)], tag="two") +add("multi_dot", [rand(10, 2), rand(2, 30), rand(30, 4)], tag="three") +add("multi_dot", [rand(2, 30), rand(30, 4), rand(4, 25)], tag="three-b") +add("multi_dot", [rand(5, 3), rand(3, 40), rand(40, 2), rand(2, 6)], tag="four") +add("multi_dot", [rand(3), rand(3, 4), rand(4, 5), rand(5)], tag="vec-vec") +add("multi_dot", [rand(3), rand(3, 4), rand(4, 5)], tag="vec-first") +add("multi_dot", [rand(3, 4), rand(4, 5), rand(5, 6), rand(6)], tag="vec-last") +add("multi_dot", [rand(3, 4, dt=C), rand(4, 5), rand(5, 2, dt=I), rand(2, 2)], tag="mixed") +add("multi_dot", [rand(3, 4)], tag="one") +add("multi_dot", [rand(3, 4), rand(5, 6), rand(6, 2)], tag="mismatch") +add("multi_dot", [rand(3, 4), rand(2, 4, 5), rand(5, 2)], tag="3d") +add("multi_dot", [rand(3, 4), rand(5, 2)], tag="two-mismatch") + +print(json.dumps({"numpy": np.__version__, "cases": cases}))