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ranocharanocha
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‎src/alefeld.jl‎

Lines changed: 43 additions & 42 deletions
Original file line numberDiff line numberDiff line change
@@ -9,18 +9,17 @@ algorithm 4.1 because, in certain sense, the second algorithm(4.2) is an optimal
99
struct Alefeld <: AbstractBracketingAlgorithm end
1010

1111
function SciMLBase.solve(prob::IntervalNonlinearProblem,
12-
alg::Alefeld, args...; abstol = nothing,
13-
reltol = nothing,
14-
maxiters = 1000, kwargs...)
15-
12+
alg::Alefeld, args...; abstol = nothing,
13+
reltol = nothing,
14+
maxiters = 1000, kwargs...)
1615
f = Base.Fix2(prob.f, prob.p)
1716
a, b = prob.tspan
1817
c = a - (b - a) / (f(b) - f(a)) * f(a)
19-
18+
2019
fc = f(c)
2120
if iszero(fc)
2221
return SciMLBase.build_solution(prob, alg, c, fc;
23-
retcode = ReturnCode.Success,
22+
retcode = ReturnCode.Success,
2423
left = a,
2524
right = b)
2625
end
@@ -33,47 +32,47 @@ function SciMLBase.solve(prob::IntervalNonlinearProblem,
3332
f₁, f₂, f₃, f₄ = f(a), f(b), f(d), f(e)
3433
if i == 2 || (f₁ == f₂ || f₁ == f₃ || f₁ == f₄ || f₂ == f₃ || f₂ == f₄ || f₃ == f₄)
3534
c = _newton_quadratic(f, a, b, d, 2)
36-
else
35+
else
3736
c = _ipzero(f, a, b, d, e)
3837
if (c - a) * (c - b) ≥ 0
3938
c = _newton_quadratic(f, a, b, d, 2)
4039
end
41-
end
40+
end
4241
ē, fc = d, f(c)
43-
(a == c || b == c) &&
42+
(a == c || b == c) &&
4443
return SciMLBase.build_solution(prob, alg, c, fc;
4544
retcode = ReturnCode.FloatingPointLimit,
46-
left = a,
47-
right = b)
45+
left = a,
46+
right = b)
4847
iszero(fc) &&
4948
return SciMLBase.build_solution(prob, alg, c, fc;
50-
retcode = ReturnCode.Success,
51-
left = a,
52-
right = b)
53-
ā, b̄, d̄ = _bracket(f, a, b, c)
49+
retcode = ReturnCode.Success,
50+
left = a,
51+
right = b)
52+
ā, b̄, d̄ = _bracket(f, a, b, c)
5453

5554
# The second bracketing block
5655
f₁, f₂, f₃, f₄ = f(ā), f(b̄), f(d̄), f(ē)
5756
if f₁ == f₂ || f₁ == f₃ || f₁ == f₄ || f₂ == f₃ || f₂ == f₄ || f₃ == f₄
5857
c = _newton_quadratic(f, ā, b̄, d̄, 3)
59-
else
58+
else
6059
c = _ipzero(f, ā, b̄, d̄, ē)
6160
if (c - ā) * (c - b̄) ≥ 0
6261
c = _newton_quadratic(f, ā, b̄, d̄, 3)
6362
end
6463
end
6564
fc = f(c)
66-
(ā == c || b̄ == c) &&
65+
(ā == c || b̄ == c) &&
6766
return SciMLBase.build_solution(prob, alg, c, fc;
6867
retcode = ReturnCode.FloatingPointLimit,
69-
left = ā,
68+
left = ā,
7069
right = b̄)
7170
iszero(fc) &&
7271
return SciMLBase.build_solution(prob, alg, c, fc;
73-
retcode = ReturnCode.Success,
74-
left = ā,
75-
right = b̄)
76-
ā, b̄, d̄ = _bracket(f, ā, b̄, c)
72+
retcode = ReturnCode.Success,
73+
left = ā,
74+
right = b̄)
75+
ā, b̄, d̄ = _bracket(f, ā, b̄, c)
7776

7877
# The third bracketing block
7978
if abs(f(ā)) < abs(f(b̄))
@@ -86,17 +85,17 @@ function SciMLBase.solve(prob::IntervalNonlinearProblem,
8685
c = 0.5 * (ā + b̄)
8786
end
8887
fc = f(c)
89-
(ā == c || b̄ == c) &&
88+
(ā == c || b̄ == c) &&
9089
return SciMLBase.build_solution(prob, alg, c, fc;
9190
retcode = ReturnCode.FloatingPointLimit,
92-
left = ā,
91+
left = ā,
9392
right = b̄)
9493
iszero(fc) &&
9594
return SciMLBase.build_solution(prob, alg, c, fc;
96-
retcode = ReturnCode.Success,
97-
left = ā,
98-
right = b̄)
99-
ā, b̄, d = _bracket(f, ā, b̄, c)
95+
retcode = ReturnCode.Success,
96+
left = ā,
97+
right = b̄)
98+
ā, b̄, d = _bracket(f, ā, b̄, c)
10099

101100
# The last bracketing block
102101
if b̄ - ā < 0.5 * (b - a)
@@ -105,14 +104,14 @@ function SciMLBase.solve(prob::IntervalNonlinearProblem,
105104
e = d
106105
c = 0.5 * (ā + b̄)
107106
fc = f(c)
108-
(ā == c || b̄ == c) &&
107+
(ā == c || b̄ == c) &&
109108
return SciMLBase.build_solution(prob, alg, c, fc;
110109
retcode = ReturnCode.FloatingPointLimit,
111-
left = ā,
110+
left = ā,
112111
right = b̄)
113112
iszero(fc) &&
114113
return SciMLBase.build_solution(prob, alg, c, fc;
115-
retcode = ReturnCode.Success,
114+
retcode = ReturnCode.Success,
116115
left = ā,
117116
right = b̄)
118117
a, b, d = _bracket(f, ā, b̄, c)
@@ -133,43 +132,45 @@ function SciMLBase.solve(prob::IntervalNonlinearProblem,
133132
end
134133

135134
# Define subrotine function bracket, check fc before bracket to return solution
136-
function _bracket(f::F, a, b, c) where F
135+
function _bracket(f::F, a, b, c) where {F}
137136
if iszero(f(c))
138137
ā, b̄, d = a, b, c
139138
else
140-
if f(a) * f(c) < 0
139+
if f(a) * f(c) < 0
141140
ā, b̄, d = a, c, b
142141
elseif f(b) * f(c) < 0
143142
ā, b̄, d = c, b, a
144143
end
145144
end
146145

147-
return ā, b̄, d
146+
return ā, b̄, d
148147
end
149148

150149
# Define subrotine function newton quadratic, return the approximation of zero
151-
function _newton_quadratic(f::F, a, b, d, k) where F
152-
A = ((f(d) - f(b)) / (d - b) - (f(b) - f(a)) / (b - a)) / (d - a)
150+
function _newton_quadratic(f::F, a, b, d, k) where {F}
151+
A = ((f(d) - f(b)) / (d - b) - (f(b) - f(a)) / (b - a)) / (d - a)
153152
B = (f(b) - f(a)) / (b - a)
154153

155154
if iszero(A)
156155
return a - (1 / B) * f(a)
157156
elseif A * f(a) > 0
158-
rᵢ₋₁ = a
159-
else
157+
rᵢ₋₁ = a
158+
else
160159
rᵢ₋₁ = b
161-
end
160+
end
162161

163162
for i in 1:k
164-
rᵢ = rᵢ₋₁ - (f(a) + B * (rᵢ₋₁ - a) + A * (rᵢ₋₁ - a) * (rᵢ₋₁ - b)) / (B + A * (2 * rᵢ₋₁ - a - b))
163+
rᵢ = rᵢ₋₁ -
164+
(f(a) + B * (rᵢ₋₁ - a) + A * (rᵢ₋₁ - a) * (rᵢ₋₁ - b)) /
165+
(B + A * (2 * rᵢ₋₁ - a - b))
165166
rᵢ₋₁ = rᵢ
166167
end
167168

168169
return rᵢ₋₁
169170
end
170171

171172
# Define subrotine function ipzero, also return the approximation of zero
172-
function _ipzero(f::F, a, b, c, d) where F
173+
function _ipzero(f::F, a, b, c, d) where {F}
173174
Q₁₁ = (c - d) * f(c) / (f(d) - f(c))
174175
Q₂₁ = (b - c) * f(b) / (f(c) - f(b))
175176
Q₃₁ = (a - b) * f(a) / (f(b) - f(a))
@@ -181,4 +182,4 @@ function _ipzero(f::F, a, b, c, d) where F
181182
Q₃₃ = (D₃₂ - Q₂₂) * f(a) / (f(d) - f(a))
182183

183184
return a + Q₃₁ + Q₃₂ + Q₃₃
184-
end
185+
end

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