1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
|
import linalgebra;
#include <catch2/catch_approx.hpp>
#include <catch2/catch_test_macros.hpp>
#include <cmath>
#include <cstddef>
#include <random>
using linalgebra::DimensionMismatchError;
using linalgebra::LinAlgError;
using linalgebra::Matrix;
using linalgebra::Vector;
namespace {
double frobenius_diff(const Matrix& A, const Matrix& B) {
double s = 0.0;
for (std::size_t i = 0; i < A.rows(); ++i)
for (std::size_t j = 0; j < A.cols(); ++j) {
const double d = A(i, j) - B(i, j);
s += d * d;
}
return std::sqrt(s);
}
Matrix make_dd(std::size_t n, std::mt19937& rng) {
std::uniform_real_distribution<double> dist(0.0, 1.0);
Matrix A(n, n);
for (std::size_t i = 0; i < n; ++i) {
double row_sum = 0.0;
for (std::size_t j = 0; j < n; ++j) {
A(i, j) = dist(rng);
if (i != j) row_sum += std::abs(A(i, j));
}
A(i, i) = row_sum + 1.0; // strictly diagonally dominant
}
return A;
}
Matrix make_spd(std::size_t n, std::mt19937& rng) {
std::uniform_real_distribution<double> dist(-1.0, 1.0);
Matrix B(n, n);
for (std::size_t i = 0; i < n; ++i)
for (std::size_t j = 0; j < n; ++j)
B(i, j) = dist(rng);
Matrix A = linalgebra::transpose(B) * B;
for (std::size_t i = 0; i < n; ++i) A(i, i) += static_cast<double>(n);
return A;
}
} // namespace
// arnoldi
TEST_CASE("arnoldi: orthonormality and AQ=QH relation", "[iterative][arnoldi]") {
Matrix A{{2.0, 1.0, 0.0},
{1.0, 3.0, 1.0},
{0.0, 1.0, 4.0}};
Vector b{1.0, 0.0, 0.0};
const int k = 2;
auto res = linalgebra::arnoldi(A, b, k);
const std::size_t n = 3;
const auto steps = static_cast<std::size_t>(res.steps_taken);
// Q columns must be orthonormal.
for (std::size_t i = 0; i <= steps; ++i) {
for (std::size_t j = 0; j <= steps; ++j) {
double dot = 0.0;
for (std::size_t r = 0; r < n; ++r) dot += res.Q(r, i) * res.Q(r, j);
const double expected = (i == j) ? 1.0 : 0.0;
REQUIRE(dot == Catch::Approx(expected).margin(1e-10));
}
}
for (std::size_t j = 0; j < steps; ++j) {
Vector qj(n);
for (std::size_t i = 0; i < n; ++i) qj[i] = res.Q(i, j);
const Vector Aqj = A * qj;
for (std::size_t i = 0; i <= steps; ++i) {
double qh = 0.0;
for (std::size_t r = 0; r < n; ++r) qh += res.Q(r, i) * res.H(i, j);
}
// Direct check: ||A*qj - Q*H[:,j]||
Vector Hcol(steps + 1);
for (std::size_t i = 0; i <= steps; ++i) Hcol[i] = res.H(i, j);
double resid = 0.0;
for (std::size_t row = 0; row < n; ++row) {
double qh_row = 0.0;
for (std::size_t i = 0; i <= steps; ++i) qh_row += res.Q(row, i) * Hcol[i];
const double d = Aqj[row] - qh_row;
resid += d * d;
}
REQUIRE(std::sqrt(resid) < 1e-10);
}
}
TEST_CASE("arnoldi: breakdown on scaled identity", "[iterative][arnoldi]") {
// A = 2*I → Krylov space is one-dimensional.
Matrix A(3, 3, 0.0);
A(0, 0) = 2.0; A(1, 1) = 2.0; A(2, 2) = 2.0;
Vector b{1.0, 0.0, 0.0};
auto res = linalgebra::arnoldi(A, b, 3);
REQUIRE(res.breakdown == true);
REQUIRE(res.steps_taken <= 3);
}
TEST_CASE("arnoldi: dimension mismatch throws", "[iterative][arnoldi]") {
Matrix A(3, 3, 0.0);
Vector b(4, 0.0);
REQUIRE_THROWS_AS(linalgebra::arnoldi(A, b, 2), DimensionMismatchError);
}
// solve_cg
TEST_CASE("solve_cg: 2x2 SPD", "[iterative][cg]") {
Matrix A{{4.0, 1.0}, {1.0, 3.0}};
Vector b{1.0, 2.0};
auto res = linalgebra::solve_cg(A, b);
REQUIRE(linalgebra::norm2(A * res.x - b) < 1e-10);
}
TEST_CASE("solve_cg: identity system converges in 1 iteration", "[iterative][cg]") {
auto I = Matrix::identity(5);
Vector b{1.0, 2.0, 3.0, 4.0, 5.0};
auto res = linalgebra::solve_cg(I, b);
for (std::size_t i = 0; i < 5; ++i)
REQUIRE(res.x[i] == Catch::Approx(b[i]).margin(1e-10));
}
TEST_CASE("solve_cg: 5x5 random SPD", "[iterative][cg]") {
std::mt19937 rng(99);
Matrix A = make_spd(5, rng);
std::uniform_real_distribution<double> dist(-2.0, 2.0);
Vector b(5);
for (std::size_t i = 0; i < 5; ++i) b[i] = dist(rng);
auto res = linalgebra::solve_cg(A, b);
REQUIRE(linalgebra::norm2(A * res.x - b) < 1e-8);
// Cross-check vs LU.
auto lu = linalgebra::lu_factor(A);
auto x_lu = linalgebra::lu_solve(lu, b);
REQUIRE(linalgebra::norm2(res.x - x_lu) < 1e-8);
}
TEST_CASE("solve_cg: non-symmetric throws", "[iterative][cg]") {
Matrix A{{1.0, 2.0}, {0.0, 1.0}};
Vector b{1.0, 1.0};
REQUIRE_THROWS_AS(linalgebra::solve_cg(A, b), LinAlgError);
}
// solve_gmres
TEST_CASE("solve_gmres: 2x2 non-symmetric", "[iterative][gmres]") {
Matrix A{{2.0, 1.0}, {1.0, 3.0}};
Vector b{5.0, 7.0};
auto res = linalgebra::solve_gmres(A, b);
REQUIRE(linalgebra::norm2(A * res.x - b) < 1e-9);
}
TEST_CASE("solve_gmres: 5x5 diagonally dominant", "[iterative][gmres]") {
std::mt19937 rng(11);
Matrix A = make_dd(5, rng);
std::uniform_real_distribution<double> dist(-3.0, 3.0);
Vector b(5);
for (std::size_t i = 0; i < 5; ++i) b[i] = dist(rng);
auto res = linalgebra::solve_gmres(A, b);
REQUIRE(linalgebra::norm2(A * res.x - b) < 1e-8);
auto lu = linalgebra::lu_factor(A);
auto x_lu = linalgebra::lu_solve(lu, b);
REQUIRE(linalgebra::norm2(res.x - x_lu) < 1e-8);
}
TEST_CASE("solve_gmres: zero rhs gives zero solution", "[iterative][gmres]") {
Matrix A{{2.0, 1.0}, {1.0, 3.0}};
Vector b(2, 0.0);
auto res = linalgebra::solve_gmres(A, b);
REQUIRE(linalgebra::norm2(res.x) < 1e-12);
}
// solve_bicgstab
TEST_CASE("solve_bicgstab: 2x2 non-symmetric", "[iterative][bicgstab]") {
Matrix A{{2.0, 1.0}, {1.0, 3.0}};
Vector b{5.0, 7.0};
auto res = linalgebra::solve_bicgstab(A, b);
REQUIRE(linalgebra::norm2(A * res.x - b) < 1e-9);
}
TEST_CASE("solve_bicgstab: 5x5 diagonally dominant", "[iterative][bicgstab]") {
std::mt19937 rng(22);
Matrix A = make_dd(5, rng);
std::uniform_real_distribution<double> dist(-3.0, 3.0);
Vector b(5);
for (std::size_t i = 0; i < 5; ++i) b[i] = dist(rng);
auto res = linalgebra::solve_bicgstab(A, b);
REQUIRE(linalgebra::norm2(A * res.x - b) < 1e-8);
auto lu = linalgebra::lu_factor(A);
auto x_lu = linalgebra::lu_solve(lu, b);
REQUIRE(linalgebra::norm2(res.x - x_lu) < 1e-8);
}
|