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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);
}
// Check orthogonality ||Q^T Q - I||_F < tol.
double ortho_error(const Matrix& Q) {
const std::size_t n = Q.rows();
const std::size_t m = Q.cols();
const Matrix Qt = linalgebra::transpose(Q);
const Matrix QtQ = Qt * Q;
const Matrix I = Matrix::identity(m);
return frobenius_diff(QtQ, I);
}
// Build a symmetric matrix from B^T B + shift * I.
Matrix make_sym(std::size_t n, double shift, 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) += shift;
return A;
}
} // namespace
// tridiagonalize
TEST_CASE("tridiagonalize: 2x2 symmetric", "[sym_eigen][tridiagonalize]") {
Matrix A{{4.0, 2.0}, {2.0, 3.0}};
auto res = linalgebra::tridiagonalize(A);
// Reconstruction: Q T Q^T == A.
const Matrix QTQT = res.Q * res.T * linalgebra::transpose(res.Q);
REQUIRE(frobenius_diff(A, QTQT) < 1e-12);
// Q must be orthogonal.
REQUIRE(ortho_error(res.Q) < 1e-12);
// T must be tridiagonal: T(i,j) == 0 for |i-j| > 1.
const std::size_t n = res.T.rows();
for (std::size_t i = 0; i < n; ++i)
for (std::size_t j = 0; j < n; ++j)
if (i > j + 1 || j > i + 1)
REQUIRE(std::abs(res.T(i, j)) < 1e-12);
}
TEST_CASE("tridiagonalize: 4x4 symmetric", "[sym_eigen][tridiagonalize]") {
Matrix A{{6.0, 2.0, 1.0, 0.0},
{2.0, 5.0, 3.0, 1.0},
{1.0, 3.0, 4.0, 2.0},
{0.0, 1.0, 2.0, 3.0}};
auto res = linalgebra::tridiagonalize(A);
const Matrix QTQT = res.Q * res.T * linalgebra::transpose(res.Q);
REQUIRE(frobenius_diff(A, QTQT) < 1e-10);
REQUIRE(ortho_error(res.Q) < 1e-12);
const std::size_t n = res.T.rows();
for (std::size_t i = 0; i < n; ++i)
for (std::size_t j = 0; j < n; ++j)
if (i > j + 1 || j > i + 1)
REQUIRE(std::abs(res.T(i, j)) < 1e-10);
}
TEST_CASE("tridiagonalize: identity", "[sym_eigen][tridiagonalize]") {
auto I = Matrix::identity(5);
auto res = linalgebra::tridiagonalize(I);
REQUIRE(frobenius_diff(I, res.Q * res.T * linalgebra::transpose(res.Q)) < 1e-12);
}
TEST_CASE("tridiagonalize: random symmetric", "[sym_eigen][tridiagonalize]") {
std::mt19937 rng(77);
Matrix A = make_sym(8, 5.0, rng);
auto res = linalgebra::tridiagonalize(A);
REQUIRE(frobenius_diff(A, res.Q * res.T * linalgebra::transpose(res.Q)) < 1e-9);
REQUIRE(ortho_error(res.Q) < 1e-11);
}
TEST_CASE("tridiagonalize: non-symmetric throws", "[sym_eigen][tridiagonalize]") {
Matrix A{{1.0, 2.0}, {3.0, 4.0}};
REQUIRE_THROWS_AS(linalgebra::tridiagonalize(A), LinAlgError);
}
TEST_CASE("tridiagonalize: non-square throws", "[sym_eigen][tridiagonalize]") {
Matrix A(2, 3);
REQUIRE_THROWS_AS(linalgebra::tridiagonalize(A), DimensionMismatchError);
}
// eigenvectors_inverse_iteration
TEST_CASE("eigenvectors_inverse_iteration: 2x2 diagonal", "[sym_eigen][inverse_iter]") {
Matrix A(2, 2, 0.0);
A(0, 0) = 3.0; A(1, 1) = 7.0;
Vector lambdas{3.0, 7.0};
auto res = linalgebra::eigenvectors_inverse_iteration(A, lambdas);
// Each column should satisfy A*v ≈ lambda*v.
for (std::size_t col = 0; col < 2; ++col) {
Vector v(2);
v[0] = res.eigenvectors(0, col);
v[1] = res.eigenvectors(1, col);
const Vector Av = A * v;
const double lam = lambdas[col];
double resid = 0.0;
for (std::size_t i = 0; i < 2; ++i) {
const double d = Av[i] - lam * v[i];
resid += d * d;
}
REQUIRE(std::sqrt(resid) < 1e-8);
}
}
TEST_CASE("eigenvectors_inverse_iteration: 3x3 symmetric", "[sym_eigen][inverse_iter]") {
// Known symmetric matrix: compute eigenvalues with francis, then eigenvectors.
Matrix A{{6.0, 2.0, 1.0},
{2.0, 3.0, 1.0},
{1.0, 1.0, 1.0}};
auto eig = linalgebra::eigenvalues_francis(A);
// Use real eigenvalues only.
auto res = linalgebra::eigenvectors_inverse_iteration(A, eig.eigenvalues_real);
for (std::size_t col = 0; col < 3; ++col) {
Vector v(3);
for (std::size_t i = 0; i < 3; ++i) v[i] = res.eigenvectors(i, col);
const Vector Av = A * v;
const double lam = eig.eigenvalues_real[col];
double resid = 0.0;
for (std::size_t i = 0; i < 3; ++i) {
const double d = Av[i] - lam * v[i];
resid += d * d;
}
REQUIRE(std::sqrt(resid) < 1e-6);
}
}
TEST_CASE("eigenvectors_inverse_iteration: non-square throws", "[sym_eigen][inverse_iter]") {
Matrix A(2, 3);
Vector lambdas{1.0};
REQUIRE_THROWS_AS(linalgebra::eigenvectors_inverse_iteration(A, lambdas),
DimensionMismatchError);
}
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