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export module linalgebra:sym_eigen;
import std;
import :error;
import :vector;
import :matrix;
import :norms;
import :lu;
// References used throughout this file:
// T&B — Trefethen & Bau, "Numerical Linear Algebra"
// GVL — Golub & Van Loan, "Matrix Computations" 4th ed.
export namespace linalgebra {
// Symmetric tridiagonalization via Householder reflections
// Reference: GVL §8.3.1; T&B Lecture 26
//
// For symmetric A, computes Q and T such that A = Q T Q^T,
// where T is symmetric tridiagonal and Q is orthogonal.
struct TridiagonalizeResult {
Matrix T; // symmetric tridiagonal (full n×n)
Matrix Q; // orthogonal: A = Q T Q^T
};
[[nodiscard]] TridiagonalizeResult tridiagonalize(const Matrix& A,
double symmetry_tolerance = 1e-12);
// Eigenvectors via inverse iteration
// Reference: GVL §7.6.1; T&B Lecture 27
//
// Given A and a vector of approximate eigenvalues, returns a matrix whose
// columns are the corresponding (approximate) eigenvectors.
struct InverseIterationOptions {
double tolerance = 1e-10;
int max_iterations = 100;
};
struct InverseIterationResult {
Matrix eigenvectors; // n × k, column j is eigenvector for eigenvalues[j]
Vector residuals; // ||A*v_j - lambda_j*v_j||_2 for each j
};
[[nodiscard]] InverseIterationResult eigenvectors_inverse_iteration(
const Matrix& A,
const Vector& eigenvalues,
InverseIterationOptions opts = {});
} // namespace linalgebra
namespace linalgebra {
TridiagonalizeResult tridiagonalize(const Matrix& A, double symmetry_tolerance) {
if (A.rows() != A.cols()) {
std::ostringstream oss;
oss << "tridiagonalize requires a square matrix, got "
<< A.rows() << "x" << A.cols();
throw DimensionMismatchError(oss.str());
}
const std::size_t n = A.rows();
for (std::size_t i = 0; i < n; ++i)
for (std::size_t j = i + 1; j < n; ++j)
if (std::abs(A(i, j) - A(j, i)) > symmetry_tolerance)
throw LinAlgError("tridiagonalize requires a symmetric matrix");
Matrix T = A;
Matrix Q = Matrix::identity(n);
for (std::size_t k = 0; k + 2 <= n; ++k) {
const std::size_t p = n - k - 1; // length of sub-column
// Form Householder vector u from T[k+1:, k].
std::vector<double> u(p);
for (std::size_t i = 0; i < p; ++i) u[i] = T(k + 1 + i, k);
double x_norm = 0.0;
for (double v : u) x_norm += v * v;
x_norm = std::sqrt(x_norm);
if (x_norm == 0.0) continue;
const double sigma = (u[0] >= 0.0 ? 1.0 : -1.0) * x_norm;
u[0] += sigma;
double utu = 0.0;
for (double v : u) utu += v * v;
const double tau = 2.0 / utu;
// Apply H = I - tau*u*u^T symmetrically: T <- H T H
// Left: T[k+1:, :] -= tau * u * (u^T T[k+1:, :])
for (std::size_t j = 0; j < n; ++j) {
double d = 0.0;
for (std::size_t i = 0; i < p; ++i) d += u[i] * T(k + 1 + i, j);
const double coeff = tau * d;
for (std::size_t i = 0; i < p; ++i) T(k + 1 + i, j) -= coeff * u[i];
}
// Right: T[:, k+1:] -= tau * (T[:, k+1:] u) * u^T
for (std::size_t j = 0; j < n; ++j) {
double d = 0.0;
for (std::size_t i = 0; i < p; ++i) d += T(j, k + 1 + i) * u[i];
const double coeff = tau * d;
for (std::size_t i = 0; i < p; ++i) T(j, k + 1 + i) -= coeff * u[i];
}
// Explicitly zero sub-subdiagonal entries for numerical cleanliness.
for (std::size_t i = 1; i < p; ++i) {
T(k + 1 + i, k) = 0.0;
T(k, k + 1 + i) = 0.0;
}
// Accumulate Q: Q[:, k+1:] -= tau * (Q[:, k+1:] u) * u^T
for (std::size_t j = 0; j < n; ++j) {
double d = 0.0;
for (std::size_t i = 0; i < p; ++i) d += Q(j, k + 1 + i) * u[i];
const double coeff = tau * d;
for (std::size_t i = 0; i < p; ++i) Q(j, k + 1 + i) -= coeff * u[i];
}
}
return TridiagonalizeResult{std::move(T), std::move(Q)};
}
InverseIterationResult eigenvectors_inverse_iteration(const Matrix& A,
const Vector& eigenvalues,
InverseIterationOptions opts) {
if (A.rows() != A.cols()) {
std::ostringstream oss;
oss << "eigenvectors_inverse_iteration requires a square matrix, got "
<< A.rows() << "x" << A.cols();
throw DimensionMismatchError(oss.str());
}
const std::size_t n = A.rows();
const std::size_t k = eigenvalues.size();
Matrix evecs(n, k);
Vector residuals(k, 0.0);
// Starting vector: uniform unit vector.
Vector v0(n, 1.0 / std::sqrt(static_cast<double>(n)));
for (std::size_t col = 0; col < k; ++col) {
double lambda = eigenvalues[col];
// Build shifted matrix B = A - lambda*I.
Matrix B = A;
for (std::size_t i = 0; i < n; ++i) B(i, i) -= lambda;
// Try to factor; if near-singular (either at factorization or solve time), perturb the shift.
auto make_lu = [&]() -> LUResult {
try {
LUResult result = lu_factor(B, 1e-14);
lu_solve(result, v0); // probe: backward_substitution validates U diagonal
return result;
} catch (const SingularMatrixError&) {
lambda += 1e-7;
for (std::size_t i = 0; i < n; ++i) B(i, i) += 1e-7;
return lu_factor(B, 1e-14);
}
};
LUResult lu_b = make_lu();
Vector v = v0;
bool converged = false;
for (int iter = 0; iter < opts.max_iterations; ++iter) {
Vector w = lu_solve(lu_b, v);
// Normalize.
double w_norm = norm2(w);
if (w_norm == 0.0) break;
v = w / w_norm;
// Residual: ||A v - lambda v||.
const Vector Av = A * v;
double res = 0.0;
for (std::size_t i = 0; i < n; ++i) {
const double d = Av[i] - eigenvalues[col] * v[i];
res += d * d;
}
res = std::sqrt(res);
if (res < opts.tolerance) {
converged = true;
residuals[col] = res;
break;
}
}
if (!converged) {
std::ostringstream oss;
oss << "eigenvectors_inverse_iteration: did not converge for eigenvalue "
<< col << " (lambda = " << eigenvalues[col] << ") in "
<< opts.max_iterations << " iterations";
throw NonConvergenceError(oss.str());
}
for (std::size_t i = 0; i < n; ++i) evecs(i, col) = v[i];
}
return InverseIterationResult{std::move(evecs), std::move(residuals)};
}
} // namespace linalgebra
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