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Diffstat (limited to 'src/qr_iteration.cpp')
| -rw-r--r-- | src/qr_iteration.cpp | 287 |
1 files changed, 98 insertions, 189 deletions
diff --git a/src/qr_iteration.cpp b/src/qr_iteration.cpp index e476781..64fd25f 100644 --- a/src/qr_iteration.cpp +++ b/src/qr_iteration.cpp @@ -1,73 +1,99 @@ -#include "qr_iteration.hpp" - +module; #include <cassert> -#include <cmath> -#include <sstream> -#include "linalg_error.hpp" -#include "matrix.hpp" -#include "qr.hpp" -#include "vector.hpp" +export module linalgebra:qr_iteration; +import std; +import :error; +import :vector; +import :matrix; +import :qr; // References used throughout this file: // T&B — Trefethen & Bau, "Numerical Linear Algebra" // GVL — Golub & Van Loan, "Matrix Computations" 4th ed. -namespace linalg { +export namespace linalgebra { -namespace { +// --------------------------------------------------------------------------- +// Options +// --------------------------------------------------------------------------- + +struct QRIterationOptions { + double tolerance = 1e-10; + int max_iterations = 1000; + bool track_convergence = false; +}; + +struct QRIterationResult { + Vector eigenvalues_real; + Vector eigenvalues_imag; + int iterations = 0; + // std::vector is used here because linalgebra::Vector has no push_back; + // convergence_history is a plain time-series container, not a math object. + std::vector<double> convergence_history; +}; + +[[nodiscard]] QRIterationResult eigenvalues_unshifted(const Matrix& A, + QRIterationOptions opts = {}); + +[[nodiscard]] QRIterationResult eigenvalues_shifted(const Matrix& A, + QRIterationOptions opts = {}); + +// Givens rotation G acting on rows/columns i and i+1: +// +// G = | c s | chosen so that G * [x; y]^T = [r; 0]^T +// | -s c | with c = x/r, s = y/r, r = hypot(x, y) +struct GivensRotation { + double c; + double s; + std::size_t i; + + [[nodiscard]] static GivensRotation make(double x, double y, std::size_t row_index); + void apply_left(Matrix& M, std::size_t col_start = 0) const; + void apply_right(Matrix& M, std::size_t row_end) const; +}; +struct HessenbergResult { + Matrix H; + Matrix Q; +}; -// Frobenius norm of the strict lower triangle of an n×n matrix. -// This is the standard convergence diagnostic for QR iteration: as A_k -// approaches the real Schur form, all entries below the main diagonal -// (excluding 2×2 block sub-diagonals) tend to zero. -// Ref: T&B §28; used as the convergence criterion in Algorithm 28.1. -double lower_triangle_norm(const Matrix& A) { +[[nodiscard]] HessenbergResult hessenberg_reduction(const Matrix& A); + +void hessenberg_qr_step(Matrix& H, double sigma); + +[[nodiscard]] QRIterationResult eigenvalues_hessenberg(const Matrix& A, + QRIterationOptions opts = {}); + +} // namespace linalgebra + +namespace { + +double lower_triangle_norm(const linalgebra::Matrix& A) { const std::size_t n = A.rows(); double s = 0.0; - for (std::size_t i = 1; i < n; ++i) // row 1 .. n-1 - for (std::size_t j = 0; j < i; ++j) // col 0 .. i-1 (strict lower) + for (std::size_t i = 1; i < n; ++i) + for (std::size_t j = 0; j < i; ++j) s += A(i, j) * A(i, j); return std::sqrt(s); } -// Extract eigenvalues from a quasi-upper-triangular matrix (real Schur form). -// -// Scans the diagonal from top-left to bottom-right. At each position i: -// — |A(i+1, i)| < tol → 1×1 block: real eigenvalue A(i,i), imag = 0. -// — otherwise → 2×2 block [A(i..i+1, i..i+1)]: eigenvalues via -// quadratic formula. When the discriminant is -// negative the result is a complex-conjugate pair, -// stored as (re, +im) and (re, -im) in the real -// and imaginary part Vectors. -// -// Fills positions 0..n-1 of `real_out` and `imag_out` (pre-sized to n). -// -// Ref: T&B Lecture 28; GVL §7.4.1. -void extract_eigenvalues(const Matrix& T, double tol, - Vector& real_out, Vector& imag_out) { +void extract_eigenvalues(const linalgebra::Matrix& T, double tol, + linalgebra::Vector& real_out, linalgebra::Vector& imag_out) { const std::size_t n = T.rows(); - std::size_t out = 0; - std::size_t i = 0; + std::size_t out = 0; + std::size_t i = 0; while (i < n) { const bool is_last = (i + 1 == n); const bool sub_small = is_last || (std::abs(T(i + 1, i)) < tol); if (sub_small) { - // 1×1 block: real eigenvalue. real_out[out] = T(i, i); imag_out[out] = 0.0; ++out; ++i; } else { - // 2×2 block: - // | a b | - // | c d | - // Characteristic polynomial: lambda^2 - (a+d)*lambda + (ad - bc) = 0. - // Discriminant: (a-d)^2 + 4*b*c. - // Ref: GVL §7.4.1. const double a = T(i, i); const double b = T(i, i + 1); const double c = T(i + 1, i); @@ -76,15 +102,12 @@ void extract_eigenvalues(const Matrix& T, double tol, const double disc = (a - d) * (a - d) + 4.0 * b * c; if (disc >= 0.0) { - // Real eigenvalues unusual in converged real Schur form, but - // handled robustly in case the block didn't fully split. const double sq = std::sqrt(disc); real_out[out] = 0.5 * (tr + sq); imag_out[out] = 0.0; real_out[out + 1] = 0.5 * (tr - sq); imag_out[out + 1] = 0.0; } else { - // Complex-conjugate pair: real part ± imaginary part. const double re = 0.5 * tr; const double im = 0.5 * std::sqrt(-disc); real_out[out] = re; @@ -100,37 +123,32 @@ void extract_eigenvalues(const Matrix& T, double tol, assert(out == n); } -void require_square(const Matrix& A, const char* fname) { +void require_square(const linalgebra::Matrix& A, const char* fname) { if (A.rows() != A.cols()) { std::ostringstream oss; oss << fname << ": requires a square matrix, got " << A.rows() << "x" << A.cols(); - throw DimensionMismatchError(oss.str()); + throw linalgebra::DimensionMismatchError(oss.str()); } } +double wilkinson_shift(const linalgebra::Matrix& A) { + const std::size_t n = A.rows(); + const double a = A(n - 2, n - 2); + const double b = A(n - 1, n - 2); + const double d = A(n - 1, n - 1); + const double delta = 0.5 * (a - d); + const double denom = std::abs(delta) + std::hypot(delta, b); + if (denom == 0.0) return d; + const double sgn = (delta >= 0.0) ? 1.0 : -1.0; + return d - sgn * (b * b) / denom; +} + } // namespace -// --- Unshifted QR iteration --- -// -// Each step performs an orthogonal similarity transformation: -// A_{k-1} = Q_k R_k (Householder QR; backward-stable) -// A_k = R_k Q_k = Q_k^T A_{k-1} Q_k -// -// Similarity preserves eigenvalues (GVL §7.3.1, Theorem 7.3.1). -// The iterates converge to the real Schur form: a quasi-upper-triangular -// matrix whose 1×1 blocks give real eigenvalues and 2×2 blocks give -// complex-conjugate pairs. -// -// Convergence rate: linear, with per-step reduction factor -// |lambda_{j+1} / lambda_j| for the (j, j+1) coupling. -// (T&B Lecture 28, Theorem 28.2; GVL §7.3.2) -// -// Each iteration costs O(n^3) due to full Householder QR; Hessenberg -// reduction (Stage 3) reduces subsequent steps to O(n^2). +namespace linalgebra { -QRIterationResult eigenvalues_unshifted(const Matrix& A, - QRIterationOptions opts) { +QRIterationResult eigenvalues_unshifted(const Matrix& A, QRIterationOptions opts) { require_square(A, "eigenvalues_unshifted"); const std::size_t n = A.rows(); @@ -153,13 +171,9 @@ QRIterationResult eigenvalues_unshifted(const Matrix& A, Matrix Ak = A; for (int k = 0; k < opts.max_iterations; ++k) { - // Factor A_{k-1} = Q R using backward-stable Householder reflections. const QRResult qr = qr_householder(Ak); - - // A_k = R Q (orthogonal similarity: Q^T A_{k-1} Q) Ak = qr.R * qr.Q; - // --- Convergence check --- const double lower_norm = lower_triangle_norm(Ak); if (opts.track_convergence) { @@ -184,52 +198,6 @@ QRIterationResult eigenvalues_unshifted(const Matrix& A, throw NonConvergenceError(oss.str()); } -// --- Wilkinson-shifted QR iteration --- -// -// The Wilkinson shift is the eigenvalue of the bottom-right 2×2 block -// | a b | -// | c d | -// that is closest to d (the trailing diagonal entry). -// -// Exact eigenvalue formula: μ_{1,2} = (a+d)/2 ± sqrt(((a-d)/2)² + b·c) -// We pick the one with |μ - d| smaller. -// -// When the discriminant is negative (complex eigenvalues), fall back to σ = d -// (Rayleigh quotient shift), which still accelerates convergence. -// -// Ref: T&B Lecture 29; GVL §7.4.2. - -namespace { - -double wilkinson_shift(const Matrix& A) { - const std::size_t n = A.rows(); - const double a = A(n - 2, n - 2); - const double b = A(n - 1, n - 2); // subdiagonal entry only - const double d = A(n - 1, n - 1); - const double delta = 0.5 * (a - d); - const double denom = std::abs(delta) + std::hypot(delta, b); - if (denom == 0.0) return d; - const double sgn = (delta >= 0.0) ? 1.0 : -1.0; - return d - sgn * (b * b) / denom; -} - -} // namespace - -// eigenvalues_shifted — Wilkinson-shifted QR with trailing deflation. -// -// After each QR step we check whether the trailing subdiagonal entry of the -// active block is negligible (relative criterion: GVL §7.4.1). If so, the -// bottom diagonal entry is accepted as a converged eigenvalue and the active -// subproblem shrinks by one. This "trailing deflation" enables the cubic -// convergence promised by the Wilkinson shift to compound across successive -// eigenvalues rather than stalling on the full lower-triangle norm. -// -// When the active size reaches 2 we extract both eigenvalues analytically -// from the 2×2 block (handling real and complex-conjugate pairs) rather than -// continuing to iterate. For symmetric inputs this is always a real pair. -// -// Ref: GVL §7.5.1; T&B Lecture 29. - QRIterationResult eigenvalues_shifted(const Matrix& A, QRIterationOptions opts) { require_square(A, "eigenvalues_shifted"); const std::size_t n = A.rows(); @@ -250,7 +218,7 @@ QRIterationResult eigenvalues_shifted(const Matrix& A, QRIterationOptions opts) Matrix Ak = A; std::size_t n_found = n; - std::size_t active = n; // live subproblem is rows/cols 0..active-1 + std::size_t active = n; auto store_real = [&](double re) { --n_found; @@ -281,16 +249,14 @@ QRIterationResult eigenvalues_shifted(const Matrix& A, QRIterationOptions opts) }; for (int k = 0; k < opts.max_iterations; ++k) { - // --- Deflation sweep --- while (active >= 2) { const double sub = std::abs(Ak(active - 1, active - 2)); const double scale = std::abs(Ak(active - 2, active - 2)) + std::abs(Ak(active - 1, active - 1)); - // Relative + absolute floor tolerance (GVL §7.4.1). const double deflation_tol = opts.tolerance * (scale > 0.0 ? scale : 1.0); if (sub > deflation_tol) break; - Ak(active - 1, active - 2) = 0.0; // enforce exact zero + Ak(active - 1, active - 2) = 0.0; store_real(Ak(active - 1, active - 1)); --active; } @@ -299,7 +265,6 @@ QRIterationResult eigenvalues_shifted(const Matrix& A, QRIterationOptions opts) if (active == 1) { store_real(Ak(0, 0)); active = 0; break; } if (active == 2) { close_2x2(); break; } - // --- Wilkinson-shifted QR step on the active × active subblock --- Matrix sub_mat(active, active); for (std::size_t i = 0; i < active; ++i) for (std::size_t j = 0; j < active; ++j) @@ -331,8 +296,6 @@ QRIterationResult eigenvalues_shifted(const Matrix& A, QRIterationOptions opts) return result; } -// --- Givens rotation --- - GivensRotation GivensRotation::make(double x, double y, std::size_t row_index) { const double r = std::hypot(x, y); if (r == 0.0) return {1.0, 0.0, row_index}; @@ -340,9 +303,6 @@ GivensRotation GivensRotation::make(double x, double y, std::size_t row_index) { } void GivensRotation::apply_left(Matrix& M, std::size_t col_start) const { - // Rows i and i+1, columns col_start..n-1. - // [ c s] [x] [cx + sy] - // [-s c] [y] = [-sx + cy] for (std::size_t j = col_start; j < M.cols(); ++j) { const double xi = M(i, j); const double xi1 = M(i + 1, j); @@ -352,10 +312,6 @@ void GivensRotation::apply_left(Matrix& M, std::size_t col_start) const { } void GivensRotation::apply_right(Matrix& M, std::size_t row_end) const { - // Columns i and i+1, rows 0..row_end-1. - // M * G^T where G^T = [c -s; s c]: - // new col i = c * old_i + s * old_{i+1} - // new col i+1 = -s * old_i + c * old_{i+1} for (std::size_t j = 0; j < row_end; ++j) { const double xi = M(j, i); const double xi1 = M(j, i + 1); @@ -364,16 +320,6 @@ void GivensRotation::apply_right(Matrix& M, std::size_t row_end) const { } } -// --- Hessenberg reduction --- -// For k = 0, 1, ..., n-3: -// Build a Householder reflector H_k that zeros A[k+2:n, k]. -// Apply from left: A[k+1:n, k:n] ← H_k * A[k+1:n, k:n] -// Apply from right: A[0:n, k+1:n] ← A[0:n, k+1:n] * H_k -// Accumulate Q: Q[0:n, k+1:n] ← Q[0:n, k+1:n] * H_k -// -// H_k is never formed explicitly; applied via rank-1 update with tau = 2/uᵀu. -// Ref: GVL §7.4.2 (Algorithm 7.4.2). - HessenbergResult hessenberg_reduction(const Matrix& A) { require_square(A, "hessenberg_reduction"); const std::size_t n = A.rows(); @@ -382,10 +328,9 @@ HessenbergResult hessenberg_reduction(const Matrix& A) { Matrix Q = Matrix::identity(n); for (std::size_t k = 0; k + 2 <= n; ++k) { - const std::size_t p = n - k - 1; // p = n - (k+1) + const std::size_t p = n - k - 1; if (p == 0) break; - // Build Householder vector u from H[k+1:n, k]. std::vector<double> u(p); for (std::size_t i = 0; i < p; ++i) u[i] = H(k + 1 + i, k); @@ -402,27 +347,24 @@ HessenbergResult hessenberg_reduction(const Matrix& A) { for (double v : u) utu += v * v; const double tau = 2.0 / utu; - // Apply H_k from the LEFT to H[k+1:n, k:n]. for (std::size_t j = k; j < n; ++j) { - double dot = 0.0; - for (std::size_t i = 0; i < p; ++i) dot += u[i] * H(k + 1 + i, j); - const double coeff = tau * dot; + double d = 0.0; + for (std::size_t i = 0; i < p; ++i) d += u[i] * H(k + 1 + i, j); + const double coeff = tau * d; for (std::size_t i = 0; i < p; ++i) H(k + 1 + i, j) -= coeff * u[i]; } - // Apply H_k from the RIGHT to H[0:n, k+1:n]. for (std::size_t j = 0; j < n; ++j) { - double dot = 0.0; - for (std::size_t i = 0; i < p; ++i) dot += H(j, k + 1 + i) * u[i]; - const double coeff = tau * dot; + double d = 0.0; + for (std::size_t i = 0; i < p; ++i) d += H(j, k + 1 + i) * u[i]; + const double coeff = tau * d; for (std::size_t i = 0; i < p; ++i) H(j, k + 1 + i) -= coeff * u[i]; } - // Accumulate Q: Q[0:n, k+1:n] ← Q[0:n, k+1:n] * H_k. for (std::size_t j = 0; j < n; ++j) { - double dot = 0.0; - for (std::size_t i = 0; i < p; ++i) dot += Q(j, k + 1 + i) * u[i]; - const double coeff = tau * dot; + 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]; } @@ -432,20 +374,6 @@ HessenbergResult hessenberg_reduction(const Matrix& A) { return HessenbergResult{std::move(H), std::move(Q)}; } -// --- Hessenberg QR step via Givens rotations --- -// -// One shifted QR step on the upper Hessenberg matrix H: -// 1. Shift: H ← H - σI. -// 2. For k = 0..n-2: compute G_k = Givens(H(k,k), H(k+1,k)); -// apply G_k from left to rows k,k+1 of H, -// starting from column k (Hessenberg: H(k+1,j)=0, j<k). -// 3. For k = 0..n-2: apply G_k^T from right to cols k,k+1 of H, -// up to row k+2 (exploits upper-triangular structure). -// 4. Unshift: H ← H + σI. -// -// After the step H is again upper Hessenberg (GVL §7.4.2, Theorem 7.4.1). -// Total cost: O(n²). Ref: GVL §7.4.2. - void hessenberg_qr_step(Matrix& H, double sigma) { const std::size_t n = H.rows(); @@ -455,10 +383,7 @@ void hessenberg_qr_step(Matrix& H, double sigma) { gs.reserve(n - 1); for (std::size_t k = 0; k + 1 < n; ++k) { - // Eliminate H(k+1, k) via a rotation on rows k and k+1. GivensRotation g = GivensRotation::make(H(k, k), H(k + 1, k), k); - // Left application: rows k, k+1; columns k..n-1. - // (Hessenberg: H(k+1, j) = 0 for j < k, so starting from col k is exact.) g.apply_left(H, k); gs.push_back(g); } @@ -467,23 +392,10 @@ void hessenberg_qr_step(Matrix& H, double sigma) { gs[k].apply_right(H, std::min(k + 2, n)); } - // Unshift. for (std::size_t j = 0; j < n; ++j) H(j, j) += sigma; } -// --- Full QR algorithm --- -// -// Same outer deflation loop as eigenvalues_shifted, but each QR step uses -// hessenberg_qr_step (O(n²) Givens rotations) instead of full Householder QR -// (O(n³)). After Hessenberg reduction the matrix stays Hessenberg throughout, -// so the O(n²) per-step cost applies for every step after the one-time O(n³) -// reduction. Total cost is thus O(n³) + O(iterations · n²), which beats -// eigenvalues_shifted's O(iterations · n³) for large n. -// -// Ref: GVL §7.4.2; T&B Lecture 29. - -QRIterationResult eigenvalues_hessenberg(const Matrix& A, - QRIterationOptions opts) { +QRIterationResult eigenvalues_hessenberg(const Matrix& A, QRIterationOptions opts) { require_square(A, "eigenvalues_hessenberg"); const std::size_t n = A.rows(); @@ -535,7 +447,6 @@ QRIterationResult eigenvalues_hessenberg(const Matrix& A, }; for (int k = 0; k < opts.max_iterations; ++k) { - // --- Deflation sweep --- while (active >= 2) { const double sub = std::abs(H(active - 1, active - 2)); const double scale = std::abs(H(active - 2, active - 2)) @@ -552,7 +463,6 @@ QRIterationResult eigenvalues_hessenberg(const Matrix& A, if (active == 1) { store_real(H(0, 0)); active = 0; break; } if (active == 2) { close_2x2(); break; } - // Wilkinson shift from trailing 2×2 of the active block. const double a_w = H(active - 2, active - 2); const double b_w = H(active - 1, active - 2); const double d_w = H(active - 1, active - 1); @@ -561,7 +471,6 @@ QRIterationResult eigenvalues_hessenberg(const Matrix& A, const double sigma = (denom == 0.0) ? d_w : d_w - ((delta >= 0.0) ? 1.0 : -1.0) * (b_w * b_w) / denom; - // O(n²) Givens step on the active×active Hessenberg subblock. Matrix sub_H(active, active); for (std::size_t ii = 0; ii < active; ++ii) for (std::size_t jj = 0; jj < active; ++jj) @@ -593,4 +502,4 @@ QRIterationResult eigenvalues_hessenberg(const Matrix& A, return result; } -} // namespace linalg +} // namespace linalgebra |