Add QR decomposition

1 files changed, 218 insertions, 0 deletions

diff --git a/src/qr.cpp b/src/qr.cpp
new file mode 100644
index 0000000..57e4c2b
--- /dev/null
+++ b/src/qr.cpp
@@ -0,0 +1,218 @@
+#include "qr.hpp"
+
+#include <cmath>
+#include <sstream>
+
+#include "linalg_error.hpp"
+
+namespace linalg {
+
+namespace {
+
+void require_tall(const Matrix& A, const char* name) {
+    if (A.rows() < A.cols()) {
+        std::ostringstream oss;
+        oss << name << " requires rows >= cols, got " << A.rows() << "x" << A.cols();
+        throw DimensionMismatchError(oss.str());
+    }
+}
+
+// ||column j of M||_2
+double col_norm(const Matrix& M, std::size_t j) {
+    double s = 0.0;
+    for (std::size_t i = 0; i < M.rows(); ++i) {
+        s += M(i, j) * M(i, j);
+    }
+    return std::sqrt(s);
+}
+
+// dot product of column j of M with column k of N (same number of rows)
+double col_dot(const Matrix& M, std::size_t j, const Matrix& N, std::size_t k) {
+    double s = 0.0;
+    for (std::size_t i = 0; i < M.rows(); ++i) {
+        s += M(i, j) * N(i, k);
+    }
+    return s;
+}
+
+}  // namespace
+
+// ---------------------------------------------------------------------------
+// Classical Gram-Schmidt
+// ---------------------------------------------------------------------------
+//
+// For column j:
+//   R[i][j] = <a_j, q_i>   for i < j   
+//   v       = a_j - sum_i R[i][j] * q_i
+//   R[j][j] = ||v||
+//   q_j     = v / R[j][j]
+//
+
+QRResult qr_classical_gs(const Matrix& A, double zero_tolerance) {
+    require_tall(A, "qr_classical_gs");
+    const std::size_t m = A.rows();
+    const std::size_t n = A.cols();
+
+    Matrix Q = Matrix::zeros(m, n);
+    Matrix R = Matrix::zeros(n, n);
+
+    for (std::size_t j = 0; j < n; ++j) {
+        // Start with column j of A.
+        for (std::size_t i = 0; i < m; ++i) Q(i, j) = A(i, j);
+
+        // Project out existing basis vectors using the *original* A column.
+        for (std::size_t k = 0; k < j; ++k) {
+            R(k, j) = col_dot(A, j, Q, k);       // <a_j, q_k>
+            for (std::size_t i = 0; i < m; ++i) {
+                Q(i, j) -= R(k, j) * Q(i, k);
+            }
+        }
+
+        const double norm = col_norm(Q, j);
+        if (norm <= zero_tolerance) {
+            std::ostringstream oss;
+            oss << "qr_classical_gs: column " << j
+                << " is (nearly) linearly dependent (norm = " << norm << ")";
+            throw SingularMatrixError(oss.str());
+        }
+        R(j, j) = norm;
+        for (std::size_t i = 0; i < m; ++i) Q(i, j) /= norm;
+    }
+
+    return QRResult{std::move(Q), std::move(R)};
+}
+
+// ---------------------------------------------------------------------------
+// Modified Gram-Schmidt
+// ---------------------------------------------------------------------------
+//
+// For column j:
+//   v = a_j
+//   For k = 0 .. j-1:
+//     R[k][j] = <v, q_k> 
+//     v       = v - R[k][j] * q_k
+//   R[j][j] = ||v||
+//   q_j     = v / R[j][j]
+//
+// Each subtraction uses the already-updated v, so round-off is re-corrected
+// at every sub-step rather than compounding into one subtraction.
+
+QRResult qr_modified_gs(const Matrix& A, double zero_tolerance) {
+    require_tall(A, "qr_modified_gs");
+    const std::size_t m = A.rows();
+    const std::size_t n = A.cols();
+
+    Matrix Q = Matrix::zeros(m, n);
+    Matrix R = Matrix::zeros(n, n);
+
+    for (std::size_t j = 0; j < n; ++j) {
+        for (std::size_t i = 0; i < m; ++i) Q(i, j) = A(i, j);
+
+        for (std::size_t k = 0; k < j; ++k) {
+            R(k, j) = col_dot(Q, j, Q, k);       // <v_running, q_k>
+            for (std::size_t i = 0; i < m; ++i) {
+                Q(i, j) -= R(k, j) * Q(i, k);
+            }
+        }
+
+        const double norm = col_norm(Q, j);
+        if (norm <= zero_tolerance) {
+            std::ostringstream oss;
+            oss << "qr_modified_gs: column " << j
+                << " is (nearly) linearly dependent (norm = " << norm << ")";
+            throw SingularMatrixError(oss.str());
+        }
+        R(j, j) = norm;
+        for (std::size_t i = 0; i < m; ++i) Q(i, j) /= norm;
+    }
+
+    return QRResult{std::move(Q), std::move(R)};
+}
+
+// ---------------------------------------------------------------------------
+// Householder QR
+// ---------------------------------------------------------------------------
+//
+// At step k, build a Householder reflector H_k that maps R[k:, k] to
+// -sign(R[k,k]) * ||R[k:,k]|| * e_1.
+//
+// H = I - (2 / (u^T u)) * u * u^T
+//   where u = x + sign(x_0) * ||x|| * e_1   (sign chosen to avoid cancellation)
+//
+// H is never formed explicitly.  It is applied via the rank-1 update:
+//   M[k:, :] -= u * (2/(u^T u) * (u^T M[k:, :]))
+//
+// After n reflections, the working copy of A has become R (upper triangular).
+// Q is accumulated by applying each H_k to an identity matrix from the left.
+// The thin Q (m x n) is the first n columns of the full m x m orthogonal Q.
+
+QRResult qr_householder(const Matrix& A) {
+    require_tall(A, "qr_householder");
+    const std::size_t m = A.rows();
+    const std::size_t n = A.cols();
+
+    // Will become R.
+    Matrix work = A;
+
+    // Q accumulated as full m x m orthogonal matrix; trim to m x n.
+    Matrix Q_full = Matrix::identity(m);
+
+    for (std::size_t k = 0; k < n; ++k) {
+        const std::size_t p = m - k;  // length of the subvector
+
+        // Build Householder vector u from the subcolumn work[k:, k].
+        std::vector<double> u(p);
+        for (std::size_t i = 0; i < p; ++i) u[i] = work(k + i, k);
+
+        const double x_norm = [&] {
+            double s = 0.0;
+            for (double v : u) s += v * v;
+            return std::sqrt(s);
+        }();
+
+        if (x_norm == 0.0) continue;
+
+        // sigma = sign(u[0]) * ||x||
+        const double sigma = (u[0] >= 0.0 ? 1.0 : -1.0) * x_norm;
+        u[0] += sigma;
+
+        const double utu = [&] {
+            double s = 0.0;
+            for (double v : u) s += v * v;
+            return s;
+        }();
+        const double tau = 2.0 / utu;
+
+        // Apply H_k to work[k:, 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] * work(k + i, j);
+            const double coeff = tau * dot;
+            for (std::size_t i = 0; i < p; ++i) work(k + i, j) -= coeff * u[i];
+        }
+
+        // Apply H_k to Q_full[k:, 0:m] 
+        for (std::size_t j = 0; j < m; ++j) {
+            double dot = 0.0;
+            for (std::size_t i = 0; i < p; ++i) dot += u[i] * Q_full(k + i, j);
+            const double coeff = tau * dot;
+            for (std::size_t i = 0; i < p; ++i) Q_full(k + i, j) -= coeff * u[i];
+        }
+    }
+
+    // Thin Q: first n columns of Q_full^T
+    Matrix Q(m, n);
+    for (std::size_t i = 0; i < m; ++i)
+        for (std::size_t j = 0; j < n; ++j)
+            Q(i, j) = Q_full(j, i);
+
+    // Thin R: first n rows of work
+    Matrix R(n, n);
+    for (std::size_t i = 0; i < n; ++i)
+        for (std::size_t j = 0; j < n; ++j)
+            R(i, j) = work(i, j);
+
+    return QRResult{std::move(Q), std::move(R)};
+}
+
+}  // namespace linalg