345 lines
15 KiB
C++
345 lines
15 KiB
C++
#pragma once
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#include "Matrix.hpp"
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/**
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* @brief library that uses Matrix.hpp and performs SVD on a matrix
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*/
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namespace SVD {
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/**
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* @brief Compute the Singular Value Decomposition (SVD) of this matrix.
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*
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* Decomposes A into U × Σ × Vᵀ where:
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* - U is an m×k orthogonal matrix (left singular vectors)
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* - Σ is a k×k diagonal matrix with non-negative singular values
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* (stored as a k×1 column vector)
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* - Vᵀ is a k×n orthogonal matrix (right singular vectors, transposed)
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* - k = min(m, n)
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*
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* The decomposition satisfies: A ≈ U × diag(Σ) × Vᵀ
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* Singular values are returned in descending order.
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*
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* @param U Output: left singular vectors (m×k orthogonal matrix)
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* @param sigma Output: singular values as k×1 column vector, sorted descending
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* @param Vt Output: right singular vectors transposed (k×n matrix)
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*
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* @note This implementation uses the Golub-Kahan-Reinsch algorithm:
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* 1. Householder bidiagonalization of A
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* 2. Implicit QR iteration on the bidiagonal matrix
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* 3. Accumulation of U and V factors throughout
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*/
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template <uint8_t rows, uint8_t columns>
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void SVD(Matrix<rows, columns> &matrixToDecompose, Matrix<rows, columns> &U,
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Matrix<columns, 1> &sigma, Matrix<columns, columns> &Vt);
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// ========================================================================
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// SVD Building Block Functions (for unit testing)
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// These operate on internal 5×5 working arrays for maximum flexibility.
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// ========================================================================
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/**
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* @brief Compute a Householder reflector vector.
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*
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* Given input vector x, computes normalized v and scalar alpha such that:
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* (I - 2·v·vᵀ) · x = [alpha, 0, 0, ...]ᵀ
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*
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* @param x Input vector (up to 5 elements)
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* @param len Number of valid elements in x
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* @param v Output: normalized Householder vector (v[0] is the first element)
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* @param alpha Output: the resulting first element after reflection
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* @return The norm of the input vector x
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*/
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static float ComputeHouseholder(const float *x, uint8_t len, float *v,
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float &alpha);
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/**
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* @brief Apply a Householder reflection from the left.
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*
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* Transforms W = (I - 2·v·vᵀ) · W where v operates on rows [startRow..endRow].
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*
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* @param W Input/output: matrix to transform (5×5 working array)
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* @param v Householder vector (length = endRow - startRow + 1)
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* @param startRow First row index
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* @param endRow Last row index
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*/
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static void ApplyHouseholderLeft(Matrix<5, 5> &W, const float *v,
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uint8_t startRow, uint8_t endRow);
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/**
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* @brief Apply a Householder reflection from the right.
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*
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* Transforms W = W · (I - 2·v·vᵀ) where v operates on columns
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* [startCol..endCol].
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*
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* @param W Input/output: matrix to transform (5×5 working array)
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* @param v Householder vector (length = endCol - startCol + 1)
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* @param startCol First column index
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* @param endCol Last column index
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*/
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static void ApplyHouseholderRight(Matrix<5, 5> &W, const float *v,
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uint8_t startCol, uint8_t endCol);
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/**
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* @brief Reduce a matrix to upper bidiagonal form using Householder reflections.
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*
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* Applies a sequence of Householder reflections to reduce the input matrix
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* W (m×q, where q ≥ p) to upper bidiagonal form B (p×q), accumulating
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* the left and right transformation matrices in QL and QR respectively.
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*
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* Algorithm (Golub-Kahan bidiagonalization):
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* For k = 0 to p-1:
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* 1. Left HH on column k, rows k..m-1: zero out subdiagonal below B[k+1][k]
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* 2. Right HH on row k, cols k+2..q-1: zero out superdiagonal above B[k][k+1]
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*
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* The accumulated transformations satisfy:
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* QLᵀ · W_original · QR = B (upper bidiagonal)
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*
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* @param W Input/output: matrix to bidiagonalize (5×5, must be at least p×q)
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* @param m Number of rows in the working matrix
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* @param q Number of columns in the working matrix (q ≥ p)
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* @param p Rank = min(m, original_columns) — number of bidiagonalization steps
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* @param QL Input/output: left Householder accumulation (initialized to identity,
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* output: QLᵀ such that QLᵀ·W = B)
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* @param QR Input/output: right Householder accumulation (initialized to identity,
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* output: QR such that W·QR = B after left apply)
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*/
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static void Bidiagonalize(Matrix<5, 5> &W,
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uint8_t m, uint8_t q, uint8_t p,
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Matrix<5, 5> &QL,
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Matrix<5, 5> &QR);
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/**
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* @brief Deflate a bidiagonal matrix by zeroing negligible superdiagonals.
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*
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* Scans the p×p upper-bidiagonal matrix stored in W and zeros out any
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* superdiagonal element W[i][i+1] whose magnitude is negligible relative to
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* the local diagonal scale (|W[i][i]| + |W[i+1][i+1]|). Deflating splits
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* the matrix into independent unreduced blocks that can each be solved
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* separately.
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*
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* @param W Input/output: bidiagonal matrix (5×5 working array, first p×p used)
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* @param p Size of the bidiagonal matrix (min(rows, columns))
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* @param tol Relative deflation tolerance (e.g. 1e-8f)
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*/
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static void DeflateBidiagonal(Matrix<5, 5> &W, uint8_t p, float tol);
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/**
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* @brief Check whether a bidiagonal matrix has fully reduced to diagonal.
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*
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* Returns true when every superdiagonal element of the p×p bidiagonal
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* matrix in W is (numerically) zero, i.e. the diagonal entries are the
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* (unsorted) singular values and no unreduced blocks remain.
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*
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* @param W Input: bidiagonal matrix (5×5 working array, first p×p used)
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* @param p Size of the bidiagonal matrix (min(rows, columns))
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* @param tol Numerical zero threshold multiplier
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* @return true when all superdiagonal elements are ~0
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*/
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static bool BidiagonalIsDiagonal(const Matrix<5, 5> &W, uint8_t p, float tol);
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/**
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* @brief Compute the full SVD of a 2×2 upper-bidiagonal block (pure).
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*
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* Decomposes B = [[a, b], [0, d]] as:
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* B = Ublock · diag(sigma[0], sigma[1]) · Vblockᵀ
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*
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* Guarantees:
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* - sigma[0] ≥ sigma[1] ≥ 0 (singular values, from eigenvalues of BᵀB)
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* - Ublock and Vblock are orthogonal (columns are the left/right
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* singular vectors respectively; Vblock = scipy's Vᵀᵀ)
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* - Ublock · diag(sigma) · Vblockᵀ == B (within float tolerance)
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*
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* Math: eigenvectors of BᵀB = [[a², ab], [ab, b²+d²]] give the right
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* singular vectors (v1 = normalize(ab, σ1²−a²) with a safe fallback when
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* that vector is ~0; v2 = (−v1y, v1x)); left singular vectors are
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* uᵢ = B·vᵢ/σᵢ with a rank-deficiency guard: when σᵢ ≈ 0 (i.e. ~1e-30),
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* that U column is filled with the signed orthogonal complement of the
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* other U column instead of dividing by ~0.
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*
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* @param a B[0][0] (first diagonal element)
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* @param b B[0][1] (superdiagonal element)
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* @param d B[1][1] (second diagonal element)
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* @param Ublock Output: 2×2 left singular vectors (columns)
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* @param Vblock Output: 2×2 right singular vectors (columns)
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* @param sigma Output: singular values, sigma[0] ≥ sigma[1] ≥ 0
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*/
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static void SolveBidiagonalBlock2x2(float a, float b, float d,
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float Ublock[2][2], float Vblock[2][2],
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float sigma[2]);
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/**
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* @brief Cyclic Jacobi eigenvalue algorithm for a symmetric matrix (pure).
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*
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* Reduces symmetric n×n matrix T to (near-)diagonal form IN PLACE using
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* cyclic Jacobi rotations, accumulating the eigenvectors in V.
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*
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* On return:
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* - T's diagonal entries are the eigenvalues (off-diagonals ~0)
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* - evals[i] = T[i][i], UNSORTED
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* - columns of V are the corresponding eigenvectors (T·V = V·Λ)
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*
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* @param T Input/output: symmetric matrix (5×5 storage, first n×n used,
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* destroyed in place)
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* @param n Matrix size (≤ 5)
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* @param evals Output: eigenvalues, unsorted (evals[i] = T[i][i])
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* @param V Output: eigenvector matrix, columns are eigenvectors
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*/
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static void JacobiEigenSymmetric(float T[5][5], uint8_t n, float evals[5],
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float V[5][5]);
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/**
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* @brief Fold a block SVD's factors into the QL/QR accumulators.
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*
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* Given the block SVD of a bidiagonal block, B = Ublock·Σ·Vblockᵀ, the
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* accumulated Householder matrices must absorb the block factors:
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* QL[:, blockStart..blockStart+blockSize−1] ← QL[:, ...] · Ublock
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* (rows 0..rowsQL−1)
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* QR[:, blockStart..blockStart+blockSize−1] ← QR[:, ...] · Vblock
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* (rows 0..rowsQR−1)
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*
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* rowsQL / rowsQR are the meaningful row extents of the accumulators
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* (e.g. for a wide matrix W = Aᵀ, QL carries n = rows(W) meaningful
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* rows while QR is read back over its first m rows).
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*
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* @param blockStart First column/row index of the block in W
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* @param blockSize Size of the block (2, or > 2 for the Jacobi path)
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* @param Ublock Left singular-vector factor of the block (blockSize×blockSize in 5×5 storage)
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* @param Vblock Right singular-vector factor of the block (blockSize×blockSize in 5×5 storage)
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* @param rowsQL Number of meaningful rows of QL
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* @param rowsQR Number of meaningful rows of QR
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* @param QL Input/output: left transformation accumulator
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* @param QR Input/output: right transformation accumulator
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*/
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static void ApplyBlockFactorsToAccumulators(uint8_t blockStart,
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uint8_t blockSize,
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const float Ublock[5][5],
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const float Vblock[5][5],
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uint8_t rowsQL, uint8_t rowsQR,
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Matrix<5, 5> &QL,
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Matrix<5, 5> &QR);
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/**
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* @brief Solve a bidiagonal block larger than 2×2 via Jacobi eigen of BᵀB.
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*
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* Computes the full SVD of the unreduced upper-bidiagonal block
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* W[blockStart..blockStart+blockSize−1] via eigendecomposition of the
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* tridiagonal T = BᵀB:
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* 1. Snapshot the ORIGINAL block diagonal/superdiagonal from W
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* 2. JacobiEigenSymmetric on T → eigenvalues (unsorted) + V
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* 3. Sort eigenvalues descending, reordering V
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* 4. Ublock = B_orig · V · Σ⁻¹ (from the snapshot, so W is not
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* overwritten before Ublock is computed)
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* 5. Fold Ublock/Vblock into QL/QR via ApplyBlockFactorsToAccumulators
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* 6. Only then write sqrt(eigenvalues) into W's diagonal and zero the
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* block's superdiagonals
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*
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* @param W Input/output: bidiagonal matrix (5×5 working array); the block's
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* diagonal holds the singular values and its superdiagonals are
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* zeroed on return
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* @param blockStart First column/row index of the block
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* @param blockSize Size of the block (> 2, ≤ 5)
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* @param rowsQL Number of meaningful rows of QL
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* @param rowsQR Number of meaningful rows of QR
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* @param QL Input/output: left transformation accumulator
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* @param QR Input/output: right transformation accumulator
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* @param tol (unused: Jacobi convergence tolerance is internal)
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*/
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static void SolveBidiagonalBlockJacobi(Matrix<5, 5> &W, uint8_t blockStart,
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uint8_t blockSize, uint8_t rowsQL,
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uint8_t rowsQR, Matrix<5, 5> &QL,
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Matrix<5, 5> &QR, float tol);
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/**
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* @brief Extract singular values from bidiagonal matrix diagonal and sort.
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*
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* Extracts absolute values of diagonal elements of W as singular values,
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* then sorts them in descending order while reordering columns of QL
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* and QR to maintain consistency.
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*
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* @param W Input: bidiagonal matrix (5×5 working array)
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* @param sigma Output: sorted singular values (5×1 column vector, only first p used)
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* @param p Number of singular values (min(rows, columns))
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* @param QL Input/output: left transformation matrix (modified during sort)
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* @param QR Input/output: right transformation matrix (modified during sort)
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*/
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static void ExtractAndSortSingularValues(Matrix<5, 5> &W,
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Matrix<5, 1> &sigma,
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uint8_t p,
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Matrix<5, 5> &QL,
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Matrix<5, 5> &QR);
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/**
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* @brief Assemble final U and Vt matrices from QL/QR.
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*
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* Computes the final left singular vectors (U) and right singular vectors
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* transposed (Vt) from the accumulated Householder transformations.
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*
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* For non-transpose case: U = QL[:,0:p], Vt = QR[:,0:p]ᵀ
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* For transpose case: U = QR[:,0:p]ᵀ, Vt = QL[:,0:p]ᵀ
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*
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* @param m Number of rows in original matrix
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* @param n Number of columns in original matrix
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* @param p Rank = min(m, n)
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* @param transposeNeeded True if we computed SVD(Aᵀ) instead of SVD(A)
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* @param QL Left Householder accumulation (5×5)
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* @param QR Right Householder accumulation (5×5)
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* @param U Output: left singular vectors (m×n matrix, only first p columns used)
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* @param Vt Output: right singular vectors transposed (n×n matrix, only first p rows used)
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*/
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static void AssembleUAndVt(uint8_t m, uint8_t n, uint8_t p,
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bool transposeNeeded,
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const Matrix<5, 5> &QL,
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const Matrix<5, 5> &QR,
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Matrix<5, 5> &U,
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Matrix<5, 5> &Vt);
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/**
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* @brief Compute a Givens rotation that zeros out y.
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*
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* Computes c, s such that:
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* [c s] [x] = [r]
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* [-s c] [y] [0]
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* where r = sqrt(x² + y²).
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*
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* @param x First element
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* @param y Second element (to be zeroed)
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* @param c Output: cosine of rotation angle
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* @param s Output: sine of rotation angle
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*/
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static void ComputeGivens(float x, float y, float &c, float &s);
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/**
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* @brief Apply a Givens rotation from the left to rows i and j.
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*
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* Applies [c s; -s c] to rows i, j of W (columns startCol..endCol).
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*
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* @param W Input/output: matrix to transform
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* @param i First row index
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* @param j Second row index
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* @param c Cosine of rotation angle
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* @param s Sine of rotation angle
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* @param startCol First column to transform
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* @param endCol Last column to transform
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*/
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static void ApplyGivensLeft(Matrix<5, 5> &W, uint8_t i, uint8_t j, float c,
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float s, uint8_t startCol, uint8_t endCol);
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/**
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* @brief Apply a Givens rotation from the right to columns i and j.
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*
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* Applies [c -s; s c]ᵀ to columns i, j of W (rows startRow..endRow).
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*
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* @param W Input/output: matrix to transform
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* @param i First column index
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* @param j Second column index
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* @param c Cosine of rotation angle
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* @param s Sine of rotation angle
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* @param startRow First row to transform
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* @param endRow Last row to transform
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*/
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static void ApplyGivensRight(Matrix<5, 5> &W, uint8_t i, uint8_t j, float c,
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float s, uint8_t startRow, uint8_t endRow);
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} // namespace SVD
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#ifndef SVD_H_
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#include "SVD.cpp"
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#endif // SVD_H_
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