JacobiSVD

JacobiSVD refers to a class of Jacobi-rotation based algorithms for computing the singular value decomposition of a real or complex matrix. The method is named after the Jacobi rotation concept and is valued for its numerical stability and accuracy. In the Jacobi SVD, a sequence of two-sided plane rotations is applied to the input matrix in order to annihilate off-diagonal elements, effectively diagonalizing the product A^T A (or A A^T). Through these rotations, orthogonal (or unitary, in the complex case) matrices U and V are accumulated, and the resulting diagonal matrix contains the singular values S, with A approximating U S V^T.

Algorithmically, the Jacobi approach targets the off-diagonal energy of A^T A by performing Givens-like rotations on pairs of columns (and corresponding rows) to drive off-diagonal entries to zero. The process may follow a cyclic or greedy pivot strategy and is repeated in sweeps until convergence criteria are met. Convergence is typically declared when the sum of squares of the off-diagonal elements falls below a tolerance or a maximum number of sweeps is reached. The singular values are the square roots of the eigenvalues of A^T A, and the corresponding right singular vectors form V; left singular vectors U are obtained from A V and S.

JacobiSVD is known for high numerical accuracy, particularly for ill-conditioned matrices, and offers robust backward stability. Its computational cost is competitive for small to medium-sized matrices and is highly amenable to parallelization. However, for very large matrices, other SVD approaches (such as Lanczos-based or randomized methods) may offer faster performance.

In software implementations, JacobiSVD appears as a function or class. For example, MATLAB provides JacobiSVD to return U, S, and V for real or complex inputs, while the Eigen C++ library includes a JacobiSVD class with options for computing full or thin U and V.