Theory - 2023.2 English

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2023.2 English

The SVD of m-by-m symmetric matrix A is given by

\[A = U \Sigma U^T\]

where \(U\) is an orthogonal (unitary) matrix and \(\Sigma\) is an m-by-m matrix with real diagonal elements.

There are two dominant categories of SVD algorithms for dense matrix: bidiagonalization methods and Jacobi methods. The classical bidiagonalization method is a long sequential calculation, the oPGA has no advantage in the calculation. In contrast, Jacobi methods apply plane rotations to the entire matrix A. Two-sided Jacobi methods iteratively apply rotations on both sides of the matrix A to bring it to diagonal form, while one-sided Hestenes Jacobi methods apply rotations on one side to orthogonalize the columns of matrix A, and bring \(A^TA\) to diagonal form. While Jacobi methods are often slow than bidiagonalization methods, they have a better potential in unrolling and pipelining.