SVD workflow: - 2023.2 English

The input parameters for the 4x4 SVD function is the 4x4 matrix \(A\), and the output matrices are \(U\), \(V\), and \(\Sigma\) respectively. As shown in the above figure, the SVD process has 4 main steps:

1. Find the max value of matrix \(A\);
2. Divide all member of A by \(max(A)\) and initiate \(U\), \(\Sigma\), \(V\);
3. The iterative process of Jacobi SVD;
4. Sort matrix \(S\), and change \(U\) and \(V\);

The iterative process of Jacobi SVD is the core function. Firstly, the matrix \(A\) will be divided into 2x3 (= \(C_{4}^{2}\)) sub-blocks of 2x2 through its diagonal elements, since for a 4x4 matrix, at the same moment, it has only 2 pairs independent 2x2 matrix, and 3 rounds of nonredundant full permutation.

Once we get the 2x2 Submatrix, the Jacobi methods or Givens rotation (module SVD 2x2) can be applied. Here we use pipelining to bind the two 2x2 SVD process. The output of 2x2 SVD is the rotation matrix Equation (2).

The next step is to decompose the rotation matrix from original matrix \(A\) and add it to matrix \(U\) and \(V\). After that, a convergence determination is performed to reduce the off-diagonal value of matrix \(A\). When the matrix \(A\) is reduced to a diagonal matrix, step 3 will be finished.