Combined Systolic Array for Matrix Portrait Computation

Abstract

Given the matrix A ∋ C ^n×n and scalars λ1; λ2; λm ∈ C, our task is to design a systolic implementation of the matrix portrait computation - i.e., the singular value decomposition of matrices A - λ _k I; k = 1; 2; :::;m. We propose the triangular-rectangular and hexagonal systolic subarrays for the recursive QR updating of matrices A - λ _k I, and another triangular subarray for the singular value decomposition of the R-factor. Let m; n and r be the number of various λs, the matrix order and the number of repeated loops in the SVD algorithm, respectively. Due to the large amount of overlap between subarrays, the time complexity of our solution is O(3mn) whereas the straightforward systolic implementation requires O([# 7/2mn]+4rm) times steps. The number of PEs and delays is O([cn²]), where c = 37/8 for our solution and c = 5/8 for the straightforward solution.