Bidiagonalization and symmetric tridiagonalization by systolic arrays

We give a systolic algorithm and array for bidiagonalization of an n x n matrix in O(nlog, n) time, using O(n2) cells. Bandedness of the input matrix may be effectively exploited. If the matrix is banded, with p nonzero subdiagonals and q nonzero superdiagonais, then 4n In(p + q) + O(n) clocks and 2n(p + q ) + O((p + q)’ + n) cells are needed. This is faster than the best previously reported result by the factor log, e = 1.44.. .. Moreover, in contrast to earlier systolic designs, which require the matrix to be preloaded into the array and the result matrix extracted after bidiagonalization, the present arrays are pipelined. Work reported herein was supported in part by Cooperative Agreement NCC 2-387 between the National Aeronautics and Space Administration (NASA) and the Universities Space Research Association (USRA) and pmally supported by Office of Naval Research (ONR) under Contract NOW 1486-K-06 10. Bidiagonalization and Symmetric Tridiagonahation by Systolic Arrays* Robert Schreibert November 29, 1988 Abstract We give a systolic algorithm and array for bidiagonalization of an n x n matrix in O ( n log, n) time, using O(nZ) cells. Bandedness of the input matrix may be effectively exploited. If the matrix is banded, with p nonzero subdiagonals and q nonzero superdiagonals, then 4n In(p + q) + O(n) clocks aad 2n(p -+ q ) + O( ( p + q)2 + n) cells are needed. This is faster than the best previously reported result by the factor log, e = 2-44 0. Moreover, in contrast to earlier systolic designs, which require the matrix to be preloaded into the array and the result matrix extracted after bidiagonaiization, the present arrays are pipelindWe give a systolic algorithm and array for bidiagonalization of an n x n matrix in O ( n log, n) time, using O(nZ) cells. Bandedness of the input matrix may be effectively exploited. If the matrix is banded, with p nonzero subdiagonals and q nonzero superdiagonals, then 4n In(p + q) + O(n) clocks aad 2n(p -+ q ) + O( ( p + q)2 + n) cells are needed. This is faster than the best previously reported result by the factor log, e = 2-44 0. Moreover, in contrast to earlier systolic designs, which require the matrix to be preloaded into the array and the result matrix extracted after bidiagonaiization, the present arrays are pipelind