An Effective Modified Bidiagonal Reduction Algorithm For Computing Singular Value Decomposition

Bidiagonal reduction is the preliminary stage for computing the singular value decomposition.The best-known bidiagonal algorithm is GKBR(GK bidiagonal reduc-tion),this result has much larger rounding error using the algorithm.In 2002,Jessel L.Barlow find RGBR(right givens bidiagonal reduction)algorithm in[1].This algo-rithm have two stages.First,a preprocessing technique uses QR factorization with column pivoting.Second,dealing with rounding error using appropriate method such that the smaller element have much higher relative accuracy in bidiagonal matrix.So that,it is reasonable to compute all of the singular values of a matrix to higher relative accuracy.In this paper,we have discovered MBR(modifying bidiagonal reduction)algorith-m for a triangular matrix R.This algorithm consists of two stages.In the first stage,we constructs TGBR algorithm such that to take advantage of the structure of R to bidiag-onalize it.The new procedure requires fewer operations and keeping accuracy.In the second stage,we utilize the superiority of higher accuracy for RGBR algorithm.Final-ly,MBR algorithm coordinate computation complexity and accuracy.So that,we can decrease the computation complexity and keep accuracy.