Accession Number : ADA206865

Title :   On the Forward Instability of the QR Transformation,

Corporate Author : CALIFORNIA UNIV BERKELEY CENTER FOR PURE AND APPLIED MATHEMATICS

Personal Author(s) : Le, J ; Parlett, B N

PDF Url : ADA206865

Report Date : Jul 1988

Pagination or Media Count : 41

Abstract : This study is in the area of matrix eigenvalue computations The Householder-QR algorithm has become the standard method for diagonalizing a symmetric matrix. First the matrix is reduced to tridiagonal form T by a technique introduced by A. Householder in 1958. Next the tridiagonal matrix T is diagonalized by successive applications of the QR transformation with shifts Moreover it is well known that the QR transformation is backward stable. That means that the computed transform is exactly orthogonally similar to a matrix close to the old one. It is also known to the experts that the QR transformation sometimes exhibits forward instability. That means the the computed output is far from the result obtained with exact arithmetic. For the purpose of computing eigenvalues and eigenvectors the property of backward stability is all that one requires. However the QR transformation has other uses and in some cases forward stability is desirable. This report analyzes the forward instability of QR and shows that it occurs only when the shift is very close to eigenvalues with a special property. (kr)

Descriptors :   *MATRICES(MATHEMATICS), *EIGENVALUES, COMPUTATIONS, EIGENVECTORS, OUTPUT, STABILITY, STANDARDIZATION

Subject Categories : Numerical Mathematics

Distribution Statement : APPROVED FOR PUBLIC RELEASE