First Order Error Analysis of a Linear System of Equations by use of Error Propagation Matrices connected to the Pseudo Inverse Solution

Abstract

The singular value decomposition (SVD) of a matrix A = U#V is a useful tool for analyzing the e#ect of errors in A of the pseudoinverse solution to Ax = b. Let E be the error propagation matrix such that the first order error propagation result dx = E A dA(:) is satisfied. Then the SVD of E is directly available from the SVD of A. It is shown how to calculate E A in di#erent cases: well-, over- and underdetermined as well as rank-deficient. Illustrative small examples are analyzed as is the connection between the singular values of E and condition numbers. A few steps are also taken towards the analysis of a regularized solution of Ax = b