Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
A New Scaling for Newton's Iteration for the Polar Decomposition and its Backward Stability
SIAM Journal on Matrix Analysis and Applications
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We present a roundoff error analysis of the method for solving the linear least squares problem min"x@?b-Ax@?"2 with full column rank matrix A, using only factors @S and V from the SVD decomposition of A=U@SV^T. This method (called SNE"S"V"D here) is an analogue of the method of seminormal equations (SNE"Q"R), where the solution is computed from R^TRx=A^Tb using only the factor R from the QR factorization of A. Such methods have practical applications when A is large and sparse and if one needs to solve least squares problems with the same matrix A and multiple right-hand sides. However, in general both SNE"Q"R and SNE"S"V"D are not forward stable. We analyze one step of fixed precision iterative refinement to improve the accuracy of the SNE"S"V"D method. We show that, under the condition O(u)@k^2(A)