Accurate Singular Values and Differential QD Algorithms
CALIFORNIA UNIV BERKELEY CENTER FOR PURE AND APPLIED MATHEMATICS
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We have discovered a new implementation of the qd algorithm that has a far wider domain of stability than Rutishausers version. Our algorithm was developed from an examination of the LR-Cholesky transformation and can be adapted to parallel computation in stark contrast to traditional qd. Our algorithm also yields useful a posteriori upper and lower bounds on the smallest singular value of a bidiagonal matrix. The zero-shift bidiagonal QR of Demmel and Kahan computes the smallest singular values to maximal relative accuracy and the others to maximal absolute accuracy with little or no degradation in efficiency when compared with the LINPACK code. Our algorithm obtains maximal relative accuracy for all the singular values and runs at least four times faster than the LINPACK code. qd, LR algorithm, Cholesky decomposition, singular values, SVD, bidiagonal matrices.
- Numerical Mathematics