Computing Small Singular Values of Bidiagonal Matrices With Guaranteed High Relative Accuracy (Classic Reprint)

High-precision singular values for bidiagonal matrices, with a faster, stable algorithm. A new approach blends the standard QR iteration with a zero-shift variant to guarantee forward stability and high relative accuracy for all singular values, regardless of their size.

This book presents the theory and the algorithm, showing how small relative changes in the data lead to small relative changes in the singular values. It also explains how the method can be used to compute eigenvalues of symmetric tridiagonal matrices and discusses convergence criteria, practical implementation details, and numerical results that compare performance with traditional methods.

• How a QR-based algorithm can achieve guaranteed high relative accuracy for every singular value of a bidiagonal matrix.


• Conditions and perturbation results that justify stability and accurate results across a range of matrix structures.


• The role of a forward-stable, zero-shift modification to the QR iteration and its practical impact on speed.


• Numerical experiments and practical considerations, including when and how the method outperforms standard approaches.

Ideal for readers of numerical linear algebra and computational mathematics seeking robust, precision-focused methods for singular value and eigenvalue problems.