William Kahan

Calculating the Singular Values and Pseudo-Inverse of a Matrix

A numerically stable and fairly fast scheme is described to compute the unitary matrices U and V which transform a given matrix A into a diagonal form

The Rotation of Eigenvectors by a Perturbation. III

\Sigma = U^ * AV

Accurate singular values of bidiagonal matrices

, thus exhibiting A’s singular values on

NORM-PRESERVING DILATIONS AND THEIR APPLICATIONS TO OPTIMAL ERROR BOUNDS*

\Sigma

Residual Bounds on Approximate Eigensystems of Nonnormal Matrices

’s diagonal. The scheme first transforms A to a bidiagonal matrix J, then diagonalizes J. The scheme described here is complicated but does not suffer from the computational difficulties which occasionally afflict some previously known methods. Some applications are mentioned, in particular the use of the pseudo-inverse

Precimonious: tuning assistant for floating-point precision

A^I = V\Sigma ^I U^*

Error bounds from extra-precise iterative refinement

to solve least squares problems in a way which dampens spurious oscillation and cancellation.

On computing givens rotations reliably and efficiently

When a Hermitian linear operator is slightly perturbed, by how much can its invariant subspaces change? Given some approximations to a cluster of neighboring eigenvalues and to the corresponding eigenvectors of a real symmetric matrix, and given an estimate for the gap that separates the cluster from all other eigenvalues, how much can the subspace spanned by the eigenvectors differ from the subspace spanned by our approximations? These questions are closely related; both are investigated here. The difference between the two subspaces is characterized in terms of certain angles through which one subspace must be rotated in order most directly to reach the other. These angles unify the treatment of natural geometric, operator-theoretic and error-analytic questions concerning those subspaces. Sharp bounds upon trigonometric functions of these angles are obtained from the gap and from bounds upon either the perturbation (1st question) or a computable residual (2nd question). An example is included.

A Proposed Radix- and Word-length-independent Standard for Floating-point Arithmetic

Computing the singular values of a bidiagonal matrix is the final phase of the standard algorithm for the singular value decomposition of a general matrix. A new algorithm that computes all the sin...

Composition constants for raising the orders of unconventional schemes for ordinary differential equations

The problem is, given A, B, C, to find D such that