More on GUPTRI and Numerical Examples

A typical user will call four of the routines, namely GUPTRI, REORDR, BOUND, and EVALBD. In the following we give a brief description of the functionality of these routines. We believe that the best way to get acquainted with singular or near to singular problems and the GUPTRI form is to start using the MATLAB interface of the GUPTRI routine (see p. ).

GUPTRI:

Reduces $A - \lambda B$ to a GUPTRI form (8.34)-(8.35) in terms of a unitary equivalence transformation. The finite nonzero eigenvalues (diagonal elements) of $A_{reg} - \lambda B_{reg}$ may appear in any order. Besides $\{A,B\}$, the user must provide three input parameters (EPSU, GAP, and ZERO) that control the computation of the GUPTRI form. These three parameters are also arguments for the MATLAB function (see p. ).

REORDR:

It is possible directly from the computed GUPTRI decomposition to produce a pair of reducing subspaces associated with any part of the spectrum $\lambda(A, B)$. The user must then call the routine REORDR and provide an integer function FTEST(ALPHA, BETA), describing the spectrum of the reducing subspace to be computed, as input parameter (see section 6 in [122] for an example).

Given that the specified regular part of $A - \lambda B$ is in upper triangular form, the $1 \times 1$ diagonal blocks (the generalized eigenvalues) are reordered using pairs of Givens rotations which are accumulated in previous transformation matrices $P$ and $Q$. After the reordering, the eigenvalues specified by the function FTEST appear in $A_{11} - \lambda B_{11}$ (8.36), i.e., at the top northwest corner of the specified regular part of $A - \lambda B$ and the corresponding pair of reducing subspaces can easily be read off from the leading columns of $P$ and $Q$.

BOUND and EVALBD:

Error bounds for reducing subspaces and generalized eigenvalues are presented in [119,122]. We have illustrated by examples that eigenvalues and reducing subspaces are ill posed, which means that extra conditions on allowable perturbations $\delta A$ and $\delta B$ of $A$ and $B$ must be imposed. BOUND and EVALBD compute error bounds for pairs of reducing subspaces, as well as for selected eigenvalues of the regular part. For the reducing subspace error bounds, it is required that $(A + \delta A) - \lambda (B + \delta B)$ has reducing subspaces of the same dimension as $A - \lambda B$. For the eigenvalue bounds, it is additionally required that $(A + \delta A) - \lambda (B + \delta B)$ has the same number of eigenvalues in its selected regular part. Both of these conditions are automatically verified by the software.

Based on input parameters, BOUND decides which perturbation theorem is applicable and computes the required quantities in error bounds for eigenvalues and reducing subspaces. EVALBD takes the output of BOUND and evaluates the reducing subspace bounds of the appropriate theorem for a given input ${\tt DELTA} = {\Vert(\delta A , \delta B) \Vert _E}$.