Computing eigenspaces with specified eigenvalues of a regular matrix pair (A, B) and condition estimation: theory, algorithms and software

Theory, algorithms and LAPACK-style software for computing a pair of deflating subspaces with specified eigenvalues of a regular matrix pair (A, B) and error bounds for computed quantities (eigenvalues and eigenspaces) are presented. Thereordering of specified eigenvalues is performed with a direct orthogonal transformation method with guaranteed numerical stability. Each swap of two adjacent diagonal blocks in the real generalized Schur form, where at least one of them corresponds to a complex conjugate pair of eigenvalues, involves solving a generalized Sylvester equation and the construction of two orthogonal transformation matrices from certain eigenspaces associated with the diagonal blocks. The swapping of two 1×1 blocks is performed using orthogonal (unitary) Givens rotations. Theerror bounds are based on estimates of condition numbers for eigenvalues and eigenspaces. The software computes reciprocal values of a condition number for an individual eigenvalue (or a cluster of eigenvalues), a condition number for an eigenvector (or eigenspace), and spectral projectors onto a selected cluster. By computing reciprocal values we avoid overflow. Changes in eigenvectors and eigenspaces are measured by their change in angle. The condition numbers yield bothasymptotic andglobal error bounds. The asymptotic bounds are only accurate for small perturbations (E, F) of (A, B), while the global bounds work for all ‖(E, F.)‖ up to a certain bound, whose size is determined by the conditioning of the problem. It is also shown how these upper bounds can be estimated. Fortran 77software that implements our algorithms for reordering eigenvalues, computing (left and right) deflating subspaces with specified eigenvalues and condition number estimation are presented. Computational experiments that illustrate the accuracy, efficiency and reliability of our software are also described.

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