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Generalized Symmetric Definite Eigenproblems


This section is concerned with the solution of the generalized eigenvalue problems tex2html_wrap_inline12875, tex2html_wrap_inline12877, and tex2html_wrap_inline12879, where A and B are real symmetric or complex Hermitian and B is positive definite. Each of these problems can be reduced to a standard symmetric eigenvalue problem, using a Cholesky factorization of B as either tex2html_wrap_inline14317 or tex2html_wrap_inline14319 (tex2html_wrap_inline14321 or tex2html_wrap_inline14323 in the Hermitian case).

With tex2html_wrap_inline14317, we have
Hence the eigenvalues of tex2html_wrap_inline12875 are those of tex2html_wrap_inline14329, where C is the symmetric matrix tex2html_wrap_inline14333 and tex2html_wrap_inline14335. In the complex case C is Hermitian with tex2html_wrap_inline14339 and tex2html_wrap_inline14341.

Table 3.11 summarizes how each of the three types of problem may be reduced to standard form   tex2html_wrap_inline14329, and how the eigenvectors z of the original problem may be recovered from the eigenvectors y of the reduced problem. The table applies to real problems; for complex problems, transposed matrices must be replaced by conjugate transposes.

Table 3.11: Reduction of generalized symmetric definite eigenproblems to standard problems

Given A and a Cholesky factorization of B, the routines PxyyGST overwrite A      with the matrix C of the corresponding standard problem tex2html_wrap_inline14329 (see table 3.12). This may then be solved by using the routines described in subsection 3.3.4. No special routines are needed to recover the eigenvectors z of the generalized problem from the eigenvectors y of the standard problem, because these computations are simple applications of Level 2 or Level 3 BLAS.

Table 3.12: Computational routines for the generalized symmetric definite eigenproblem

Susan Blackford
Tue May 13 09:21:01 EDT 1997