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Generalized Symmetric Definite Eigenproblems

This section is concerned with the solution of the generalized eigenvalue problems , , and , 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 B=LLT or B=UTU (LLH or UHU in the Hermitian case). In the case , if A and B are banded then this may also be exploited to get a faster algorithm.

With B = LLT, we have

Hence the eigenvalues of are those of , where C is the symmetric matrix C = L-1 A L-T and y = LT z. In the complex case C is Hermitian with C = L-1 A L-H and y = LH z.

Table 2.13 summarizes how each of the three types of problem may be reduced to standard form , 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 2.13: Reduction of generalized symmetric definite eigenproblems to standard problems
| Type of | Factorization | Reduction | Recovery of
| problem | of B |  | eigenvectors
1. |  | B = LLT | C = L-1 A L-T | z = L-T y
|  | B = UTU | C = U-T A U-1 | z = U-1 y
2. |  | B = LLT | C = LT A L | z = L-T y
|  | B = UTU | C = U A UT | z = U-1 y
3. |  | B = LLT | C = LT A L | z = L y
|  | B = UTU | C = U A UT | z = UT y

Given A and a Cholesky factorization of B, the routines xyyGST overwrite A with the matrix C of the corresponding standard problem (see Table 2.14). This may then be solved using the routines described in subsection 2.4.4. 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.

If the problem is and the matrices A and B are banded, the matrix C as defined above is, in general, full. We can reduce the problem to a banded standard problem by modifying the definition of C thus:

where Q is an orthogonal matrix chosen to ensure that C has bandwidth no greater than that of A. Q is determined as a product of Givens rotations. This is known as Crawford's algorithm (see Crawford [19]). If X is required, it must be formed explicitly by the reduction routine.

A further refinement is possible when A and B are banded, which halves the amount of work required to form C (see Wilkinson [104]). Instead of the standard Cholesky factorization of B as UT U or L LT, we use a ``split Cholesky'' factorization B = ST S (SH S if B is complex), where:

with U11 upper triangular and L22 lower triangular of order approximately n/2; S has the same bandwidth as B. After B has been factorized in this way by the routine xPBSTF, the reduction of the banded generalized problem to a banded standard problem is performed by the routine xSBGST (or xHBGST for complex matrices). This routine implements a vectorizable form of the algorithm, suggested by Kaufman [77].

Table 2.14: Computational routines for the generalized symmetric definite eigenproblem
Type of matrix | Operation | Single precision | Double precision
and storage scheme |  | real | complex | real | complex
symmetric/Hermitian | reduction | SSYGST | CHEGST | DSYGST | ZHEGST
symmetric/Hermitian | reduction | SSPGST | CHPGST | DSPGST | ZHPGST
(packed storage) |  |  |  |  |
symmetric/Hermitian | split Cholesky | SPBSTF | CPBSTF | DPBSTF | ZPBSTF
banded | factorization |  |  |  |
| reduction | SSBGST | DSBGST | CHBGST | ZHBGST

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