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Generalized Symmetric Definite Eigenproblems

This section is concerned with the solution of the generalized eigenvalue problems $A z = \lambda B z$, $A B z = \lambda z$, and $B A z = \lambda z$, 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 $Ax = \lambda Bz$, if A and B are banded then this may also be exploited to get a faster algorithm.

With B = LLT, we have

\begin{displaymath}
Az=\lambda Bz \quad \Rightarrow \quad (L ^{-1}AL ^{-T})(L^Tz)= \lambda(L^Tz).
\end{displaymath}

Hence the eigenvalues of $A z = \lambda B z$ are those of $Cy=\lambda y$, 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 $Cy=\lambda y$, 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. $A z = \lambda B z$ 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. $A B z = \lambda z$ B = LLT C = LT A L z = L-T y
    B = UTU C = U A UT z = U-1 y
3. $B A z = \lambda z$ 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 $Cy=\lambda y$ (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 $A z = \lambda B z$ 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:

\begin{displaymath}
C = X^T A X, \quad \mbox{ where } X = U^{-1} Q \quad \mbox{ or } \quad L^{-T} Q,
\end{displaymath}

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:

\begin{displaymath}
S = \left( \begin{array}{cc}
U_{11} & \\
M_{21} & L_{22} \\
\end{array} \right)
\end{displaymath}

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 $A z = \lambda B z$ to a banded standard problem $Cy=\lambda y$ 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


next up previous contents index
Next: Generalized Nonsymmetric Eigenproblems Up: Computational Routines Previous: Singular Value Decomposition   Contents   Index
Susan Blackford
1999-10-01