 
  
  
  
  
 
 This section is concerned with the solution of the generalized eigenvalue
 problems  ,
,  , and
, 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
, 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
  or
 or  (
 ( or
 or  in the Hermitian case).
 in the Hermitian case).
 With  , we have
, we have
 
 Hence the eigenvalues of  are those of
 are those of  ,
 where C is the symmetric matrix
,
 where C is the symmetric matrix  and
 and  .
 In the complex case C is Hermitian with
.
 In the complex case C is Hermitian with  and
 and  .
.
 Table 3.11 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.
, 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
 (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.
 (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