# ?hegst

Reduces a complex Hermitian-definite generalized eigenvalue problem to the standard form.

## Syntax

FORTRAN 77:

call chegst(itype, uplo, n, a, lda, b, ldb, info)

call zhegst(itype, uplo, n, a, lda, b, ldb, info)

Fortran 95:

call hegst(a, b [,itype] [,uplo] [,info])

## Description

This routine reduces complex Hermitian-definite generalized eigenvalue problems

`A*x = λ*B*x, A*B*x = λ*x`, or `B*A*x = λ*x`.

to the standard form `Cy = λy`. Here the matrix A is complex Hermitian, and B is complex Hermitian positive-definite. Before calling this routine, you must call ?potrf to compute the Cholesky factorization: `B = UH*U` or `B = L*LH`.

## Input Parameters

itype

  INTEGER. Must be 1 or 2 or 3.

  If `itype = 1`, the generalized eigenproblem is `A*z = lambda*B*z`

  for `uplo = 'U'`: `C = inv(UH)*A`*inv(U)``, `z = inv(U)*y`;

  for `uplo = 'L'`: `C = inv(L)*A*inv(LH)`, `z = inv(LH)*y`.

  If `itype = 2`, the generalized eigenproblem is `A*B*z = lambda*z`

  for `uplo = 'U'`: `C = U*A*UH`, ``z = inv(U)*y``;

  for `uplo = 'L'`: `C = LH*A*L`, `z = inv(LH)*y`.

  If `itype = 3`, the generalized eigenproblem is `B*A*z = lambda*z`

  for `uplo = 'U'`: `C = U*A*UH`, `z = UH*y`;

  for `uplo = 'L'`: `C = LH*A*L`, `z = L*y`.

uplo

  CHARACTER*1. Must be 'U' or 'L'.

  If `uplo = 'U'`, the array a stores the upper triangle of A; you must supply B in the factored form `B = UH*U`.

  If `uplo = 'L'`, the array a stores the lower triangle of A; you must supply B in the factored form `B = L*LH`.

n

  INTEGER. The order of the matrices A and B (`n ≥ 0`).

a, b

  COMPLEX for chegst DOUBLE COMPLEX for zhegst.

  Arrays:

  a(lda,*) contains the upper or lower triangle of A.

  The second dimension of a must be at least max(1, n).

  b(ldb,*) contains the Cholesky-factored matrix B:

  `B = UH*U` or `B = L*LH` (as returned by ?potrf).

  The second dimension of b must be at least max(1, n).

lda

  INTEGER. The first dimension of a; at least max(1, n).

ldb

  INTEGER. The first dimension of b; at least max(1, n).

## Output Parameters

a

  The upper or lower triangle of A is overwritten by the upper or lower triangle of C, as specified by the arguments itype and uplo.

info

  INTEGER.

  If `info = 0`, the execution is successful.

  If `info = -i`, the i-th parameter had an illegal value.

## Fortran 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see Fortran 95 Interface Conventions.

Specific details for the routine hegst interface are the following:

a

  Holds the matrix A of size (n,n).

b

  Holds the matrix B of size (n,n).

itype

  Must be 1, 2, or 3. The default value is 1.

uplo

  Must be 'U' or 'L'. The default value is 'U'.

## Application Notes

Forming the reduced matrix C is a stable procedure. However, it involves implicit multiplication by `inv(B)` (if `itype = 1`) or B (if `itype = 2` or 3). When the routine is used as a step in the computation of eigenvalues and eigenvectors of the original problem, there may be a significant loss of accuracy if B is ill-conditioned with respect to inversion.

The approximate number of floating-point operations is n3.

Parent topic: Generalized Symmetric-Definite Eigenvalue Problems