?hegst

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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?hegst

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

Syntax

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

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

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

Include Files

mkl.fi, lapack.f90

Description

The routine reduces a complex Hermitian positive-definite generalized eigenvalue problem to standard form.

itype	Problem	Result
1	`Ax = λB*x`	`A` overwritten by `inv(U^H)Ainv(U)` or `inv(L)Ainv(L^H)`
2	`ABx = λ*x`	`A` overwritten by `UAU^H` or `L^HAL`
3	`BAx = λ*x`	`A` overwritten by `UAU^H` or `L^HAL`

Before calling this routine, you must call ?potrf to compute the Cholesky factorization: B = U^H*U or B = L*L^H.

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 = (U^H)^-1*A*U^-1;

for uplo = 'L': C = L^-1*A*(L^H)^-1.

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

for uplo = 'U': C = U*A*U^H;

for uplo = 'L': C = L^H*A*L.

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

for uplo = 'U': C = U*A*U^H;

for uplo = 'L': C = L^H*A*L.

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 = U^H*U.

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

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 = U^H*U or B = L*L^H (as returned by ?potrf).

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

lda

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

ldb

INTEGER. The leading 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.

LAPACK 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 LAPACK 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 B^-1 (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: LAPACK Computational Routines

itype	Problem	Result
1	`Ax = λB*x`	`A` overwritten by `inv(U^H)Ainv(U)` or `inv(L)Ainv(L^H)`
2	`ABx = λ*x`	`A` overwritten by `UAU^H` or `L^HAL`
3	`BAx = λ*x`	`A` overwritten by `UAU^H` or `L^HAL`

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?hegst