?spgst

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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?spgst

Reduces a real symmetric-definite generalized eigenvalue problem to the standard form using packed storage.

Syntax

call sspgst(itype, uplo, n, ap, bp, info)

call dspgst(itype, uplo, n, ap, bp, info)

call spgst(ap, bp [,itype] [,uplo] [,info])

Include Files

mkl.fi, lapack.f90

Description

The routine reduces real symmetric-definite generalized eigenproblems

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

to the standard form C*y = λ*y, using packed matrix storage. Here A is a real symmetric matrix, and B is a real symmetric positive-definite matrix. Before calling this routine, call ?pptrf to compute the Cholesky factorization: B = U^T*U or B = L*L^T.

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(U^T)*A*inv(U), z = inv(U)*y;

for uplo = 'L': C = inv(L)*A*inv(L^T), z = inv(L^T)*y.

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

for uplo = 'U': C = U*A*U^T, z = inv(U)*y;

for uplo = 'L': C = L^T*A*L, z = inv(L^T)*y.

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

for uplo = 'U': C = U*A*U^T, z = U^T*y;

for uplo = 'L': C = L^T*A*L, z = L*y.

uplo

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

If uplo = 'U', ap stores the packed upper triangle of A;

you must supply B in the factored form B = U^T*U.

If uplo = 'L', ap stores the packed lower triangle of A;

you must supply B in the factored form B = L*L^T.

n

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

ap, bp

REAL for sspgst

DOUBLE PRECISION for dspgst.

Arrays:

ap(*) contains the packed upper or lower triangle of A.

The dimension of ap must be at least max(1, n*(n+1)/2).

bp(*) contains the packed Cholesky factor of B (as returned by ?pptrf with the same uplo value).

The dimension of bp must be at least max(1, n*(n+1)/2).

Output Parameters

ap

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 spgst interface are the following:

ap: Holds the array A of size (n*(n+1)/2).
bp: Holds the array B of size (n*(n+1)/2).
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: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?spgst