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# chpgvx.3lapack

Langue: *en*

Version: *286269 (debian - 07/07/09)*

Section: *3 (Bibliothèques de fonctions)*

## NAME

CHPGVX - computes selected eigenvalues and, optionally, eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x## SYNOPSIS

- SUBROUTINE CHPGVX(
- ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK, IFAIL, INFO )

- CHARACTER JOBZ, RANGE, UPLO

- INTEGER IL, INFO, ITYPE, IU, LDZ, M, N

- REAL ABSTOL, VL, VU

- INTEGER IFAIL( * ), IWORK( * )

- REAL RWORK( * ), W( * )

- COMPLEX AP( * ), BP( * ), WORK( * ), Z( LDZ, * )

## PURPOSE

CHPGVX computes selected eigenvalues and, optionally, eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are assumed to be Hermitian, stored in packed format, and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.## ARGUMENTS

- ITYPE (input) INTEGER
- Specifies the problem type to be solved:

= 1: A*x = (lambda)*B*x

= 2: A*B*x = (lambda)*x

= 3: B*A*x = (lambda)*x - JOBZ (input) CHARACTER*1

= 'N': Compute eigenvalues only;

= 'V': Compute eigenvalues and eigenvectors.- RANGE (input) CHARACTER*1

= 'A': all eigenvalues will be found;

= 'V': all eigenvalues in the half-open interval (VL,VU] will be found; = 'I': the IL-th through IU-th eigenvalues will be found.- UPLO (input) CHARACTER*1

= 'U': Upper triangles of A and B are stored;

= 'L': Lower triangles of A and B are stored.- N (input) INTEGER
- The order of the matrices A and B. N >= 0.
- AP (input/output) COMPLEX array, dimension (N*(N+1)/2)
- On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. On exit, the contents of AP are destroyed.
- BP (input/output) COMPLEX array, dimension (N*(N+1)/2)
- On entry, the upper or lower triangle of the Hermitian matrix B, packed columnwise in a linear array. The j-th column of B is stored in the array BP as follows: if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j; if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n. On exit, the triangular factor U or L from the Cholesky factorization B = U**H*U or B = L*L**H, in the same storage format as B.
- VL (input) REAL
- VU (input) REAL If RANGE='V', the lower and upper bounds of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.
- IL (input) INTEGER
- IU (input) INTEGER If RANGE='I', the indices (in ascending order) of the smallest and largest eigenvalues to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.
- ABSTOL (input) REAL
- The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to ABSTOL + EPS * max( |a|,|b| ) , where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing AP to tridiagonal form. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*SLAMCH('S').
- M (output) INTEGER
- The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
- W (output) REAL array, dimension (N)
- On normal exit, the first M elements contain the selected eigenvalues in ascending order.
- Z (output) COMPLEX array, dimension (LDZ, N)
- If JOBZ = 'N', then Z is not referenced. If JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized as follows: if ITYPE = 1 or 2, Z**H*B*Z = I; if ITYPE = 3, Z**H*inv(B)*Z = I. If an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL. Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = 'V', the exact value of M is not known in advance and an upper bound must be used.
- LDZ (input) INTEGER
- The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
- WORK (workspace) COMPLEX array, dimension (2*N)
- RWORK (workspace) REAL array, dimension (7*N)
- IWORK (workspace) INTEGER array, dimension (5*N)
- IFAIL (output) INTEGER array, dimension (N)
- If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.
- INFO (output) INTEGER
- = 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: CPPTRF or CHPEVX returned an error code:

<= N: if INFO = i, CHPEVX failed to converge; i eigenvectors failed to converge. Their indices are stored in array IFAIL. > N: if INFO = N + i, for 1 <= i <= n, then the leading minor of order i of B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.

## FURTHER DETAILS

Based on contributions byMark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Contenus ©2006-2020 Benjamin Poulain

Design ©2006-2020 Maxime Vantorre