?hbgvd

Computes all eigenvalues and, optionally, eigenvectors of a complex generalized Hermitian positive-definite eigenproblem with banded matrices. If eigenvectors are desired, it uses a divide and conquer method.

Syntax

FORTRAN 77:

call chbgvd(jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, w, z, ldz, work, lwork, rwork, lrwork, iwork, liwork, info)

call zhbgvd(jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, w, z, ldz, work, lwork, rwork, lrwork, iwork, liwork, info)

FORTRAN 95:

call hbgvd(ab, bb, w [,uplo] [,z] [,info])

C:

lapack_int LAPACKE_chbgvd( int matrix_layout, char jobz, char uplo, lapack_int n, lapack_int ka, lapack_int kb, lapack_complex_float* ab, lapack_int ldab, lapack_complex_float* bb, lapack_int ldbb, float* w, lapack_complex_float* z, lapack_int ldz );

lapack_int LAPACKE_zhbgvd( int matrix_layout, char jobz, char uplo, lapack_int n, lapack_int ka, lapack_int kb, lapack_complex_double* ab, lapack_int ldab, lapack_complex_double* bb, lapack_int ldbb, double* w, lapack_complex_double* z, lapack_int ldz );

Include Files

  • Fortran: mkl.fi
  • Fortran 95: lapack.f90
  • C: mkl.h

Description

The routine computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian positive-definite banded eigenproblem, of the form A*x = λ*B*x. Here A and B are assumed to be Hermitian and banded, and B is also positive definite.

If eigenvectors are desired, it uses a divide and conquer algorithm.

Input Parameters

The data types are given for the Fortran interface. A <datatype> placeholder, if present, is used for the C interface data types in the C interface section above. See C Interface Conventions for the C interface principal conventions and type definitions.

jobz

CHARACTER*1. Must be 'N' or 'V'.

If jobz = 'N', then compute eigenvalues only.

If jobz = 'V', then compute eigenvalues and eigenvectors.

uplo

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

If uplo = 'U', arrays ab and bb store the upper triangles of A and B;

If uplo = 'L', arrays ab and bb store the lower triangles of A and B.

n

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

ka

INTEGER. The number of super- or sub-diagonals in A

(ka0).

kb

INTEGER. The number of super- or sub-diagonals in B (kb 0).

ab, bb, work

COMPLEX for chbgvd

DOUBLE COMPLEX for zhbgvd

Arrays:

ab (ldab,*) is an array containing either upper or lower triangular part of the Hermitian matrix A (as specified by uplo) in band storage format.

The second dimension of the array ab must be at least max(1, n).

bb(ldbb,*) is an array containing either upper or lower triangular part of the Hermitian matrix B (as specified by uplo) in band storage format.

The second dimension of the array bb must be at least max(1, n).

work is a workspace array, its dimension max(1, lwork).

ldab

INTEGER. The leading dimension of the array ab; must be at least ka+1.

ldbb

INTEGER. The leading dimension of the array bb; must be at least kb+1.

ldz

INTEGER. The leading dimension of the output array z; ldz 1. If jobz = 'V', ldz max(1, n).

lwork

INTEGER.

The dimension of the array work.

Constraints:

If n 1, lwork 1;

If jobz = 'N' and n>1, lwork n;

If jobz = 'V' and n>1, lwork 2n2.

If lwork = -1, then a workspace query is assumed; the routine only calculates the optimal size of the work, rwork and iwork arrays, returns these values as the first entries of the work, rwork and iwork arrays, and no error message related to lwork or lrwork or liwork is issued by xerbla. See Application Notes for details.

rwork

REAL for chbgvd

DOUBLE PRECISION for zhbgvd.

Workspace array, DIMENSION max(1, lrwork).

lrwork

INTEGER.

The dimension of the array rwork.

Constraints:

If n 1, lrwork 1;

If jobz = 'N' and n>1, lrwork n;

If jobz = 'V' and n>1, lrwork 2n2+5n +1.

If lrwork = -1, then a workspace query is assumed; the routine only calculates the optimal size of the work, rwork and iwork arrays, returns these values as the first entries of the work, rwork and iwork arrays, and no error message related to lwork or lrwork or liwork is issued by xerbla. See Application Notes for details.

iwork

INTEGER.

Workspace array, DIMENSION max(1, liwork).

liwork

INTEGER.

The dimension of the array iwork.

Constraints:

If n 1, lwork 1;

If jobz = 'N' and n>1, liwork 1;

If jobz = 'V' and n>1, liwork 5n+3.

If liwork = -1, then a workspace query is assumed; the routine only calculates the optimal size of the work, rwork and iwork arrays, returns these values as the first entries of the work, rwork and iwork arrays, and no error message related to lwork or lrwork or liwork is issued by xerbla. See Application Notes for details.

Output Parameters

ab

On exit, the contents of ab are overwritten.

bb

On exit, contains the factor S from the split Cholesky factorization B = SH*S, as returned by pbstf/pbstf.

w

REAL for chbgvd

DOUBLE PRECISION for zhbgvd.

Array, DIMENSION at least max(1, n) .

If info = 0, contains the eigenvalues in ascending order.

z

COMPLEX for chbgvd

DOUBLE COMPLEX for zhbgvd

Array z(ldz,*) .

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

If jobz = 'V', then if info = 0, z contains the matrix Z of eigenvectors, with the i-th column of z holding the eigenvector associated with w(i). The eigenvectors are normalized so that ZH*B*Z = I.

If jobz = 'N', then z is not referenced.

work(1)

On exit, if info = 0, then work(1) returns the required minimal size of lwork.

rwork(1)

On exit, if info = 0, then rwork(1) returns the required minimal size of lrwork.

iwork(1)

On exit, if info = 0, then iwork(1) returns the required minimal size of liwork.

info

INTEGER.

If info = 0, the execution is successful.

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

If info > 0, and

if i n, the algorithm failed to converge, and i off-diagonal elements of an intermediate tridiagonal did not converge to zero;

if info = n + i, for 1 i n, then pbstf/pbstf returned info = i and B is not positive-definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.

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 hbgvd interface are the following:

ab

Holds the array A of size (ka+1,n).

bb

Holds the array B of size (kb+1,n).

w

Holds the vector with the number of elements n.

z

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

uplo

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

jobz

Restored based on the presence of the argument z as follows:

jobz = 'V', if z is present,

jobz = 'N', if z is omitted.

Application Notes

If you are in doubt how much workspace to supply, use a generous value of lwork (liwork or lrwork) for the first run or set lwork = -1 (liwork = -1, lrwork = -1).

If you choose the first option and set any of admissible lwork (liwork or lrwork) sizes, which is no less than the minimal value described, the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array (work, iwork, rwork) on exit. Use this value (work(1), iwork(1), rwork(1)) for subsequent runs.

If you set lwork = -1 (liwork = -1, lrwork = -1), the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work, iwork, rwork). This operation is called a workspace query.

Note that if you set lwork (liwork, lrwork) to less than the minimal required value and not -1, the routine returns immediately with an error exit and does not provide any information on the recommended workspace.

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