?hbevd

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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Date 6/24/2024

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?hbevd

Computes all eigenvalues and, optionally, all eigenvectors of a complex Hermitian band matrix using divide and conquer algorithm.

Syntax

call chbevd(jobz, uplo, n, kd, ab, ldab, w, z, ldz, work, lwork, rwork, lrwork, iwork, liwork, info)

call zhbevd(jobz, uplo, n, kd, ab, ldab, w, z, ldz, work, lwork, rwork, lrwork, iwork, liwork, info)

call hbevd(ab, w [,uplo] [,z] [,info])

Include Files

mkl.fi, lapack.f90

Description

The routine computes all the eigenvalues, and optionally all the eigenvectors, of a complex Hermitian band matrix A. In other words, it can compute the spectral factorization of A as: A = Z*Λ*Z^H.

Here Λ is a real diagonal matrix whose diagonal elements are the eigenvalues λ_i, and Z is the (complex) unitary matrix whose columns are the eigenvectors z_i. Thus,

A*z_i = λ_i*z_i for i = 1, 2, ..., n.

If the eigenvectors are requested, then this routine uses a divide and conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal-Walker-Kahan variant of the QL or QR algorithm.

Input Parameters

jobz

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

If jobz = 'N', then only eigenvalues are computed.

If jobz = 'V', then eigenvalues and eigenvectors are computed.

uplo

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

If uplo = 'U', ab stores the upper triangular part of A.

If uplo = 'L', ab stores the lower triangular part of A.

n

INTEGER. The order of the matrix A (n≥ 0).

kd

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

(kd≥ 0).

ab, work

COMPLEX for chbevd

DOUBLE COMPLEX for zhbevd.

Arrays:

ab(lda,*) 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 ab must be at least max(1, n).

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

ldab

INTEGER. The leading dimension of ab; must be at least kd+1.

ldz

INTEGER. The leading dimension of the output array z.

Constraints:

if jobz = 'N', then ldz≥ 1;

if jobz = 'V', then ldz≥ max(1, n) .

lwork

INTEGER.

The dimension of the array work.

Constraints:

if n≤ 1, then lwork≥ 1;

if jobz = 'N' and n > 1, then lwork≥n;

if jobz = 'V' and n > 1, then lwork≥ 2*n².

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 chbevd

DOUBLE PRECISION for zhbevd

Workspace array, size at least lrwork.

lrwork

INTEGER.

The dimension of the array rwork.

Constraints:

if n≤ 1, then lrwork≥ 1;

if jobz = 'N' and n > 1, then lrwork≥n;

if jobz = 'V' and n > 1, then lrwork≥ 2*n² + 5*n + 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, size max(1, liwork).

liwork

INTEGER.

The dimension of the array iwork.

Constraints:

if jobz = 'N' or n≤ 1, then liwork≥ 1;

if jobz = 'V' and n > 1, then liwork≥ 5*n+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

w

REAL for chbevd

DOUBLE PRECISION for zhbevd

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

If info = 0, contains the eigenvalues of the matrix A in ascending order. See also info.

z

COMPLEX for chbevd

DOUBLE COMPLEX for zhbevd

Array, size (ldz,*).

The second dimension of z must be:

at least 1 if jobz = 'N';

at least max(1, n) if jobz = 'V'.

If jobz = 'V', then this array is overwritten by the unitary matrix Z which contains the eigenvectors of A. The i-th column of Z contains the eigenvector which corresponds to the eigenvalue w(i).

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

ab

On exit, this array is overwritten by the values generated during the reduction to tridiagonal form.

work(1)

On exit, if lwork > 0, then the real part of work(1) returns the required minimal size of lwork.

rwork(1)

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

iwork(1)

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

info

INTEGER.

If info = 0, the execution is successful.

If info = i, then the algorithm failed to converge; i indicates the number of elements of an intermediate tridiagonal form which did not converge to zero.

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 hbevd interface are the following:

ab

Holds the array A of size (kd+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

The computed eigenvalues and eigenvectors are exact for a matrix A + E such that ||E||₂ = O(ε)||A||₂, where ε is the machine precision.

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.

The real analogue of this routine is sbevd.

See also heevd for matrices held in full storage, and hpevd for matrices held in packed storage.

Parent topic: Symmetric Eigenvalue Problems: LAPACK Driver Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?hbevd

Syntax

Include Files

Description

Input Parameters

Output Parameters

LAPACK 95 Interface Notes

Application Notes