?heevx

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?heevx

Computes selected eigenvalues and, optionally, eigenvectors of a Hermitian matrix.

Syntax

call cheevx(jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, rwork, iwork, ifail, info)

call zheevx(jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, rwork, iwork, ifail, info)

call heevx(a, w [,uplo] [,z] [,vl] [,vu] [,il] [,iu] [,m] [,ifail] [,abstol] [,info])

Include Files

mkl.fi, lapack.f90

Description

The routine computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

Note that for most cases of complex Hermetian eigenvalue problems the default choice should be heevr function as its underlying algorithm is faster and uses less workspace. ?heevx is faster for a few selected eigenvalues.

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.

range

CHARACTER*1. Must be 'A', 'V', or 'I'.

If range = 'A', all eigenvalues will be found.

If range = 'V', all eigenvalues in the half-open interval (vl, vu] will be found.

If range = 'I', the eigenvalues with indices il through iu will be found.

uplo

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

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

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

n

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

a, work

COMPLEX for cheevx

DOUBLE COMPLEX for zheevx.

Arrays:

a(lda,*) is an array containing either upper or lower triangular part of the Hermitian matrix A, as specified by uplo.

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

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

lda

INTEGER. The leading dimension of the array a. Must be at least max(1, n).

vl, vu

REAL for cheevx

DOUBLE PRECISION for zheevx.

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, iu

INTEGER.

If range = 'I', the indices of the smallest and largest eigenvalues to be returned. Constraints:

1 ≤il≤iu≤n, if n > 0;il = 1 and iu = 0, if n = 0. Not referenced if range = 'A'or 'V'.

abstol

REAL for cheevx

DOUBLE PRECISION for zheevx. The absolute error tolerance for the eigenvalues. See Application Notes for more information.

ldz

INTEGER. The leading dimension of the output array z; ldz≥ 1.

If jobz = 'V', then ldz≥max(1, n).

lwork

INTEGER.

The dimension of the array work.

lwork≥ 1 if n≤1; otherwise at least 2*n.

If lwork = -1, then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued by xerbla.

See Application Notes for the suggested value of lwork.

rwork

REAL for cheevx

DOUBLE PRECISION for zheevx.

Workspace array, size at least max(1, 7n).

iwork

INTEGER. Workspace array, size at least max(1, 5n).

Output Parameters

a

On exit, the lower triangle (if uplo = 'L') or the upper triangle (if uplo = 'U') of A, including the diagonal, is overwritten.

m

INTEGER. The total number of eigenvalues found; 0 ≤m≤n.

If range = 'A', m = n, and if range = 'I', m = iu-il+1.

w

REAL for cheevx

DOUBLE PRECISION for zheevx

Array, size max(1, n). The first m elements contain the selected eigenvalues of the matrix A in ascending order.

z

COMPLEX for cheevx

DOUBLE COMPLEX for zheevx.

Array z(ldz,*) contains eigenvectors.

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

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).

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.

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

work(1)

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

ifail

INTEGER.

Array, size at least max(1, 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 = 'V', then ifail is not referenced.

info

INTEGER.

If info = 0, the execution is successful.

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

If info = i, then i eigenvectors failed to converge; their indices are stored in the array ifail.

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 heevx interface are the following:

a: Holds the matrix A of size (n, n).
w: Holds the vector of length n.
z: Holds the matrix Z of size (n, n).
ifail: Holds the vector of length n.
uplo: Must be 'U' or 'L'. The default value is 'U'.
vl: Default value for this element is vl = -HUGE(vl).
vu: Default value for this element is vu = HUGE(vl).
il: Default value for this argument is il = 1.
iu: Default value for this argument is iu = n.
abstol: Default value for this element is abstol = 0.0_WP.
jobz: Restored based on the presence of the argument z as follows: jobz = 'V', if z is present, jobz = 'N', if z is omitted Note that there will be an error condition if ifail is present and z is omitted.
range: Restored based on the presence of arguments vl, vu, il, iu as follows: range = 'V', if one of or both vl and vu are present, range = 'I', if one of or both il and iu are present, range = 'A', if none of vl, vu, il, iu is present, Note that there will be an error condition if one of or both vl and vu are present and at the same time one of or both il and iu are present.

Application Notes

For optimum performance use lwork≥ (nb+1)*n, where nb is the maximum of the blocksize for ?hetrd and ?unmtr returned by ilaenv.

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

If you choose the first option and set any of admissible lwork 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 on exit. Use this value (work(1)) for subsequent runs.

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

Note that if you set lwork 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.

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+ε*max(|a|,|b|), where ε is the machine precision.

If abstol is less than or equal to zero, then ε*||T|| will be used in its place, where ||T|| is the 1-norm of the tridiagonal matrix obtained by reducing A to tridiagonal form. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold 2*?lamch('S'), not zero.

If this routine returns with info > 0, indicating that some eigenvectors did not converge, try setting abstol to 2*?lamch('S').

Parent topic: Symmetric Eigenvalue Problems: LAPACK Driver Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?heevx