?spevd

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?spevd

Uses divide and conquer algorithm to compute all eigenvalues and (optionally) all eigenvectors of a real symmetric matrix held in packed storage.

Syntax

call sspevd(jobz, uplo, n, ap, w, z, ldz, work, lwork, iwork, liwork, info)

call dspevd(jobz, uplo, n, ap, w, z, ldz, work, lwork, iwork, liwork, info)

call spevd(ap, w [,uplo] [,z] [,info])

Include Files

mkl.fi, lapack.f90

Description

The routine computes all the eigenvalues, and optionally all the eigenvectors, of a real symmetric matrix A (held in packed storage). In other words, it can compute the spectral factorization of A as:

A = Z*Λ*Z^T.

Here Λ is a diagonal matrix whose diagonal elements are the eigenvalues λ_i, and Z is the orthogonal 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', ap stores the packed upper triangular part of A.

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

n

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

ap, work

REAL for sspevd

DOUBLE PRECISION for dspevd

Arrays:

ap(*) contains the packed upper or lower triangle of symmetric matrix A, as specified by uplo.

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

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

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≥ 2*n;

if jobz = 'V' and n > 1, then

lwork≥n²+ 6*n + 1.

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

iwork

INTEGER. Workspace array, its dimension max(1, liwork).

liwork

INTEGER.

The dimension of the array iwork.

Constraints:

if n≤ 1, then liwork≥ 1;

if jobz = 'N' and 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 required sizes of the work and iwork arrays, returns these values as the first entries of the work and iwork arrays, and no error message related to lwork or liwork is issued by xerbla. See Application Notes for details.

Output Parameters

w, z

REAL for sspevd

DOUBLE PRECISION for dspevd

Arrays:

w(*), size at least max(1, n).

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

z(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 orthogonal matrix Z which contains the eigenvectors of A. If jobz = 'N', then z is not referenced.

ap

On exit, this array is overwritten by the values generated during the reduction to tridiagonal form. The elements of the diagonal and the off-diagonal of the tridiagonal matrix overwrite the corresponding elements of A.

work(1)

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

iwork(1)

On exit, if info = 0, then iwork(1) returns the required 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 spevd interface are the following:

ap

Holds the array A of size (n*(n+1)/2).

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 it is not clear how much workspace to supply, use a generous value of lwork (or liwork) for the first run or set lwork = -1 (liwork = -1).

If lwork (or liwork) has any of admissible 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) on exit. Use this value (work(1), iwork(1)) for subsequent runs.

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

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

The complex analogue of this routine is hpevd.

See also syevd for matrices held in full storage, and sbevd for banded matrices.

Parent topic: Symmetric Eigenvalue Problems: LAPACK Driver Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?spevd