?stevd

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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?stevd

Computes all eigenvalues and, optionally, all eigenvectors of a real symmetric tridiagonal matrix using divide and conquer algorithm.

Syntax

call sstevd(jobz, n, d, e, z, ldz, work, lwork, iwork, liwork, info)

call dstevd(jobz, n, d, e, z, ldz, work, lwork, iwork, liwork, info)

call stevd(d, e [,z] [,info])

Include Files

mkl.fi, lapack.f90

Description

The routine computes all the eigenvalues, and optionally all the eigenvectors, of a real symmetric tridiagonal matrix T. In other words, the routine can compute the spectral factorization of T as: T = 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,

T*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.

There is no complex analogue of this routine.

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.

n

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

d, e, work

REAL for sstevd

DOUBLE PRECISION for dstevd.

Arrays:

d(*) contains the n diagonal elements of the tridiagonal matrix T.

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

e(*) contains the n-1 off-diagonal elements of T.

The dimension of e must be at least max(1, n). The n-th element of this array is used as workspace.

work(*) is a workspace array.

The dimension of work must be at least lwork.

ldz

INTEGER. The leading dimension of the output array z. Constraints:

ldz≥ 1 if job = 'N';

ldz≥ max(1, n) if job = 'V'.

lwork

INTEGER.

The dimension of the array work.

Constraints:

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

if jobz = 'V' and n > 1, then lwork ≥n²+ 4*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 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 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

d

On exit, if info = 0, contains the eigenvalues of the matrix T in ascending order.

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?stevd