?steqr

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?steqr

Computes all eigenvalues and eigenvectors of a symmetric or Hermitian matrix reduced to tridiagonal form (QR algorithm).

Syntax

call ssteqr(compz, n, d, e, z, ldz, work, info)

call dsteqr(compz, n, d, e, z, ldz, work, info)

call csteqr(compz, n, d, e, z, ldz, work, info)

call zsteqr(compz, n, d, e, z, ldz, work, info)

call rsteqr(d, e [,z] [,compz] [,info])

call steqr(d, e [,z] [,compz] [,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: T = Z*Λ*Z^T. Here Λ is a diagonal matrix whose diagonal elements are the eigenvalues λ_i; Z is an orthogonal matrix whose columns are eigenvectors. Thus,

T*z_i = λ_i*z_i for i = 1, 2, ..., n.

The routine normalizes the eigenvectors so that ||z_i||₂ = 1.

You can also use the routine for computing the eigenvalues and eigenvectors of an arbitrary real symmetric (or complex Hermitian) matrix A reduced to tridiagonal form T: A = Q*T*Q^H. In this case, the spectral factorization is as follows: A = Q*T*Q^H = (Q*Z)*Λ*(Q*Z)^H. Before calling ?steqr, you must reduce A to tridiagonal form and generate the explicit matrix Q by calling the following routines:

	for real matrices:	for complex matrices:
full storage	?sytrd, ?orgtr	?hetrd, ?ungtr
packed storage	?sptrd, ?opgtr	?hptrd, ?upgtr
band storage	?sbtrd`(vect`='V')	?hbtrd`(vect`='V')

If you need eigenvalues only, it's more efficient to call sterf. If T is positive-definite, pteqr can compute small eigenvalues more accurately than ?steqr.

To solve the problem by a single call, use one of the divide and conquer routines stevd, syevd, spevd, or sbevd for real symmetric matrices or heevd, hpevd, or hbevd for complex Hermitian matrices.

Input Parameters

compz

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

If compz = 'N', the routine computes eigenvalues only.

If compz = 'I', the routine computes the eigenvalues and eigenvectors of the tridiagonal matrix T.

If compz = 'V', the routine computes the eigenvalues and eigenvectors of the original symmetric matrix. On entry, z must contain the orthogonal matrix used to reduce the original matrix to tridiagonal form.

n

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

d, e, work

REAL for single-precision flavors

DOUBLE PRECISION for double-precision flavors.

Arrays:

d(*) contains the diagonal elements of T.

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

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

The size of e must be at least max(1, n-1).

work(*) is a workspace array.

The size of work must be:

at least 1 if compz = 'N';

at least max(1, 2*n-2) if compz = 'V' or 'I'.

z

REAL for ssteqr

DOUBLE PRECISION for dsteqr

COMPLEX for csteqr

DOUBLE COMPLEX for zsteqr.

Array, size (ldz, *).

If compz = 'N' or 'I', z need not be set.

If vect = 'V', z must contain the orthogonal matrix used in the reduction to tridiagonal form.

The second dimension of z must be:

at least 1 if compz = 'N';

at least max(1, n) if compz = 'V' or 'I'.

work (lwork) is a workspace array.

ldz

INTEGER. The leading dimension of z. Constraints:

ldz≥ 1 if compz = 'N';

ldz≥ max(1, n) if compz = 'V' or 'I'.

Output Parameters

d

The n eigenvalues in ascending order, unless info > 0.

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?steqr