?steqr

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 11/07/2023

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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.

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 steqr interface are the following:

d

Holds the vector of length n.

e

Holds the vector of length (n-1).

z

Holds the matrix Z of size (n,n).

compz

If omitted, this argument is restored based on the presence of argument z as follows:

compz = 'I', if z is present,

compz = 'N', if z is omitted.

If present, compz must be equal to 'I' or 'V' and the argument z must also be present. Note that there will be an error condition if compz is present and z omitted.

Note that two variants of Fortran 95 interface for steqr routine are needed because of an ambiguous choice between real and complex cases appear when z is omitted. Thus, the name rsteqr is used in real cases (single or double precision), and the name steqr is used in complex cases (single or double precision).

Application Notes

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

If λ_i is an exact eigenvalue, and μ_i is the corresponding computed value, then

|μ_i - λ_i| ≤c(n)*ε*||T||₂

where c(n) is a modestly increasing function of n.

If z_i is the corresponding exact eigenvector, and w_i is the corresponding computed vector, then the angle θ(z_i, w_i) between them is bounded as follows:

θ(z_i, w_i) ≤c(n)*ε*||T||₂ / min_i≠j|λ_i - λ_j|.

The total number of floating-point operations depends on how rapidly the algorithm converges. Typically, it is about

24n² if compz = 'N';

7n³ (for complex flavors, 14n³) if compz = 'V' or 'I'.

Parent topic: Symmetric Eigenvalue Problems: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?steqr

Syntax

Include Files

Description

Input Parameters

Output Parameters

LAPACK 95 Interface Notes

Application Notes