?stein

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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?stein

Computes the eigenvectors corresponding to specified eigenvalues of a real symmetric tridiagonal matrix.

Syntax

call sstein(n, d, e, m, w, iblock, isplit, z, ldz, work, iwork, ifailv, info)

call dstein(n, d, e, m, w, iblock, isplit, z, ldz, work, iwork, ifailv, info)

call cstein(n, d, e, m, w, iblock, isplit, z, ldz, work, iwork, ifailv, info)

call zstein(n, d, e, m, w, iblock, isplit, z, ldz, work, iwork, ifailv, info)

call stein(d, e, w, iblock, isplit, z [,ifailv] [,info])

Include Files

mkl.fi, lapack.f90

Description

The routine computes the eigenvectors of a real symmetric tridiagonal matrix T corresponding to specified eigenvalues, by inverse iteration. It is designed to be used in particular after the specified eigenvalues have been computed by ?stebz with order = 'B', but may also be used when the eigenvalues have been computed by other routines.

If you use this routine after ?stebz, it can take advantage of the block structure by performing inverse iteration on each block T_i separately, which is more efficient than using the whole matrix T.

If T has been formed by reduction of a full symmetric or Hermitian matrix A to tridiagonal form, you can transform eigenvectors of T to eigenvectors of A by calling ?ormtr or ?opmtr (for real flavors) or by calling ?unmtr or ?upmtr (for complex flavors).

Input Parameters

n

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

m

INTEGER. The number of eigenvectors to be returned.

d, e, w

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 sub-diagonal elements of T stored in elements 1 to n-1

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

w(*) contains the eigenvalues of T, stored in w(1) to w(m) (as returned by stebz). Eigenvalues of T₁ must be supplied first, in non-decreasing order; then those of T₂, again in non-decreasing order, and so on. Constraint:

if iblock(i) = iblock(i+1), w(i) ≤w(i+1).

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

iblock, isplit

INTEGER.

Arrays, size at least max(1, n). The arrays iblock and isplit, as returned by ?stebz with order = 'B'.

If you did not call ?stebz with order = 'B', set all elements of iblock to 1, and isplit(1) to n.)

ldz

INTEGER. The leading dimension of the output array z; ldz≥ max(1, n).

work

REAL for single-precision flavors

DOUBLE PRECISION for double-precision flavors.

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

iwork

INTEGER.

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

Output Parameters

z

REAL for sstein

DOUBLE PRECISION for dstein

COMPLEX for cstein

DOUBLE COMPLEX for zstein.

Array, size (ldz, *).

If info = 0, z contains an n-by-n matrix the columns of which are orthonormal eigenvectors. (The i-th column corresponds to the ith eigenvalue.)

ifailv

INTEGER.

Array, size at least max(1, m).

If info = i > 0, the first i elements of ifailv contain the indices of any eigenvectors that failed to converge.

info

INTEGER.

If info = 0, the execution is successful.

If info = i, then i eigenvectors (as indicated by the parameter ifailv) each failed to converge in 5 iterations. The current iterates are stored in the corresponding columns of the array z.

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 stein interface are the following:

d: Holds the vector of length n.
e: Holds the vector of length n.
w: Holds the vector of length n.
iblock: Holds the vector of length n.
isplit: Holds the vector of length n.
z: Holds the matrix Z of size (n,m).
ifailv: Holds the vector of length (m).

Application Notes

Each computed eigenvector z_i is an exact eigenvector of a matrix T+E_i, where ||E_i||₂ = O(ε)*||T||₂. However, a set of eigenvectors computed by this routine may not be orthogonal to so high a degree of accuracy as those computed by ?steqr.

Parent topic: Symmetric Eigenvalue Problems: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?stein