Computes selected eigenvalues and eigenvectors of a real symmetric tridiagonal matrix.

Syntax

lapack_int LAPACKE_sstevx (int matrix_layout, char jobz, char range, lapack_int n, float* d, float* e, float vl, float vu, lapack_int il, lapack_int iu, float abstol, lapack_int* m, float* w, float* z, lapack_int ldz, lapack_int* ifail);

lapack_int LAPACKE_dstevx (int matrix_layout, char jobz, char range, lapack_int n, double* d, double* e, double vl, double vu, lapack_int il, lapack_int iu, double abstol, lapack_int* m, double* w, double* z, lapack_int ldz, lapack_int* ifail);

Include Files

  • mkl.h

Description

The routine computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

Input Parameters

matrix_layout

Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).

jobz

Must be 'N' or 'V'.

If job = 'N', then only eigenvalues are computed.

If job = 'V', then eigenvalues and eigenvectors are computed.

range

Must be 'A' or 'V' or 'I'.

If range = 'A', the routine computes all eigenvalues.

If range = 'V', the routine computes eigenvalues w[i] in the half-open interval: vl<w[i]vu.

If range = 'I', the routine computes eigenvalues with indices il to iu.

n

The order of the matrix A (n 0).

d, e

Arrays:

d contains the n diagonal elements of the tridiagonal matrix A.

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

e contains the n-1 subdiagonal elements of A.

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

vl, vu

If range = 'V', the lower and upper bounds of the interval to be searched for eigenvalues.

Constraint: vl< vu.

If range = 'A' or 'I', vl and vu are not referenced.

il, iu

If range = 'I', the indices in ascending order of the smallest and largest eigenvalues to be returned.

Constraint: 1 iliun, if n > 0; il=1 and iu=0 if n = 0.

If range = 'A' or 'V', il and iu are not referenced.

abstol
ldz

The leading dimensions of the output array z; ldz 1. If jobz = 'V', then ldz max(1, n) for column major layout and ldz max(1, m) for row major layout.

Output Parameters

m

The total number of eigenvalues found,

0 mn.

If range = 'A', m = n, if range = 'I', m = iu-il+1, and if range = 'V' the exact value of m is unknown.

w, z

Arrays:

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

The first m elements of w contain the selected eigenvalues of the matrix A in ascending order.

z(size at least max(1, ldz*m) for column major layout and max(1, ldz*n) for row major layout) .

If jobz = 'V', then if info = 0, the first m columns of z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of z holding the eigenvector associated with w[i - 1].

If an eigenvector fails to converge, then that column of z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in ifail.

If jobz = 'N', then z is not referenced.

d, e

On exit, these arrays may be multiplied by a constant factor chosen to avoid overflow or underflow in computing the eigenvalues.

ifail

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

If jobz = 'V', then if info = 0, the first m elements of ifail are zero; if info > 0, the ifail contains the indices of the eigenvectors that failed to converge.

If jobz = 'N', then ifail is not referenced.

Return Values

This function returns a value info.

If info=0, the execution is successful.

If info = -i, the i-th parameter had an illegal value.

If info = i, then i eigenvectors failed to converge; their indices are stored in the array ifail.

Application Notes

An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to abstol+ε*max(|a|,|b|), where ε is the machine precision.

If abstol is less than or equal to zero, then ε*|A|1 is used instead. Eigenvalues are computed most accurately when abstol is set to twice the underflow threshold 2*?lamch('S'), not zero.

If this routine returns with info > 0, indicating that some eigenvectors did not converge, set abstol to 2*?lamch('S').

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