# ?stevx

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

## Syntax

FORTRAN 77:

call sstevx(jobz, range, n, d, e, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info)

call dstevx(jobz, range, n, d, e, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info)

Fortran 95:

call stevx(d, e, w [, z] [,vl] [,vu] [,il] [,iu] [,m] [,ifail] [,abstol] [,info])

## Description

This 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

jobz

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

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

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

range

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

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

  If `range = 'V'`, the routine computes eigenvalues lambda`(i)` in the half-open interval: `vl<lambda`(i)`≤ vu`.

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

n

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

d, e, work

  REAL for sstevx

  DOUBLE PRECISION for dstevx.

  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.

  work(*) is a workspace array.

  The dimension of work must be at least max(1, 5n).

vl, vu

  REAL for sstevx

  DOUBLE PRECISION for dstevx.

  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

  INTEGER.

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

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

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

abstol

  REAL for sstevx

  DOUBLE PRECISION for dstevx. The absolute error tolerance to which each eigenvalue is required. See Application notes for details on error tolerance.

ldz

  INTEGER. The leading dimensions of the output array z; `ldz ≥ 1`. If `jobz = 'V'`, then `ldz ≥ max`(1, n).

iwork

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

## Output Parameters

m

  INTEGER. The total number of eigenvalues found,

  `0 ≤ m ≤ n`.

  If `range = 'A'`, `m = n`, and if `range = 'I'`, `m = iu-il+1`.

w, z

  REAL for sstevx

  DOUBLE PRECISION for dstevx.

  Arrays:

  w(*), DIMENSION at least max(1, n).

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

  z(ldz,*).

  The second dimension of z must be at least max(1, m).

  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).

  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.

  Note: you must ensure that at least max(1,m) columns are supplied in the array z; if `range = 'V'`, the exact value of m is not known in advance and an upper bound must be used.

d, e

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

ifail

  INTEGER.

  Array, DIMENSION 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.

info

  INTEGER.

  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.

## Fortran 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 Fortran 95 Interface Conventions.

Specific details for the routine stevx 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).

z

  Holds the matrix Z of size (n, n), where the values n and m are significant.

ifail

  Holds the vector of length (n).

vl

  Default value for this element is vl = -HUGE(vl).

vu

  Default value for this element is vu = HUGE(vl).

il

  Default value for this argument is `il = 1`.

iu

  Default value for this argument is `iu = n`.

abstol

  Default value for this element is `abstol = 0.0_WP`.

jobz

  Restored based on the presence of the argument z as follows:

  `jobz = 'V'`, if z is present,

  `jobz = 'N'`, if z is omitted

  Note that there will be an error condition if ifail is present and z is omitted.

range

  Restored based on the presence of arguments vl, vu, il, iu as follows:

  `range = 'V'`, if one of or both vl and vu are present,

  `range = 'I'`, if one of or both il and iu are present,

  `range = 'A'`, if none of vl, vu, il, iu is present,

  Note that there will be an error condition if one of or both vl and vu are present and at the same time one of or both il and iu are present.

## 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').

Parent topic: Symmetric Eigenproblems