# ?stevr

Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix using the Relatively Robust Representations.

## Syntax

FORTRAN 77:

call sstevr(jobz, range, n, d, e, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, iwork, liwork, info)

call dstevr(jobz, range, n, d, e, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, iwork, liwork, info)

Fortran 95:

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

## Description

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

Whenever possible, the routine calls ?stemr to compute the eigenspectrum using Relatively Robust Representations. ?stegr computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various “good” L*D*LT representations (also known as Relatively Robust Representations). Gram-Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows. For the i-th unreduced block of T:

1. Compute `T - σi = Li*Di*LiT`, such that Li*Di*LiT is a relatively robust representation.

2. Compute the eigenvalues, λj, of Li*Di*LiT to high relative accuracy by the dqds algorithm.

3. If there is a cluster of close eigenvalues, “choose” σi close to the cluster, and go to Step (a).

4. Given the approximate eigenvalue λj of Li*Di*LiT, compute the corresponding eigenvector by forming a rank-revealing twisted factorization.

The desired accuracy of the output can be specified by the input parameter abstol.

The routine ?stevr calls ?stemr when the full spectrum is requested on machines which conform to the IEEE-754 floating point standard. ?stevr calls ?stebz and ?stein on non-IEEE machines and when partial spectrum requests are made.

## Input Parameters

jobz

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

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

  If `jobz = '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.

  For `range = 'V'`or 'I' and iu-il < n-1, sstebz/dstebz and sstein/dstein are called.

n

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

d, e, work

  REAL for sstevr

  DOUBLE PRECISION for dstevr.

  Arrays:

  d(*) contains the n diagonal elements of the tridiagonal matrix T.

  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, its dimension `max(1, lwork)`.
vl, vu

  REAL for sstevr

  DOUBLE PRECISION for dstevr.

  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 ssyevr

  DOUBLE PRECISION for dsyevr.

  The absolute error tolerance to which each eigenvalue/eigenvector is required.

  If `jobz = 'V'`, the eigenvalues and eigenvectors output have residual norms bounded by abstol, and the dot products between different eigenvectors are bounded by abstol. If `abstol < n *eps*|T|`, then `n *eps*|T|` will be used in its place, where `eps` is the machine precision, and `|T|` is the 1-norm of the matrix T. The eigenvalues are computed to an accuracy of`eps*|T|` irrespective of abstol.

  If high relative accuracy is important, set abstol to ?lamch('S').

ldz

  INTEGER. The leading dimension of the output array z.

  Constraints:

  `ldz ≥ 1` if `jobz = 'N'`;

  `ldz ≥ max(1, n)` if `jobz = 'V'`.

lwork

  INTEGER.

  The dimension of the array work. Constraint:

  `lwork ≥ max(1, 20*n)`.

  If `lwork = -1`, then a workspace query is assumed; the routine only calculates the required sizes of the work and iwork arrays, returns these values as the first entries of the work and iwork arrays, and no error message related to lwork or liwork is issued by xerbla. See Application Notes for details.

iwork

  INTEGER.

  Workspace array, its dimension `max(1, liwork)`.

liwork

  INTEGER.

  The dimension of the array iwork,

  `lwork ≥ max(1, 10*n)`.

  If `liwork = -1`, then a workspace query is assumed; the routine only calculates the required sizes of the work and iwork arrays, returns these values as the first entries of the work and iwork arrays, and no error message related to lwork or liwork is issued by xerbla. See Application Notes for details.

## 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 sstevr

  DOUBLE PRECISION for dstevr.

  Arrays:

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

  The first m elements of w contain the selected eigenvalues of the matrix T 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 T corresponding to the selected eigenvalues, with the i-th column of z holding the eigenvector associated with w(i).

  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.

isuppz

  INTEGER.

  Array, DIMENSION at least 2 *max(1, m).

  The support of the eigenvectors in z, i.e., the indices indicating the nonzero elements in z. The i-th eigenvector is nonzero only in elements isuppz( 2i-1) through isuppz( 2i ).

  Implemented only for `range = 'A'` or 'I' and `iu-il = n-1`.

work`(1)`

  On exit, if `info = 0`, then work`(1)` returns the required minimal size of lwork.

iwork`(1)`

  On exit, if `info = 0`, then iwork`(1)` returns the required minimal size of liwork.

info

  INTEGER.

  If `info = 0`, the execution is successful.

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

  If `info = i`, an internal error has occurred.

## 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 stevr 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.

isuppz

  Holds the vector of length (2*n), where the values (2*m) are significant.

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

Normal execution of the routine ?stegr may create `NaN`s and infinities and hence may abort due to a floating point exception in environments which do not handle NaNs and infinities in the IEEE standard default manner.

If it is not clear how much workspace to supply, use a generous value of lwork (or liwork) for the first run, or set `lwork = -1` (`liwork = -1`).

If lwork (or liwork) has any of admissible sizes, which is no less than the minimal value described, then the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array (work, iwork) on exit. Use this value (`work(1)`, `iwork(1)`) for subsequent runs.

If `lwork = -1` (`liwork = -1`), then the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work, iwork). This operation is called a workspace query.

Note that if lwork (liwork) is less than the minimal required value and is not equal to -1, then the routine returns immediately with an error exit and does not provide any information on the recommended workspace.

Parent topic: Symmetric Eigenproblems