sstevr.f −

**Functions/Subroutines**

subroutine **sstevr** (JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO)

SSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

**subroutine sstevr (characterJOBZ, characterRANGE, integerN, real, dimension( * )D, real, dimension( * )E, realVL, realVU, integerIL, integerIU, realABSTOL, integerM, real, dimension( * )W, real, dimension( ldz, * )Z, integerLDZ, integer, dimension( * )ISUPPZ, real, dimension( * )WORK, integerLWORK, integer, dimension( * )IWORK, integerLIWORK, integerINFO)
SSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices**

**Purpose:**

SSTEVR 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, SSTEVR calls SSTEMR to compute the

eigenspectrum using Relatively Robust Representations. SSTEMR

computes eigenvalues by the dqds algorithm, while orthogonal

eigenvectors are computed from various "good" L D L^T 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,

(a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T

is a relatively robust representation,

(b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high

relative accuracy by the dqds algorithm,

(c) If there is a cluster of close eigenvalues, "choose" sigma_i

close to the cluster, and go to step (a),

(d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T,

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.

For more details, see "A new O(n^2) algorithm for the symmetric

tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon,

Computer Science Division Technical Report No. UCB//CSD-97-971,

UC Berkeley, May 1997.

Note 1 : SSTEVR calls SSTEMR when the full spectrum is requested

on machines which conform to the ieee-754 floating point standard.

SSTEVR calls SSTEBZ and SSTEIN on non-ieee machines and

when partial spectrum requests are made.

Normal execution of SSTEMR may create NaNs 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.

**Parameters:**

*JOBZ*

JOBZ is CHARACTER*1

= ’N’: Compute eigenvalues only;

= ’V’: Compute eigenvalues and eigenvectors.

*RANGE*

RANGE is CHARACTER*1

= ’A’: all eigenvalues will be found.

= ’V’: all eigenvalues in the half-open interval (VL,VU]

will be found.

= ’I’: the IL-th through IU-th eigenvalues will be found.

For RANGE = ’V’ or ’I’ and IU - IL < N - 1, SSTEBZ and

SSTEIN are called

*N*

N is INTEGER

The order of the matrix. N >= 0.

*D*

D is REAL array, dimension (N)

On entry, the n diagonal elements of the tridiagonal matrix

A.

On exit, D may be multiplied by a constant factor chosen

to avoid over/underflow in computing the eigenvalues.

*E*

E is REAL array, dimension (max(1,N-1))

On entry, the (n-1) subdiagonal elements of the tridiagonal

matrix A in elements 1 to N-1 of E.

On exit, E may be multiplied by a constant factor chosen

to avoid over/underflow in computing the eigenvalues.

*VL*

VL is REAL

*VU*

VU is REAL

If RANGE=’V’, the lower and upper bounds of the interval to

be searched for eigenvalues. VL < VU.

Not referenced if RANGE = ’A’ or ’I’.

*IL*

IL is INTEGER

*IU*

IU is INTEGER

If RANGE=’I’, the indices (in ascending order) of the

smallest and largest eigenvalues to be returned.

1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.

Not referenced if RANGE = ’A’ or ’V’.

*ABSTOL*

ABSTOL is REAL

The absolute error tolerance for the eigenvalues.

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 + EPS * max( |a|,|b| ) ,

where EPS is the machine precision. If ABSTOL is less than

or equal to zero, then EPS*|T| will be used in its place,

where |T| is the 1-norm of the tridiagonal matrix obtained

by reducing A to tridiagonal form.

See "Computing Small Singular Values of Bidiagonal Matrices

with Guaranteed High Relative Accuracy," by Demmel and

Kahan, LAPACK Working Note #3.

If high relative accuracy is important, set ABSTOL to

SLAMCH( ’Safe minimum’ ). Doing so will guarantee that

eigenvalues are computed to high relative accuracy when

possible in future releases. The current code does not

make any guarantees about high relative accuracy, but

future releases will. See J. Barlow and J. Demmel,

"Computing Accurate Eigensystems of Scaled Diagonally

Dominant Matrices", LAPACK Working Note #7, for a discussion

of which matrices define their eigenvalues to high relative

accuracy.

*M*

M is INTEGER

The total number of eigenvalues found. 0 <= M <= N.

If RANGE = ’A’, M = N, and if RANGE = ’I’, M = IU-IL+1.

*W*

W is REAL array, dimension (N)

The first M elements contain the selected eigenvalues in

ascending order.

*Z*

Z is REAL array, dimension (LDZ, 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).

Note: the user 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.

*LDZ*

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if

JOBZ = ’V’, LDZ >= max(1,N).

*ISUPPZ*

ISUPPZ is INTEGER array, dimension ( 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( 2*i-1 ) through

ISUPPZ( 2*i ).

Implemented only for RANGE = ’A’ or ’I’ and IU - IL = N - 1

*WORK*

WORK is REAL array, dimension (MAX(1,LWORK))

On exit, if INFO = 0, WORK(1) returns the optimal (and

minimal) LWORK.

*LWORK*

LWORK is INTEGER

The dimension of the array WORK. LWORK >= 20*N.

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the optimal 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.

*IWORK*

IWORK is INTEGER array, dimension (MAX(1,LIWORK))

On exit, if INFO = 0, IWORK(1) returns the optimal (and

minimal) LIWORK.

*LIWORK*

LIWORK is INTEGER

The dimension of the array IWORK. LIWORK >= 10*N.

If LIWORK = -1, then a workspace query is assumed; the

routine only calculates the optimal 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.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: Internal error

**Author:**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

November 2011

**Contributors:**

Inderjit Dhillon, IBM Almaden, USA

Osni Marques, LBNL/NERSC, USA

Ken Stanley, Computer Science Division, University of California at Berkeley, USA

Jason Riedy, Computer Science Division, University of California at Berkeley, USA

Definition at line 298 of file sstevr.f.

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