ssygvx.f −

**Functions/Subroutines**

subroutine **ssygvx** (ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO)

SSYGST

**subroutine ssygvx (integerITYPE, characterJOBZ, characterRANGE, characterUPLO, integerN, real, dimension( lda, * )A, integerLDA, real, dimension( ldb, * )B, integerLDB, realVL, realVU, integerIL, integerIU, realABSTOL, integerM, real, dimension( * )W, real, dimension( ldz, * )Z, integerLDZ, real, dimension( * )WORK, integerLWORK, integer, dimension( * )IWORK, integer, dimension( * )IFAIL, integerINFO)
SSYGST**

**Purpose:**

SSYGVX computes selected eigenvalues, and optionally, eigenvectors

of a real generalized symmetric-definite eigenproblem, of the form

A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A

and B are assumed to be symmetric and B is also positive definite.

Eigenvalues and eigenvectors can be selected by specifying either a

range of values or a range of indices for the desired eigenvalues.

**Parameters:**

*ITYPE*

ITYPE is INTEGER

Specifies the problem type to be solved:

= 1: A*x = (lambda)*B*x

= 2: A*B*x = (lambda)*x

= 3: B*A*x = (lambda)*x

*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.

*UPLO*

UPLO is CHARACTER*1

= ’U’: Upper triangle of A and B are stored;

= ’L’: Lower triangle of A and B are stored.

*N*

N is INTEGER

The order of the matrix pencil (A,B). N >= 0.

*A*

A is REAL array, dimension (LDA, N)

On entry, the symmetric matrix A. If UPLO = ’U’, the

leading N-by-N upper triangular part of A contains the

upper triangular part of the matrix A. If UPLO = ’L’,

the leading N-by-N lower triangular part of A contains

the lower triangular part of the matrix A.

On exit, the lower triangle (if UPLO=’L’) or the upper

triangle (if UPLO=’U’) of A, including the diagonal, is

destroyed.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,N).

*B*

B is REAL array, dimension (LDA, N)

On entry, the symmetric matrix B. If UPLO = ’U’, the

leading N-by-N upper triangular part of B contains the

upper triangular part of the matrix B. If UPLO = ’L’,

the leading N-by-N lower triangular part of B contains

the lower triangular part of the matrix B.

On exit, if INFO <= N, the part of B containing the matrix is

overwritten by the triangular factor U or L from the Cholesky

factorization B = U**T*U or B = L*L**T.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

*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 C to tridiagonal form, where C is the symmetric

matrix of the standard symmetric problem to which the

generalized problem is transformed.

Eigenvalues will be computed most accurately when ABSTOL is

set to twice the underflow threshold 2*DLAMCH(’S’), not zero.

If this routine returns with INFO>0, indicating that some

eigenvectors did not converge, try setting ABSTOL to

2*SLAMCH(’S’).

*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)

On normal exit, the first M elements contain the selected

eigenvalues in ascending order.

*Z*

Z is REAL array, dimension (LDZ, max(1,M))

If JOBZ = ’N’, then Z is not referenced.

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).

The eigenvectors are normalized as follows:

if ITYPE = 1 or 2, Z**T*B*Z = I;

if ITYPE = 3, Z**T*inv(B)*Z = 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.

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).

*WORK*

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

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

*LWORK*

LWORK is INTEGER

The length of the array WORK. LWORK >= max(1,8*N).

For optimal efficiency, LWORK >= (NB+3)*N,

where NB is the blocksize for SSYTRD returned by ILAENV.

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

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*IWORK*

IWORK is INTEGER array, dimension (5*N)

*IFAIL*

IFAIL is INTEGER array, dimension (N)

If JOBZ = ’V’, then if INFO = 0, the first M elements of

IFAIL are zero. If INFO > 0, then IFAIL contains the

indices of the eigenvectors that failed to converge.

If JOBZ = ’N’, then IFAIL is not referenced.

*INFO*

INFO is INTEGER

= 0: successful exit

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

> 0: SPOTRF or SSYEVX returned an error code:

<= N: if INFO = i, SSYEVX failed to converge;

i eigenvectors failed to converge. Their indices

are stored in array IFAIL.

> N: if INFO = N + i, for 1 <= i <= N, then the leading

minor of order i of B is not positive definite.

The factorization of B could not be completed and

no eigenvalues or eigenvectors were computed.

**Author:**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

November 2011

**Contributors:**

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Definition at line 289 of file ssygvx.f.

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