ssygvd.f −

**Functions/Subroutines**

subroutine **ssygvd** (ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK, LWORK, IWORK, LIWORK, INFO)

SSYGST

**subroutine ssygvd (integerITYPE, characterJOBZ, characterUPLO, integerN, real, dimension( lda, * )A, integerLDA, real, dimension( ldb, * )B, integerLDB, real, dimension( * )W, real, dimension( * )WORK, integerLWORK, integer, dimension( * )IWORK, integerLIWORK, integerINFO)
SSYGST**

**Purpose:**

SSYGVD computes all the eigenvalues, and optionally, the 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.

If eigenvectors are desired, it uses a divide and conquer algorithm.

The divide and conquer algorithm makes very mild assumptions about

floating point arithmetic. It will work on machines with a guard

digit in add/subtract, or on those binary machines without guard

digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or

Cray-2. It could conceivably fail on hexadecimal or decimal machines

without guard digits, but we know of none.

**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.

*UPLO*

UPLO is CHARACTER*1

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

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

*N*

N is INTEGER

The order of the matrices A and 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, if JOBZ = ’V’, then if INFO = 0, A contains the

matrix Z of eigenvectors. 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 JOBZ = ’N’, then on exit the upper triangle (if UPLO=’U’)

or the lower triangle (if UPLO=’L’) 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 (LDB, 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).

*W*

W is REAL array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*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 dimension of the array WORK.

If N <= 1, LWORK >= 1.

If JOBZ = ’N’ and N > 1, LWORK >= 2*N+1.

If JOBZ = ’V’ and N > 1, LWORK >= 1 + 6*N + 2*N**2.

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 LIWORK.

*LIWORK*

LIWORK is INTEGER

The dimension of the array IWORK.

If N <= 1, LIWORK >= 1.

If JOBZ = ’N’ and N > 1, LIWORK >= 1.

If JOBZ = ’V’ and N > 1, LIWORK >= 3 + 5*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: SPOTRF or SSYEVD returned an error code:

<= N: if INFO = i and JOBZ = ’N’, then the algorithm

failed to converge; i off-diagonal elements of an

intermediate tridiagonal form did not converge to

zero;

if INFO = i and JOBZ = ’V’, then the algorithm

failed to compute an eigenvalue while working on

the submatrix lying in rows and columns INFO/(N+1)

through mod(INFO,N+1);

> 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

**Further Details:**

Modified so that no backsubstitution is performed if SSYEVD fails to

converge (NEIG in old code could be greater than N causing out of

bounds reference to A - reported by Ralf Meyer). Also corrected the

description of INFO and the test on ITYPE. Sven, 16 Feb 05.

**Contributors:**

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

Definition at line 227 of file ssygvd.f.

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