sspgvx.f −

**Functions/Subroutines**

subroutine **sspgvx** (ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO)

SSPGST

**subroutine sspgvx (integerITYPE, characterJOBZ, characterRANGE, characterUPLO, integerN, real, dimension( * )AP, real, dimension( * )BP, realVL, realVU, integerIL, integerIU, realABSTOL, integerM, real, dimension( * )W, real, dimension( ldz, * )Z, integerLDZ, real, dimension( * )WORK, integer, dimension( * )IWORK, integer, dimension( * )IFAIL, integerINFO)
SSPGST**

**Purpose:**

SSPGVX 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, stored in packed storage, 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.

*AP*

AP is REAL array, dimension (N*(N+1)/2)

On entry, the upper or lower triangle of the symmetric matrix

A, packed columnwise in a linear array. The j-th column of A

is stored in the array AP as follows:

if UPLO = ’U’, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;

if UPLO = ’L’, AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

On exit, the contents of AP are destroyed.

*BP*

BP is REAL array, dimension (N*(N+1)/2)

On entry, the upper or lower triangle of the symmetric matrix

B, packed columnwise in a linear array. The j-th column of B

is stored in the array BP as follows:

if UPLO = ’U’, BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;

if UPLO = ’L’, BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.

On exit, the triangular factor U or L from the Cholesky

factorization B = U**T*U or B = L*L**T, in the same storage

format as B.

*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.

Eigenvalues will be computed most accurately when ABSTOL is

set to twice the underflow threshold 2*SLAMCH(’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 (8*N)

*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: SPPTRF or SSPEVX returned an error code:

<= N: if INFO = i, SSPEVX 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 262 of file sspgvx.f.

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