---
name: "sspgvx"
section: "3lapack"
platform: "linux"
package: "liblapack-doc"
version: "3.2.1-8"
release: "Debian 6.0 (Squeeze)"
platforms: ["linux"]
url: "https://new.carta.tech/sspgvx/3lapack"
---

# sspgvx(3lapack)

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

## NAME

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

## SYNOPSIS

- SUBROUTINE SSPGVX(

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

- CHARACTER

  JOBZ, RANGE, UPLO

- INTEGER

  IL, INFO, ITYPE, IU, LDZ, M, N

- REAL

  ABSTOL, VL, VU

- INTEGER

  IFAIL( \* ), IWORK( \* )

- REAL

  AP( \* ), BP( \* ), W( \* ), WORK( \* ), Z( LDZ, \* )

## 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.\

## ARGUMENTS

- ITYPE (input) 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 (input) CHARACTER\*1

  \= 'N':  Compute eigenvalues only;\
   \= 'V':  Compute eigenvalues and eigenvectors.

- RANGE (input) 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 (input) CHARACTER\*1

  \= 'U':  Upper triangle of A and B are stored;\
   \= 'L':  Lower triangle of A and B are stored.

- N (input) INTEGER

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

- AP (input/output) 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 (input/output) 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 (input) REAL

  VU      (input) 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 (input) INTEGER

  IU      (input) 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 (input) 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 (output) INTEGER

  The total number of eigenvalues found.  0 \<= M \<= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

- W (output) REAL array, dimension (N)

  On normal exit, the first M elements contain the selected eigenvalues in ascending order.

- Z (output) 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 (input) INTEGER

  The leading dimension of the array Z.  LDZ \>= 1, and if JOBZ = 'V', LDZ \>= max(1,N).

- WORK (workspace) REAL array, dimension (8\*N)

  - IWORK (workspace) INTEGER array, dimension (5\*N)

  - IFAIL (output) 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 (output) 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.

## FURTHER DETAILS

Based on contributions by\
\
    Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA