table of contents
SSPGVD(1) | LAPACK driver routine (version 3.2) | SSPGVD(1) |
NAME¶
SSPGVD - 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
SYNOPSIS¶
- SUBROUTINE SSPGVD(
- ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO )
CHARACTER JOBZ, UPLO INTEGER INFO, ITYPE, LDZ, LIWORK, LWORK, N INTEGER IWORK( * ) REAL AP( * ), BP( * ), W( * ), WORK( * ), Z( LDZ, * )
PURPOSE¶
SSPGVD 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, stored in packed format, 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.
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. - UPLO (input) CHARACTER*1
-
= 'U': Upper triangles of A and B are stored;
= 'L': Lower triangles of A and B are stored. - N (input) INTEGER
- The order of the matrices A and 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.
- W (output) REAL array, dimension (N)
- If INFO = 0, the eigenvalues in ascending order.
- Z (output) REAL array, dimension (LDZ, N)
- If JOBZ = 'V', then if INFO = 0, Z 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 Z is not referenced.
- LDZ (input) INTEGER
- The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
- WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
- On exit, if INFO = 0, WORK(1) returns the required LWORK.
- LWORK (input) INTEGER
- The dimension of the array WORK. If N <= 1, LWORK >= 1. If JOBZ = 'N' and N > 1, LWORK >= 2*N. 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 required 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 (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
- On exit, if INFO = 0, IWORK(1) returns the required LIWORK.
- LIWORK (input) INTEGER
- The dimension of the array IWORK. If JOBZ = 'N' or 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 required 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 (output) INTEGER
- = 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: SPPTRF or SSPEVD returned an error code:
<= N: if INFO = i, SSPEVD failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero; > 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
November 2008 | LAPACK driver routine (version 3.2) |