.TH DSBGVX 1 "November 2006" " LAPACK driver routine (version 3.1) " " LAPACK driver routine (version 3.1) " .SH NAME DSBGVX - selected eigenvalues, and optionally, eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form A*x=(lambda)*B*x .SH SYNOPSIS .TP 19 SUBROUTINE DSBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO ) .TP 19 .ti +4 CHARACTER JOBZ, RANGE, UPLO .TP 19 .ti +4 INTEGER IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M, N .TP 19 .ti +4 DOUBLE PRECISION ABSTOL, VL, VU .TP 19 .ti +4 INTEGER IFAIL( * ), IWORK( * ) .TP 19 .ti +4 DOUBLE PRECISION AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ), W( * ), WORK( * ), Z( LDZ, * ) .SH PURPOSE DSBGVX computes selected eigenvalues, and optionally, eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric and banded, and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either all eigenvalues, a range of values or a range of indices for the desired eigenvalues. .SH ARGUMENTS .TP 8 JOBZ (input) CHARACTER*1 = \(aqN\(aq: Compute eigenvalues only; .br = \(aqV\(aq: Compute eigenvalues and eigenvectors. .TP 8 RANGE (input) CHARACTER*1 .br = \(aqA\(aq: all eigenvalues will be found. .br = \(aqV\(aq: all eigenvalues in the half-open interval (VL,VU] will be found. = \(aqI\(aq: the IL-th through IU-th eigenvalues will be found. .TP 8 UPLO (input) CHARACTER*1 .br = \(aqU\(aq: Upper triangles of A and B are stored; .br = \(aqL\(aq: Lower triangles of A and B are stored. .TP 8 N (input) INTEGER The order of the matrices A and B. N >= 0. .TP 8 KA (input) INTEGER The number of superdiagonals of the matrix A if UPLO = \(aqU\(aq, or the number of subdiagonals if UPLO = \(aqL\(aq. KA >= 0. .TP 8 KB (input) INTEGER The number of superdiagonals of the matrix B if UPLO = \(aqU\(aq, or the number of subdiagonals if UPLO = \(aqL\(aq. KB >= 0. .TP 8 AB (input/output) DOUBLE PRECISION array, dimension (LDAB, N) On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first ka+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = \(aqU\(aq, AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; if UPLO = \(aqL\(aq, AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). On exit, the contents of AB are destroyed. .TP 8 LDAB (input) INTEGER The leading dimension of the array AB. LDAB >= KA+1. .TP 8 BB (input/output) DOUBLE PRECISION array, dimension (LDBB, N) On entry, the upper or lower triangle of the symmetric band matrix B, stored in the first kb+1 rows of the array. The j-th column of B is stored in the j-th column of the array BB as follows: if UPLO = \(aqU\(aq, BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; if UPLO = \(aqL\(aq, BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). On exit, the factor S from the split Cholesky factorization B = S**T*S, as returned by DPBSTF. .TP 8 LDBB (input) INTEGER The leading dimension of the array BB. LDBB >= KB+1. .TP 8 Q (output) DOUBLE PRECISION array, dimension (LDQ, N) If JOBZ = \(aqV\(aq, the n-by-n matrix used in the reduction of A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x, and consequently C to tridiagonal form. If JOBZ = \(aqN\(aq, the array Q is not referenced. .TP 8 LDQ (input) INTEGER The leading dimension of the array Q. If JOBZ = \(aqN\(aq, LDQ >= 1. If JOBZ = \(aqV\(aq, LDQ >= max(1,N). .TP 8 VL (input) DOUBLE PRECISION VU (input) DOUBLE PRECISION If RANGE=\(aqV\(aq, the lower and upper bounds of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = \(aqA\(aq or \(aqI\(aq. .TP 8 IL (input) INTEGER IU (input) INTEGER If RANGE=\(aqI\(aq, 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 = \(aqA\(aq or \(aqV\(aq. .TP 8 ABSTOL (input) DOUBLE PRECISION 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*DLAMCH(\(aqS\(aq), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*DLAMCH(\(aqS\(aq). .TP 8 M (output) INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = \(aqA\(aq, M = N, and if RANGE = \(aqI\(aq, M = IU-IL+1. .TP 8 W (output) DOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order. .TP 8 Z (output) DOUBLE PRECISION array, dimension (LDZ, N) If JOBZ = \(aqV\(aq, then if INFO = 0, Z contains the matrix Z of eigenvectors, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized so Z**T*B*Z = I. If JOBZ = \(aqN\(aq, then Z is not referenced. .TP 8 LDZ (input) INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = \(aqV\(aq, LDZ >= max(1,N). .TP 8 WORK (workspace/output) DOUBLE PRECISION array, dimension (7*N) .TP 8 IWORK (workspace/output) INTEGER array, dimension (5*N) .TP 8 IFAIL (output) INTEGER array, dimension (M) If JOBZ = \(aqV\(aq, then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvalues that failed to converge. If JOBZ = \(aqN\(aq, then IFAIL is not referenced. .TP 8 INFO (output) INTEGER = 0 : successful exit .br < 0 : if INFO = -i, the i-th argument had an illegal value .br <= N: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in IFAIL. > N : DPBSTF returned an error code; i.e., 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. .SH FURTHER DETAILS Based on contributions by .br Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA