SUBROUTINE ZHBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, \$ Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, \$ LIWORK, INFO ) * * -- LAPACK driver routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. CHARACTER JOBZ, UPLO INTEGER INFO, KA, KB, LDAB, LDBB, LDZ, LIWORK, LRWORK, \$ LWORK, N * .. * .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION RWORK( * ), W( * ) COMPLEX*16 AB( LDAB, * ), BB( LDBB, * ), WORK( * ), \$ Z( LDZ, * ) * .. * * Purpose * ======= * * ZHBGVD computes all the eigenvalues, and optionally, the eigenvectors * of a complex generalized Hermitian-definite banded eigenproblem, of * the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian * and banded, 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 * ========= * * 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. * * KA (input) INTEGER * The number of superdiagonals of the matrix A if UPLO = 'U', * or the number of subdiagonals if UPLO = 'L'. KA >= 0. * * KB (input) INTEGER * The number of superdiagonals of the matrix B if UPLO = 'U', * or the number of subdiagonals if UPLO = 'L'. KB >= 0. * * AB (input/output) COMPLEX*16 array, dimension (LDAB, N) * On entry, the upper or lower triangle of the Hermitian 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 = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). * * On exit, the contents of AB are destroyed. * * LDAB (input) INTEGER * The leading dimension of the array AB. LDAB >= KA+1. * * BB (input/output) COMPLEX*16 array, dimension (LDBB, N) * On entry, the upper or lower triangle of the Hermitian 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 = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; * if UPLO = 'L', 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**H*S, as returned by ZPBSTF. * * LDBB (input) INTEGER * The leading dimension of the array BB. LDBB >= KB+1. * * W (output) DOUBLE PRECISION array, dimension (N) * If INFO = 0, the eigenvalues in ascending order. * * Z (output) COMPLEX*16 array, dimension (LDZ, N) * If JOBZ = 'V', 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 that Z**H*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 >= N. * * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK)) * On exit, if INFO=0, WORK(1) returns the optimal LWORK. * * LWORK (input) INTEGER * The dimension of the array WORK. * If N <= 1, LWORK >= 1. * If JOBZ = 'N' and N > 1, LWORK >= N. * If JOBZ = 'V' and N > 1, LWORK >= 2*N**2. * * If LWORK = -1, then a workspace query is assumed; the routine * only calculates the optimal sizes of the WORK, RWORK and * IWORK arrays, returns these values as the first entries of * the WORK, RWORK and IWORK arrays, and no error message * related to LWORK or LRWORK or LIWORK is issued by XERBLA. * * RWORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LRWORK)) * On exit, if INFO=0, RWORK(1) returns the optimal LRWORK. * * LRWORK (input) INTEGER * The dimension of array RWORK. * If N <= 1, LRWORK >= 1. * If JOBZ = 'N' and N > 1, LRWORK >= N. * If JOBZ = 'V' and N > 1, LRWORK >= 1 + 5*N + 2*N**2. * * If LRWORK = -1, then a workspace query is assumed; the * routine only calculates the optimal sizes of the WORK, RWORK * and IWORK arrays, returns these values as the first entries * of the WORK, RWORK and IWORK arrays, and no error message * related to LWORK or LRWORK or LIWORK is issued by XERBLA. * * IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK)) * On exit, if INFO=0, IWORK(1) returns the optimal LIWORK. * * LIWORK (input) INTEGER * The dimension of 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 optimal sizes of the WORK, RWORK * and IWORK arrays, returns these values as the first entries * of the WORK, RWORK and IWORK arrays, and no error message * related to LWORK or LRWORK 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: if INFO = i, and i is: * <= N: the algorithm 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 ZPBSTF * returned INFO = i: 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 * * ===================================================================== * * .. Parameters .. COMPLEX*16 CONE, CZERO PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ), \$ CZERO = ( 0.0D+0, 0.0D+0 ) ) * .. * .. Local Scalars .. LOGICAL LQUERY, UPPER, WANTZ CHARACTER VECT INTEGER IINFO, INDE, INDWK2, INDWRK, LIWMIN, LLRWK, \$ LLWK2, LRWMIN, LWMIN * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL DSTERF, XERBLA, ZGEMM, ZHBGST, ZHBTRD, ZLACPY, \$ ZPBSTF, ZSTEDC * .. * .. Executable Statements .. * * Test the input parameters. * WANTZ = LSAME( JOBZ, 'V' ) UPPER = LSAME( UPLO, 'U' ) LQUERY = ( LWORK.EQ.-1 .OR. LRWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) * INFO = 0 IF( N.LE.1 ) THEN LWMIN = 1 LRWMIN = 1 LIWMIN = 1 ELSE IF( WANTZ ) THEN LWMIN = 2*N**2 LRWMIN = 1 + 5*N + 2*N**2 LIWMIN = 3 + 5*N ELSE LWMIN = N LRWMIN = N LIWMIN = 1 END IF IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( KA.LT.0 ) THEN INFO = -4 ELSE IF( KB.LT.0 .OR. KB.GT.KA ) THEN INFO = -5 ELSE IF( LDAB.LT.KA+1 ) THEN INFO = -7 ELSE IF( LDBB.LT.KB+1 ) THEN INFO = -9 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN INFO = -12 END IF * IF( INFO.EQ.0 ) THEN WORK( 1 ) = LWMIN RWORK( 1 ) = LRWMIN IWORK( 1 ) = LIWMIN * IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -14 ELSE IF( LRWORK.LT.LRWMIN .AND. .NOT.LQUERY ) THEN INFO = -16 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN INFO = -18 END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZHBGVD', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) \$ RETURN * * Form a split Cholesky factorization of B. * CALL ZPBSTF( UPLO, N, KB, BB, LDBB, INFO ) IF( INFO.NE.0 ) THEN INFO = N + INFO RETURN END IF * * Transform problem to standard eigenvalue problem. * INDE = 1 INDWRK = INDE + N INDWK2 = 1 + N*N LLWK2 = LWORK - INDWK2 + 2 LLRWK = LRWORK - INDWRK + 2 CALL ZHBGST( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Z, LDZ, \$ WORK, RWORK( INDWRK ), IINFO ) * * Reduce Hermitian band matrix to tridiagonal form. * IF( WANTZ ) THEN VECT = 'U' ELSE VECT = 'N' END IF CALL ZHBTRD( VECT, UPLO, N, KA, AB, LDAB, W, RWORK( INDE ), Z, \$ LDZ, WORK, IINFO ) * * For eigenvalues only, call DSTERF. For eigenvectors, call ZSTEDC. * IF( .NOT.WANTZ ) THEN CALL DSTERF( N, W, RWORK( INDE ), INFO ) ELSE CALL ZSTEDC( 'I', N, W, RWORK( INDE ), WORK, N, WORK( INDWK2 ), \$ LLWK2, RWORK( INDWRK ), LLRWK, IWORK, LIWORK, \$ INFO ) CALL ZGEMM( 'N', 'N', N, N, N, CONE, Z, LDZ, WORK, N, CZERO, \$ WORK( INDWK2 ), N ) CALL ZLACPY( 'A', N, N, WORK( INDWK2 ), N, Z, LDZ ) END IF * WORK( 1 ) = LWMIN RWORK( 1 ) = LRWMIN IWORK( 1 ) = LIWMIN RETURN * * End of ZHBGVD * END