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SUBROUTINE DSBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W,
$ Z, LDZ, WORK, LWORK, 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, LWORK, N
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION AB( LDAB, * ), BB( LDBB, * ), W( * ),
$ WORK( * ), Z( LDZ, * )
* ..
*
* Purpose
* =======
*
* DSBGVD computes all the eigenvalues, and optionally, the 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. 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) 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 = '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) 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 = 'U', BB(ka+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**T*S, as returned by DPBSTF.
*
* 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) DOUBLE PRECISION 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 Z**T*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) DOUBLE PRECISION 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 >= 3*N.
* If JOBZ = 'V' and N > 1, LWORK >= 1 + 5*N + 2*N**2.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal 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 LIWORK > 0, IWORK(1) returns the optimal 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 optimal 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: 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 DPBSTF
* 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 ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, UPPER, WANTZ
CHARACTER VECT
INTEGER IINFO, INDE, INDWK2, INDWRK, LIWMIN, LLWRK2,
$ LWMIN
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DLACPY, DPBSTF, DSBGST, DSBTRD, DSTEDC,
$ DSTERF, XERBLA
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
UPPER = LSAME( UPLO, 'U' )
LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
INFO = 0
IF( N.LE.1 ) THEN
LIWMIN = 1
LWMIN = 1
ELSE IF( WANTZ ) THEN
LIWMIN = 3 + 5*N
LWMIN = 1 + 5*N + 2*N**2
ELSE
LIWMIN = 1
LWMIN = 2*N
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
IWORK( 1 ) = LIWMIN
*
IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -14
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
INFO = -16
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSBGVD', -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 DPBSTF( 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 = INDWRK + N*N
LLWRK2 = LWORK - INDWK2 + 1
CALL DSBGST( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Z, LDZ,
$ WORK( INDWRK ), IINFO )
*
* Reduce to tridiagonal form.
*
IF( WANTZ ) THEN
VECT = 'U'
ELSE
VECT = 'N'
END IF
CALL DSBTRD( VECT, UPLO, N, KA, AB, LDAB, W, WORK( INDE ), Z, LDZ,
$ WORK( INDWRK ), IINFO )
*
* For eigenvalues only, call DSTERF. For eigenvectors, call SSTEDC.
*
IF( .NOT.WANTZ ) THEN
CALL DSTERF( N, W, WORK( INDE ), INFO )
ELSE
CALL DSTEDC( 'I', N, W, WORK( INDE ), WORK( INDWRK ), N,
$ WORK( INDWK2 ), LLWRK2, IWORK, LIWORK, INFO )
CALL DGEMM( 'N', 'N', N, N, N, ONE, Z, LDZ, WORK( INDWRK ), N,
$ ZERO, WORK( INDWK2 ), N )
CALL DLACPY( 'A', N, N, WORK( INDWK2 ), N, Z, LDZ )
END IF
*
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
*
RETURN
*
* End of DSBGVD
*
END
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