SUBROUTINE ZHEEVD( JOBZ, UPLO, N, A, LDA, W, 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, LDA, LIWORK, LRWORK, LWORK, N
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION RWORK( * ), W( * )
COMPLEX*16 A( LDA, * ), WORK( * )
* ..
*
* Purpose
* =======
*
* ZHEEVD computes all eigenvalues and, optionally, eigenvectors of a
* complex Hermitian matrix A. 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 triangle of A is stored;
* = 'L': Lower triangle of A is stored.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* A (input/output) COMPLEX*16 array, dimension (LDA, N)
* On entry, the Hermitian matrix A. If UPLO = 'U', the
* leading N-by-N upper triangular part of A contains the
* upper triangular part of the matrix A. If UPLO = 'L',
* the leading N-by-N lower triangular part of A contains
* the lower triangular part of the matrix A.
* On exit, if JOBZ = 'V', then if INFO = 0, A contains the
* orthonormal eigenvectors of the matrix A.
* If JOBZ = 'N', then on exit the lower triangle (if UPLO='L')
* or the upper triangle (if UPLO='U') of A, including the
* diagonal, is destroyed.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* W (output) DOUBLE PRECISION array, dimension (N)
* If INFO = 0, the eigenvalues in ascending order.
*
* 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 length of the array WORK.
* If N <= 1, LWORK must be at least 1.
* If JOBZ = 'N' and N > 1, LWORK must be at least N + 1.
* If JOBZ = 'V' and N > 1, LWORK must be at least 2*N + 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 (LRWORK)
* On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
*
* LRWORK (input) INTEGER
* The dimension of the array RWORK.
* If N <= 1, LRWORK must be at least 1.
* If JOBZ = 'N' and N > 1, LRWORK must be at least N.
* If JOBZ = 'V' and N > 1, LRWORK must be at least
* 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 the array IWORK.
* If N <= 1, LIWORK must be at least 1.
* If JOBZ = 'N' and N > 1, LIWORK must be at least 1.
* If JOBZ = 'V' and N > 1, LIWORK must be at least 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 JOBZ = 'N', then the algorithm failed
* to converge; i off-diagonal elements of an intermediate
* tridiagonal form did not converge to zero;
* if INFO = i and JOBZ = 'V', then the algorithm failed
* to compute an eigenvalue while working on the submatrix
* lying in rows and columns INFO/(N+1) through
* mod(INFO,N+1).
*
* Further Details
* ===============
*
* Based on contributions by
* Jeff Rutter, Computer Science Division, University of California
* at Berkeley, USA
*
* Modified description of INFO. Sven, 16 Feb 05.
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
COMPLEX*16 CONE
PARAMETER ( CONE = ( 1.0D0, 0.0D0 ) )
* ..
* .. Local Scalars ..
LOGICAL LOWER, LQUERY, WANTZ
INTEGER IINFO, IMAX, INDE, INDRWK, INDTAU, INDWK2,
$ INDWRK, ISCALE, LIOPT, LIWMIN, LLRWK, LLWORK,
$ LLWRK2, LOPT, LROPT, LRWMIN, LWMIN
DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
$ SMLNUM
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, ZLANHE
EXTERNAL LSAME, ILAENV, DLAMCH, ZLANHE
* ..
* .. External Subroutines ..
EXTERNAL DSCAL, DSTERF, XERBLA, ZHETRD, ZLACPY, ZLASCL,
$ ZSTEDC, ZUNMTR
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
LOWER = LSAME( UPLO, 'L' )
LQUERY = ( LWORK.EQ.-1 .OR. LRWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
INFO = 0
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
END IF
*
IF( INFO.EQ.0 ) THEN
IF( N.LE.1 ) THEN
LWMIN = 1
LRWMIN = 1
LIWMIN = 1
LOPT = LWMIN
LROPT = LRWMIN
LIOPT = LIWMIN
ELSE
IF( WANTZ ) THEN
LWMIN = 2*N + N*N
LRWMIN = 1 + 5*N + 2*N**2
LIWMIN = 3 + 5*N
ELSE
LWMIN = N + 1
LRWMIN = N
LIWMIN = 1
END IF
LOPT = MAX( LWMIN, N +
$ ILAENV( 1, 'ZHETRD', UPLO, N, -1, -1, -1 ) )
LROPT = LRWMIN
LIOPT = LIWMIN
END IF
WORK( 1 ) = LOPT
RWORK( 1 ) = LROPT
IWORK( 1 ) = LIOPT
*
IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -8
ELSE IF( LRWORK.LT.LRWMIN .AND. .NOT.LQUERY ) THEN
INFO = -10
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
INFO = -12
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZHEEVD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( N.EQ.1 ) THEN
W( 1 ) = A( 1, 1 )
IF( WANTZ )
$ A( 1, 1 ) = CONE
RETURN
END IF
*
* Get machine constants.
*
SAFMIN = DLAMCH( 'Safe minimum' )
EPS = DLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
BIGNUM = ONE / SMLNUM
RMIN = SQRT( SMLNUM )
RMAX = SQRT( BIGNUM )
*
* Scale matrix to allowable range, if necessary.
*
ANRM = ZLANHE( 'M', UPLO, N, A, LDA, RWORK )
ISCALE = 0
IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
ISCALE = 1
SIGMA = RMIN / ANRM
ELSE IF( ANRM.GT.RMAX ) THEN
ISCALE = 1
SIGMA = RMAX / ANRM
END IF
IF( ISCALE.EQ.1 )
$ CALL ZLASCL( UPLO, 0, 0, ONE, SIGMA, N, N, A, LDA, INFO )
*
* Call ZHETRD to reduce Hermitian matrix to tridiagonal form.
*
INDE = 1
INDTAU = 1
INDWRK = INDTAU + N
INDRWK = INDE + N
INDWK2 = INDWRK + N*N
LLWORK = LWORK - INDWRK + 1
LLWRK2 = LWORK - INDWK2 + 1
LLRWK = LRWORK - INDRWK + 1
CALL ZHETRD( UPLO, N, A, LDA, W, RWORK( INDE ), WORK( INDTAU ),
$ WORK( INDWRK ), LLWORK, IINFO )
*
* For eigenvalues only, call DSTERF. For eigenvectors, first call
* ZSTEDC to generate the eigenvector matrix, WORK(INDWRK), of the
* tridiagonal matrix, then call ZUNMTR to multiply it to the
* Householder transformations represented as Householder vectors in
* A.
*
IF( .NOT.WANTZ ) THEN
CALL DSTERF( N, W, RWORK( INDE ), INFO )
ELSE
CALL ZSTEDC( 'I', N, W, RWORK( INDE ), WORK( INDWRK ), N,
$ WORK( INDWK2 ), LLWRK2, RWORK( INDRWK ), LLRWK,
$ IWORK, LIWORK, INFO )
CALL ZUNMTR( 'L', UPLO, 'N', N, N, A, LDA, WORK( INDTAU ),
$ WORK( INDWRK ), N, WORK( INDWK2 ), LLWRK2, IINFO )
CALL ZLACPY( 'A', N, N, WORK( INDWRK ), N, A, LDA )
END IF
*
* If matrix was scaled, then rescale eigenvalues appropriately.
*
IF( ISCALE.EQ.1 ) THEN
IF( INFO.EQ.0 ) THEN
IMAX = N
ELSE
IMAX = INFO - 1
END IF
CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
END IF
*
WORK( 1 ) = LOPT
RWORK( 1 ) = LROPT
IWORK( 1 ) = LIOPT
*
RETURN
*
* End of ZHEEVD
*
END