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SUBROUTINE CHBEVD( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK,
$ LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO )
*
* -- LAPACK driver routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* September 30, 1994
*
* .. Scalar Arguments ..
CHARACTER JOBZ, UPLO
INTEGER INFO, KD, LDAB, LDZ, LIWORK, LRWORK, LWORK, N
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
REAL RWORK( * ), W( * )
COMPLEX AB( LDAB, * ), WORK( * ), Z( LDZ, * )
* ..
*
* Purpose
* =======
*
* CHBEVD computes all the eigenvalues and, optionally, eigenvectors of
* a complex Hermitian band 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.
*
* KD (input) INTEGER
* The number of superdiagonals of the matrix A if UPLO = 'U',
* or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*
* AB (input/output) COMPLEX array, dimension (LDAB, N)
* On entry, the upper or lower triangle of the Hermitian band
* matrix A, stored in the first KD+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(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
* if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
*
* On exit, AB is overwritten by values generated during the
* reduction to tridiagonal form. If UPLO = 'U', the first
* superdiagonal and the diagonal of the tridiagonal matrix T
* are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
* the diagonal and first subdiagonal of T are returned in the
* first two rows of AB.
*
* LDAB (input) INTEGER
* The leading dimension of the array AB. LDAB >= KD + 1.
*
* W (output) REAL array, dimension (N)
* If INFO = 0, the eigenvalues in ascending order.
*
* Z (output) COMPLEX array, dimension (LDZ, N)
* If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
* eigenvectors of the matrix A, with the i-th column of Z
* holding the eigenvector associated with W(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) COMPLEX array, dimension (LWORK)
* On exit, if LWORK > 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension 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.
* If JOBZ = 'V' and N > 1, LWORK must be at least 2*N**2.
*
* RWORK (workspace/output) REAL array,
* dimension (LRWORK)
* On exit, if LRWORK > 0, RWORK(1) returns the optimal LRWORK.
*
* LRWORK (input) INTEGER
* The dimension of 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 + 4*N + 2*N*lg N + 3*N**2 ,
* where lg( N ) = smallest integer k such
* that 2**k >= N .
*
* IWORK (workspace/output) INTEGER array, dimension (LIWORK)
* On exit, if LIWORK > 0, IWORK(1) returns the optimal LIWORK.
*
* LIWORK (input) INTEGER
* The dimension of array IWORK.
* If JOBZ = 'N' or N <= 1, LIWORK must be at least 1.
* If JOBZ = 'V' and N > 1, LIWORK must be at least 2 + 5*N .
*
* INFO (output) INTEGER
* = 0: successful exit.
* < 0: if INFO = -i, the i-th argument had an illegal value.
* > 0: if INFO = i, the algorithm failed to converge; i
* off-diagonal elements of an intermediate tridiagonal
* form did not converge to zero.
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0, TWO = 2.0E0 )
COMPLEX CZERO, CONE
PARAMETER ( CZERO = ( 0.0E0, 0.0E0 ),
$ CONE = ( 1.0E0, 0.0E0 ) )
* ..
* .. Local Scalars ..
LOGICAL LOWER, WANTZ
INTEGER IINFO, IMAX, INDE, INDWK2, INDWRK, ISCALE, LGN,
$ LIWMIN, LLRWK, LLWK2, LRWMIN, LWMIN
REAL ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
$ SMLNUM
* ..
* .. External Functions ..
LOGICAL LSAME
REAL CLANHB, SLAMCH
EXTERNAL LSAME, CLANHB, SLAMCH
* ..
* .. External Subroutines ..
EXTERNAL CGEMM, CHBTRD, CLACPY, CLASCL, CSTEDC, SSCAL,
$ SSTERF, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, LOG, REAL, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
LOWER = LSAME( UPLO, 'L' )
*
INFO = 0
IF( N.LE.1 ) THEN
LGN = 0
LWMIN = 1
LRWMIN = 1
LIWMIN = 1
ELSE
LGN = INT( LOG( REAL( N ) ) / LOG( TWO ) )
IF( 2**LGN.LT.N )
$ LGN = LGN + 1
IF( 2**LGN.LT.N )
$ LGN = LGN + 1
IF( WANTZ ) THEN
LWMIN = 2*N**2
LRWMIN = 1 + 4*N + 2*N*LGN + 3*N**2
LIWMIN = 2 + 5*N
ELSE
LWMIN = N
LRWMIN = N
LIWMIN = 1
END IF
END IF
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( KD.LT.0 ) THEN
INFO = -4
ELSE IF( LDAB.LT.KD+1 ) THEN
INFO = -6
ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
INFO = -9
ELSE IF( LWORK.LT.LWMIN ) THEN
INFO = -11
ELSE IF( LRWORK.LT.LRWMIN ) THEN
INFO = -13
ELSE IF( LIWORK.LT.LIWMIN ) THEN
INFO = -15
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CHBEVD ', -INFO )
GO TO 10
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ GO TO 10
*
IF( N.EQ.1 ) THEN
W( 1 ) = AB( 1, 1 )
IF( WANTZ )
$ Z( 1, 1 ) = CONE
GO TO 10
END IF
*
* Get machine constants.
*
SAFMIN = SLAMCH( 'Safe minimum' )
EPS = SLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
BIGNUM = ONE / SMLNUM
RMIN = SQRT( SMLNUM )
RMAX = SQRT( BIGNUM )
*
* Scale matrix to allowable range, if necessary.
*
ANRM = CLANHB( 'M', UPLO, N, KD, AB, LDAB, 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 ) THEN
IF( LOWER ) THEN
CALL CLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
ELSE
CALL CLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
END IF
END IF
*
* Call CHBTRD to reduce Hermitian band matrix to tridiagonal form.
*
INDE = 1
INDWRK = INDE + N
INDWK2 = 1 + N*N
LLWK2 = LWORK - INDWK2 + 1
LLRWK = LRWORK - INDWRK + 1
CALL CHBTRD( JOBZ, UPLO, N, KD, AB, LDAB, W, RWORK( INDE ), Z,
$ LDZ, WORK, IINFO )
*
* For eigenvalues only, call SSTERF. For eigenvectors, call CSTEDC.
*
IF( .NOT.WANTZ ) THEN
CALL SSTERF( N, W, RWORK( INDE ), INFO )
ELSE
CALL CSTEDC( 'I', N, W, RWORK( INDE ), WORK, N, WORK( INDWK2 ),
$ LLWK2, RWORK( INDWRK ), LLRWK, IWORK, LIWORK,
$ INFO )
CALL CGEMM( 'N', 'N', N, N, N, CONE, Z, LDZ, WORK, N, CZERO,
$ WORK( INDWK2 ), N )
CALL CLACPY( 'A', N, N, WORK( INDWK2 ), N, Z, LDZ )
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 SSCAL( IMAX, ONE / SIGMA, W, 1 )
END IF
*
10 CONTINUE
IF( LWORK.GT.0 )
$ WORK( 1 ) = LWMIN
IF( LRWORK.GT.0 )
$ RWORK( 1 ) = LRWMIN
IF( LIWORK.GT.0 )
$ IWORK( 1 ) = LIWMIN
RETURN
*
* End of CHBEVD
*
END
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