422 lines
14 KiB
FortranFixed
422 lines
14 KiB
FortranFixed
SUBROUTINE ZHBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL,
|
|
$ VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK,
|
|
$ IWORK, IFAIL, INFO )
|
|
*
|
|
* -- LAPACK driver routine (version 3.1) --
|
|
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
|
|
* November 2006
|
|
*
|
|
* .. Scalar Arguments ..
|
|
CHARACTER JOBZ, RANGE, UPLO
|
|
INTEGER IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N
|
|
DOUBLE PRECISION ABSTOL, VL, VU
|
|
* ..
|
|
* .. Array Arguments ..
|
|
INTEGER IFAIL( * ), IWORK( * )
|
|
DOUBLE PRECISION RWORK( * ), W( * )
|
|
COMPLEX*16 AB( LDAB, * ), Q( LDQ, * ), WORK( * ),
|
|
$ Z( LDZ, * )
|
|
* ..
|
|
*
|
|
* Purpose
|
|
* =======
|
|
*
|
|
* ZHBEVX computes selected eigenvalues and, optionally, eigenvectors
|
|
* of a complex Hermitian band matrix A. Eigenvalues and eigenvectors
|
|
* can be selected by specifying either a range of values or a range of
|
|
* indices for the desired eigenvalues.
|
|
*
|
|
* Arguments
|
|
* =========
|
|
*
|
|
* JOBZ (input) CHARACTER*1
|
|
* = 'N': Compute eigenvalues only;
|
|
* = 'V': Compute eigenvalues and eigenvectors.
|
|
*
|
|
* RANGE (input) CHARACTER*1
|
|
* = 'A': all eigenvalues will be found;
|
|
* = 'V': all eigenvalues in the half-open interval (VL,VU]
|
|
* will be found;
|
|
* = 'I': the IL-th through IU-th eigenvalues will be found.
|
|
*
|
|
* 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*16 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.
|
|
*
|
|
* LDAB (input) INTEGER
|
|
* The leading dimension of the array AB. LDAB >= KD + 1.
|
|
*
|
|
* Q (output) COMPLEX*16 array, dimension (LDQ, N)
|
|
* If JOBZ = 'V', the N-by-N unitary matrix used in the
|
|
* reduction to tridiagonal form.
|
|
* If JOBZ = 'N', the array Q is not referenced.
|
|
*
|
|
* LDQ (input) INTEGER
|
|
* The leading dimension of the array Q. If JOBZ = 'V', then
|
|
* LDQ >= max(1,N).
|
|
*
|
|
* VL (input) DOUBLE PRECISION
|
|
* VU (input) DOUBLE PRECISION
|
|
* If RANGE='V', the lower and upper bounds of the interval to
|
|
* be searched for eigenvalues. VL < VU.
|
|
* Not referenced if RANGE = 'A' or 'I'.
|
|
*
|
|
* IL (input) INTEGER
|
|
* IU (input) INTEGER
|
|
* If RANGE='I', 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 = 'A' or 'V'.
|
|
*
|
|
* 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 AB to tridiagonal form.
|
|
*
|
|
* Eigenvalues will be computed most accurately when ABSTOL is
|
|
* set to twice the underflow threshold 2*DLAMCH('S'), not zero.
|
|
* If this routine returns with INFO>0, indicating that some
|
|
* eigenvectors did not converge, try setting ABSTOL to
|
|
* 2*DLAMCH('S').
|
|
*
|
|
* See "Computing Small Singular Values of Bidiagonal Matrices
|
|
* with Guaranteed High Relative Accuracy," by Demmel and
|
|
* Kahan, LAPACK Working Note #3.
|
|
*
|
|
* M (output) INTEGER
|
|
* The total number of eigenvalues found. 0 <= M <= N.
|
|
* If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
|
|
*
|
|
* W (output) DOUBLE PRECISION array, dimension (N)
|
|
* The first M elements contain the selected eigenvalues in
|
|
* ascending order.
|
|
*
|
|
* Z (output) COMPLEX*16 array, dimension (LDZ, max(1,M))
|
|
* If JOBZ = 'V', then if INFO = 0, the first M columns of Z
|
|
* contain the orthonormal eigenvectors of the matrix A
|
|
* corresponding to the selected eigenvalues, with the i-th
|
|
* column of Z holding the eigenvector associated with W(i).
|
|
* If an eigenvector fails to converge, then that column of Z
|
|
* contains the latest approximation to the eigenvector, and the
|
|
* index of the eigenvector is returned in IFAIL.
|
|
* If JOBZ = 'N', then Z is not referenced.
|
|
* Note: the user must ensure that at least max(1,M) columns are
|
|
* supplied in the array Z; if RANGE = 'V', the exact value of M
|
|
* is not known in advance and an upper bound must be used.
|
|
*
|
|
* LDZ (input) INTEGER
|
|
* The leading dimension of the array Z. LDZ >= 1, and if
|
|
* JOBZ = 'V', LDZ >= max(1,N).
|
|
*
|
|
* WORK (workspace) COMPLEX*16 array, dimension (N)
|
|
*
|
|
* RWORK (workspace) DOUBLE PRECISION array, dimension (7*N)
|
|
*
|
|
* IWORK (workspace) INTEGER array, dimension (5*N)
|
|
*
|
|
* IFAIL (output) INTEGER array, dimension (N)
|
|
* If JOBZ = 'V', then if INFO = 0, the first M elements of
|
|
* IFAIL are zero. If INFO > 0, then IFAIL contains the
|
|
* indices of the eigenvectors that failed to converge.
|
|
* If JOBZ = 'N', then IFAIL is not referenced.
|
|
*
|
|
* INFO (output) INTEGER
|
|
* = 0: successful exit
|
|
* < 0: if INFO = -i, the i-th argument had an illegal value
|
|
* > 0: if INFO = i, then i eigenvectors failed to converge.
|
|
* Their indices are stored in array IFAIL.
|
|
*
|
|
* =====================================================================
|
|
*
|
|
* .. Parameters ..
|
|
DOUBLE PRECISION ZERO, ONE
|
|
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
|
|
COMPLEX*16 CZERO, CONE
|
|
PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ),
|
|
$ CONE = ( 1.0D0, 0.0D0 ) )
|
|
* ..
|
|
* .. Local Scalars ..
|
|
LOGICAL ALLEIG, INDEIG, LOWER, TEST, VALEIG, WANTZ
|
|
CHARACTER ORDER
|
|
INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
|
|
$ INDISP, INDIWK, INDRWK, INDWRK, ISCALE, ITMP1,
|
|
$ J, JJ, NSPLIT
|
|
DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
|
|
$ SIGMA, SMLNUM, TMP1, VLL, VUU
|
|
COMPLEX*16 CTMP1
|
|
* ..
|
|
* .. External Functions ..
|
|
LOGICAL LSAME
|
|
DOUBLE PRECISION DLAMCH, ZLANHB
|
|
EXTERNAL LSAME, DLAMCH, ZLANHB
|
|
* ..
|
|
* .. External Subroutines ..
|
|
EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTERF, XERBLA, ZCOPY,
|
|
$ ZGEMV, ZHBTRD, ZLACPY, ZLASCL, ZSTEIN, ZSTEQR,
|
|
$ ZSWAP
|
|
* ..
|
|
* .. Intrinsic Functions ..
|
|
INTRINSIC DBLE, MAX, MIN, SQRT
|
|
* ..
|
|
* .. Executable Statements ..
|
|
*
|
|
* Test the input parameters.
|
|
*
|
|
WANTZ = LSAME( JOBZ, 'V' )
|
|
ALLEIG = LSAME( RANGE, 'A' )
|
|
VALEIG = LSAME( RANGE, 'V' )
|
|
INDEIG = LSAME( RANGE, 'I' )
|
|
LOWER = LSAME( UPLO, 'L' )
|
|
*
|
|
INFO = 0
|
|
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
|
|
INFO = -1
|
|
ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
|
|
INFO = -2
|
|
ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
|
|
INFO = -3
|
|
ELSE IF( N.LT.0 ) THEN
|
|
INFO = -4
|
|
ELSE IF( KD.LT.0 ) THEN
|
|
INFO = -5
|
|
ELSE IF( LDAB.LT.KD+1 ) THEN
|
|
INFO = -7
|
|
ELSE IF( WANTZ .AND. LDQ.LT.MAX( 1, N ) ) THEN
|
|
INFO = -9
|
|
ELSE
|
|
IF( VALEIG ) THEN
|
|
IF( N.GT.0 .AND. VU.LE.VL )
|
|
$ INFO = -11
|
|
ELSE IF( INDEIG ) THEN
|
|
IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
|
|
INFO = -12
|
|
ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
|
|
INFO = -13
|
|
END IF
|
|
END IF
|
|
END IF
|
|
IF( INFO.EQ.0 ) THEN
|
|
IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
|
|
$ INFO = -18
|
|
END IF
|
|
*
|
|
IF( INFO.NE.0 ) THEN
|
|
CALL XERBLA( 'ZHBEVX', -INFO )
|
|
RETURN
|
|
END IF
|
|
*
|
|
* Quick return if possible
|
|
*
|
|
M = 0
|
|
IF( N.EQ.0 )
|
|
$ RETURN
|
|
*
|
|
IF( N.EQ.1 ) THEN
|
|
M = 1
|
|
IF( LOWER ) THEN
|
|
CTMP1 = AB( 1, 1 )
|
|
ELSE
|
|
CTMP1 = AB( KD+1, 1 )
|
|
END IF
|
|
TMP1 = DBLE( CTMP1 )
|
|
IF( VALEIG ) THEN
|
|
IF( .NOT.( VL.LT.TMP1 .AND. VU.GE.TMP1 ) )
|
|
$ M = 0
|
|
END IF
|
|
IF( M.EQ.1 ) THEN
|
|
W( 1 ) = CTMP1
|
|
IF( WANTZ )
|
|
$ Z( 1, 1 ) = CONE
|
|
END IF
|
|
RETURN
|
|
END IF
|
|
*
|
|
* Get machine constants.
|
|
*
|
|
SAFMIN = DLAMCH( 'Safe minimum' )
|
|
EPS = DLAMCH( 'Precision' )
|
|
SMLNUM = SAFMIN / EPS
|
|
BIGNUM = ONE / SMLNUM
|
|
RMIN = SQRT( SMLNUM )
|
|
RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
|
|
*
|
|
* Scale matrix to allowable range, if necessary.
|
|
*
|
|
ISCALE = 0
|
|
ABSTLL = ABSTOL
|
|
IF( VALEIG ) THEN
|
|
VLL = VL
|
|
VUU = VU
|
|
ELSE
|
|
VLL = ZERO
|
|
VUU = ZERO
|
|
END IF
|
|
ANRM = ZLANHB( 'M', UPLO, N, KD, AB, LDAB, RWORK )
|
|
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 ZLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
|
|
ELSE
|
|
CALL ZLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
|
|
END IF
|
|
IF( ABSTOL.GT.0 )
|
|
$ ABSTLL = ABSTOL*SIGMA
|
|
IF( VALEIG ) THEN
|
|
VLL = VL*SIGMA
|
|
VUU = VU*SIGMA
|
|
END IF
|
|
END IF
|
|
*
|
|
* Call ZHBTRD to reduce Hermitian band matrix to tridiagonal form.
|
|
*
|
|
INDD = 1
|
|
INDE = INDD + N
|
|
INDRWK = INDE + N
|
|
INDWRK = 1
|
|
CALL ZHBTRD( JOBZ, UPLO, N, KD, AB, LDAB, RWORK( INDD ),
|
|
$ RWORK( INDE ), Q, LDQ, WORK( INDWRK ), IINFO )
|
|
*
|
|
* If all eigenvalues are desired and ABSTOL is less than or equal
|
|
* to zero, then call DSTERF or ZSTEQR. If this fails for some
|
|
* eigenvalue, then try DSTEBZ.
|
|
*
|
|
TEST = .FALSE.
|
|
IF (INDEIG) THEN
|
|
IF (IL.EQ.1 .AND. IU.EQ.N) THEN
|
|
TEST = .TRUE.
|
|
END IF
|
|
END IF
|
|
IF ((ALLEIG .OR. TEST) .AND. (ABSTOL.LE.ZERO)) THEN
|
|
CALL DCOPY( N, RWORK( INDD ), 1, W, 1 )
|
|
INDEE = INDRWK + 2*N
|
|
IF( .NOT.WANTZ ) THEN
|
|
CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
|
|
CALL DSTERF( N, W, RWORK( INDEE ), INFO )
|
|
ELSE
|
|
CALL ZLACPY( 'A', N, N, Q, LDQ, Z, LDZ )
|
|
CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
|
|
CALL ZSTEQR( JOBZ, N, W, RWORK( INDEE ), Z, LDZ,
|
|
$ RWORK( INDRWK ), INFO )
|
|
IF( INFO.EQ.0 ) THEN
|
|
DO 10 I = 1, N
|
|
IFAIL( I ) = 0
|
|
10 CONTINUE
|
|
END IF
|
|
END IF
|
|
IF( INFO.EQ.0 ) THEN
|
|
M = N
|
|
GO TO 30
|
|
END IF
|
|
INFO = 0
|
|
END IF
|
|
*
|
|
* Otherwise, call DSTEBZ and, if eigenvectors are desired, ZSTEIN.
|
|
*
|
|
IF( WANTZ ) THEN
|
|
ORDER = 'B'
|
|
ELSE
|
|
ORDER = 'E'
|
|
END IF
|
|
INDIBL = 1
|
|
INDISP = INDIBL + N
|
|
INDIWK = INDISP + N
|
|
CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
|
|
$ RWORK( INDD ), RWORK( INDE ), M, NSPLIT, W,
|
|
$ IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ),
|
|
$ IWORK( INDIWK ), INFO )
|
|
*
|
|
IF( WANTZ ) THEN
|
|
CALL ZSTEIN( N, RWORK( INDD ), RWORK( INDE ), M, W,
|
|
$ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
|
|
$ RWORK( INDRWK ), IWORK( INDIWK ), IFAIL, INFO )
|
|
*
|
|
* Apply unitary matrix used in reduction to tridiagonal
|
|
* form to eigenvectors returned by ZSTEIN.
|
|
*
|
|
DO 20 J = 1, M
|
|
CALL ZCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 )
|
|
CALL ZGEMV( 'N', N, N, CONE, Q, LDQ, WORK, 1, CZERO,
|
|
$ Z( 1, J ), 1 )
|
|
20 CONTINUE
|
|
END IF
|
|
*
|
|
* If matrix was scaled, then rescale eigenvalues appropriately.
|
|
*
|
|
30 CONTINUE
|
|
IF( ISCALE.EQ.1 ) THEN
|
|
IF( INFO.EQ.0 ) THEN
|
|
IMAX = M
|
|
ELSE
|
|
IMAX = INFO - 1
|
|
END IF
|
|
CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
|
|
END IF
|
|
*
|
|
* If eigenvalues are not in order, then sort them, along with
|
|
* eigenvectors.
|
|
*
|
|
IF( WANTZ ) THEN
|
|
DO 50 J = 1, M - 1
|
|
I = 0
|
|
TMP1 = W( J )
|
|
DO 40 JJ = J + 1, M
|
|
IF( W( JJ ).LT.TMP1 ) THEN
|
|
I = JJ
|
|
TMP1 = W( JJ )
|
|
END IF
|
|
40 CONTINUE
|
|
*
|
|
IF( I.NE.0 ) THEN
|
|
ITMP1 = IWORK( INDIBL+I-1 )
|
|
W( I ) = W( J )
|
|
IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
|
|
W( J ) = TMP1
|
|
IWORK( INDIBL+J-1 ) = ITMP1
|
|
CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
|
|
IF( INFO.NE.0 ) THEN
|
|
ITMP1 = IFAIL( I )
|
|
IFAIL( I ) = IFAIL( J )
|
|
IFAIL( J ) = ITMP1
|
|
END IF
|
|
END IF
|
|
50 CONTINUE
|
|
END IF
|
|
*
|
|
RETURN
|
|
*
|
|
* End of ZHBEVX
|
|
*
|
|
END
|