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*> \brief \b DSTEIN
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSTEIN + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstein.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstein.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstein.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSTEIN( N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK,
* IWORK, IFAIL, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDZ, M, N
* ..
* .. Array Arguments ..
* INTEGER IBLOCK( * ), IFAIL( * ), ISPLIT( * ),
* $ IWORK( * )
* DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSTEIN computes the eigenvectors of a real symmetric tridiagonal
*> matrix T corresponding to specified eigenvalues, using inverse
*> iteration.
*>
*> The maximum number of iterations allowed for each eigenvector is
*> specified by an internal parameter MAXITS (currently set to 5).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix. N >= 0.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The n diagonal elements of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) subdiagonal elements of the tridiagonal matrix
*> T, in elements 1 to N-1.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of eigenvectors to be found. 0 <= M <= N.
*> \endverbatim
*>
*> \param[in] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (N)
*> The first M elements of W contain the eigenvalues for
*> which eigenvectors are to be computed. The eigenvalues
*> should be grouped by split-off block and ordered from
*> smallest to largest within the block. ( The output array
*> W from DSTEBZ with ORDER = 'B' is expected here. )
*> \endverbatim
*>
*> \param[in] IBLOCK
*> \verbatim
*> IBLOCK is INTEGER array, dimension (N)
*> The submatrix indices associated with the corresponding
*> eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to
*> the first submatrix from the top, =2 if W(i) belongs to
*> the second submatrix, etc. ( The output array IBLOCK
*> from DSTEBZ is expected here. )
*> \endverbatim
*>
*> \param[in] ISPLIT
*> \verbatim
*> ISPLIT is INTEGER array, dimension (N)
*> The splitting points, at which T breaks up into submatrices.
*> The first submatrix consists of rows/columns 1 to
*> ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
*> through ISPLIT( 2 ), etc.
*> ( The output array ISPLIT from DSTEBZ is expected here. )
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ, M)
*> The computed eigenvectors. The eigenvector associated
*> with the eigenvalue W(i) is stored in the i-th column of
*> Z. Any vector which fails to converge is set to its current
*> iterate after MAXITS iterations.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (5*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] IFAIL
*> \verbatim
*> IFAIL is INTEGER array, dimension (M)
*> On normal exit, all elements of IFAIL are zero.
*> If one or more eigenvectors fail to converge after
*> MAXITS iterations, then their indices are stored in
*> array IFAIL.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is 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
*> in MAXITS iterations. Their indices are stored in
*> array IFAIL.
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> MAXITS INTEGER, default = 5
*> The maximum number of iterations performed.
*>
*> EXTRA INTEGER, default = 2
*> The number of iterations performed after norm growth
*> criterion is satisfied, should be at least 1.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2015
*
*> \ingroup doubleOTHERcomputational
*
* =====================================================================
SUBROUTINE DSTEIN( N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK,
$ IWORK, IFAIL, INFO )
*
* -- LAPACK computational routine (version 3.6.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2015
*
* .. Scalar Arguments ..
INTEGER INFO, LDZ, M, N
* ..
* .. Array Arguments ..
INTEGER IBLOCK( * ), IFAIL( * ), ISPLIT( * ),
$ IWORK( * )
DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TEN, ODM3, ODM1
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TEN = 1.0D+1,
$ ODM3 = 1.0D-3, ODM1 = 1.0D-1 )
INTEGER MAXITS, EXTRA
PARAMETER ( MAXITS = 5, EXTRA = 2 )
* ..
* .. Local Scalars ..
INTEGER B1, BLKSIZ, BN, GPIND, I, IINFO, INDRV1,
$ INDRV2, INDRV3, INDRV4, INDRV5, ITS, J, J1,
$ JBLK, JMAX, NBLK, NRMCHK
DOUBLE PRECISION DTPCRT, EPS, EPS1, NRM, ONENRM, ORTOL, PERTOL,
$ SCL, SEP, TOL, XJ, XJM, ZTR
* ..
* .. Local Arrays ..
INTEGER ISEED( 4 )
* ..
* .. External Functions ..
INTEGER IDAMAX
DOUBLE PRECISION DASUM, DDOT, DLAMCH, DNRM2
EXTERNAL IDAMAX, DASUM, DDOT, DLAMCH, DNRM2
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DLAGTF, DLAGTS, DLARNV, DSCAL,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
DO 10 I = 1, M
IFAIL( I ) = 0
10 CONTINUE
*
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( M.LT.0 .OR. M.GT.N ) THEN
INFO = -4
ELSE IF( LDZ.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE
DO 20 J = 2, M
IF( IBLOCK( J ).LT.IBLOCK( J-1 ) ) THEN
INFO = -6
GO TO 30
END IF
IF( IBLOCK( J ).EQ.IBLOCK( J-1 ) .AND. W( J ).LT.W( J-1 ) )
$ THEN
INFO = -5
GO TO 30
END IF
20 CONTINUE
30 CONTINUE
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSTEIN', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. M.EQ.0 ) THEN
RETURN
ELSE IF( N.EQ.1 ) THEN
Z( 1, 1 ) = ONE
RETURN
END IF
*
* Get machine constants.
*
EPS = DLAMCH( 'Precision' )
*
* Initialize seed for random number generator DLARNV.
*
DO 40 I = 1, 4
ISEED( I ) = 1
40 CONTINUE
*
* Initialize pointers.
*
INDRV1 = 0
INDRV2 = INDRV1 + N
INDRV3 = INDRV2 + N
INDRV4 = INDRV3 + N
INDRV5 = INDRV4 + N
*
* Compute eigenvectors of matrix blocks.
*
J1 = 1
DO 160 NBLK = 1, IBLOCK( M )
*
* Find starting and ending indices of block nblk.
*
IF( NBLK.EQ.1 ) THEN
B1 = 1
ELSE
B1 = ISPLIT( NBLK-1 ) + 1
END IF
BN = ISPLIT( NBLK )
BLKSIZ = BN - B1 + 1
IF( BLKSIZ.EQ.1 )
$ GO TO 60
GPIND = J1
*
* Compute reorthogonalization criterion and stopping criterion.
*
ONENRM = ABS( D( B1 ) ) + ABS( E( B1 ) )
ONENRM = MAX( ONENRM, ABS( D( BN ) )+ABS( E( BN-1 ) ) )
DO 50 I = B1 + 1, BN - 1
ONENRM = MAX( ONENRM, ABS( D( I ) )+ABS( E( I-1 ) )+
$ ABS( E( I ) ) )
50 CONTINUE
ORTOL = ODM3*ONENRM
*
DTPCRT = SQRT( ODM1 / BLKSIZ )
*
* Loop through eigenvalues of block nblk.
*
60 CONTINUE
JBLK = 0
DO 150 J = J1, M
IF( IBLOCK( J ).NE.NBLK ) THEN
J1 = J
GO TO 160
END IF
JBLK = JBLK + 1
XJ = W( J )
*
* Skip all the work if the block size is one.
*
IF( BLKSIZ.EQ.1 ) THEN
WORK( INDRV1+1 ) = ONE
GO TO 120
END IF
*
* If eigenvalues j and j-1 are too close, add a relatively
* small perturbation.
*
IF( JBLK.GT.1 ) THEN
EPS1 = ABS( EPS*XJ )
PERTOL = TEN*EPS1
SEP = XJ - XJM
IF( SEP.LT.PERTOL )
$ XJ = XJM + PERTOL
END IF
*
ITS = 0
NRMCHK = 0
*
* Get random starting vector.
*
CALL DLARNV( 2, ISEED, BLKSIZ, WORK( INDRV1+1 ) )
*
* Copy the matrix T so it won't be destroyed in factorization.
*
CALL DCOPY( BLKSIZ, D( B1 ), 1, WORK( INDRV4+1 ), 1 )
CALL DCOPY( BLKSIZ-1, E( B1 ), 1, WORK( INDRV2+2 ), 1 )
CALL DCOPY( BLKSIZ-1, E( B1 ), 1, WORK( INDRV3+1 ), 1 )
*
* Compute LU factors with partial pivoting ( PT = LU )
*
TOL = ZERO
CALL DLAGTF( BLKSIZ, WORK( INDRV4+1 ), XJ, WORK( INDRV2+2 ),
$ WORK( INDRV3+1 ), TOL, WORK( INDRV5+1 ), IWORK,
$ IINFO )
*
* Update iteration count.
*
70 CONTINUE
ITS = ITS + 1
IF( ITS.GT.MAXITS )
$ GO TO 100
*
* Normalize and scale the righthand side vector Pb.
*
JMAX = IDAMAX( BLKSIZ, WORK( INDRV1+1 ), 1 )
SCL = BLKSIZ*ONENRM*MAX( EPS,
$ ABS( WORK( INDRV4+BLKSIZ ) ) ) /
$ ABS( WORK( INDRV1+JMAX ) )
CALL DSCAL( BLKSIZ, SCL, WORK( INDRV1+1 ), 1 )
*
* Solve the system LU = Pb.
*
CALL DLAGTS( -1, BLKSIZ, WORK( INDRV4+1 ), WORK( INDRV2+2 ),
$ WORK( INDRV3+1 ), WORK( INDRV5+1 ), IWORK,
$ WORK( INDRV1+1 ), TOL, IINFO )
*
* Reorthogonalize by modified Gram-Schmidt if eigenvalues are
* close enough.
*
IF( JBLK.EQ.1 )
$ GO TO 90
IF( ABS( XJ-XJM ).GT.ORTOL )
$ GPIND = J
IF( GPIND.NE.J ) THEN
DO 80 I = GPIND, J - 1
ZTR = -DDOT( BLKSIZ, WORK( INDRV1+1 ), 1, Z( B1, I ),
$ 1 )
CALL DAXPY( BLKSIZ, ZTR, Z( B1, I ), 1,
$ WORK( INDRV1+1 ), 1 )
80 CONTINUE
END IF
*
* Check the infinity norm of the iterate.
*
90 CONTINUE
JMAX = IDAMAX( BLKSIZ, WORK( INDRV1+1 ), 1 )
NRM = ABS( WORK( INDRV1+JMAX ) )
*
* Continue for additional iterations after norm reaches
* stopping criterion.
*
IF( NRM.LT.DTPCRT )
$ GO TO 70
NRMCHK = NRMCHK + 1
IF( NRMCHK.LT.EXTRA+1 )
$ GO TO 70
*
GO TO 110
*
* If stopping criterion was not satisfied, update info and
* store eigenvector number in array ifail.
*
100 CONTINUE
INFO = INFO + 1
IFAIL( INFO ) = J
*
* Accept iterate as jth eigenvector.
*
110 CONTINUE
SCL = ONE / DNRM2( BLKSIZ, WORK( INDRV1+1 ), 1 )
JMAX = IDAMAX( BLKSIZ, WORK( INDRV1+1 ), 1 )
IF( WORK( INDRV1+JMAX ).LT.ZERO )
$ SCL = -SCL
CALL DSCAL( BLKSIZ, SCL, WORK( INDRV1+1 ), 1 )
120 CONTINUE
DO 130 I = 1, N
Z( I, J ) = ZERO
130 CONTINUE
DO 140 I = 1, BLKSIZ
Z( B1+I-1, J ) = WORK( INDRV1+I )
140 CONTINUE
*
* Save the shift to check eigenvalue spacing at next
* iteration.
*
XJM = XJ
*
150 CONTINUE
160 CONTINUE
*
RETURN
*
* End of DSTEIN
*
END
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