SUBROUTINE SSTEIN( N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK,
$ IWORK, IFAIL, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER INFO, LDZ, M, N
* ..
* .. Array Arguments ..
INTEGER IBLOCK( * ), IFAIL( * ), ISPLIT( * ),
$ IWORK( * )
REAL D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
* Purpose
* =======
*
* SSTEIN 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).
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the matrix. N >= 0.
*
* D (input) REAL array, dimension (N)
* The n diagonal elements of the tridiagonal matrix T.
*
* E (input) REAL array, dimension (N-1)
* The (n-1) subdiagonal elements of the tridiagonal matrix
* T, in elements 1 to N-1.
*
* M (input) INTEGER
* The number of eigenvectors to be found. 0 <= M <= N.
*
* W (input) REAL 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 SSTEBZ with ORDER = 'B' is expected here. )
*
* IBLOCK (input) 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 SSTEBZ is expected here. )
*
* ISPLIT (input) 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 SSTEBZ is expected here. )
*
* Z (output) REAL 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.
*
* LDZ (input) INTEGER
* The leading dimension of the array Z. LDZ >= max(1,N).
*
* WORK (workspace) REAL array, dimension (5*N)
*
* IWORK (workspace) INTEGER array, dimension (N)
*
* IFAIL (output) 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.
*
* 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
* in MAXITS iterations. Their indices are stored in
* array IFAIL.
*
* Internal Parameters
* ===================
*
* 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.
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE, TEN, ODM3, ODM1
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, TEN = 1.0E+1,
$ ODM3 = 1.0E-3, ODM1 = 1.0E-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
REAL CTR, EPS, EPS1, NRM, ONENRM, ORTOL, PERTOL,
$ SCL, SEP, STPCRT, TOL, XJ, XJM
* ..
* .. Local Arrays ..
INTEGER ISEED( 4 )
* ..
* .. External Functions ..
INTEGER ISAMAX
REAL SASUM, SDOT, SLAMCH, SNRM2
EXTERNAL ISAMAX, SASUM, SDOT, SLAMCH, SNRM2
* ..
* .. External Subroutines ..
EXTERNAL SAXPY, SCOPY, SLAGTF, SLAGTS, SLARNV, SSCAL,
$ 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( 'SSTEIN', -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 = SLAMCH( 'Precision' )
*
* Initialize seed for random number generator SLARNV.
*
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 = B1
*
* 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
*
STPCRT = 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 SLARNV( 2, ISEED, BLKSIZ, WORK( INDRV1+1 ) )
*
* Copy the matrix T so it won't be destroyed in factorization.
*
CALL SCOPY( BLKSIZ, D( B1 ), 1, WORK( INDRV4+1 ), 1 )
CALL SCOPY( BLKSIZ-1, E( B1 ), 1, WORK( INDRV2+2 ), 1 )
CALL SCOPY( BLKSIZ-1, E( B1 ), 1, WORK( INDRV3+1 ), 1 )
*
* Compute LU factors with partial pivoting ( PT = LU )
*
TOL = ZERO
CALL SLAGTF( 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.
*
SCL = BLKSIZ*ONENRM*MAX( EPS,
$ ABS( WORK( INDRV4+BLKSIZ ) ) ) /
$ SASUM( BLKSIZ, WORK( INDRV1+1 ), 1 )
CALL SSCAL( BLKSIZ, SCL, WORK( INDRV1+1 ), 1 )
*
* Solve the system LU = Pb.
*
CALL SLAGTS( -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
CTR = -SDOT( BLKSIZ, WORK( INDRV1+1 ), 1, Z( B1, I ),
$ 1 )
CALL SAXPY( BLKSIZ, CTR, Z( B1, I ), 1,
$ WORK( INDRV1+1 ), 1 )
80 CONTINUE
END IF
*
* Check the infinity norm of the iterate.
*
90 CONTINUE
JMAX = ISAMAX( BLKSIZ, WORK( INDRV1+1 ), 1 )
NRM = ABS( WORK( INDRV1+JMAX ) )
*
* Continue for additional iterations after norm reaches
* stopping criterion.
*
IF( NRM.LT.STPCRT )
$ 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 / SNRM2( BLKSIZ, WORK( INDRV1+1 ), 1 )
JMAX = ISAMAX( BLKSIZ, WORK( INDRV1+1 ), 1 )
IF( WORK( INDRV1+JMAX ).LT.ZERO )
$ SCL = -SCL
CALL SSCAL( 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 SSTEIN
*
END