sstein2.f

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*
*
      SUBROUTINE sstein2( N, D, E, M, W, IBLOCK, ISPLIT, ORFAC, Z, LDZ,
     $                    WORK, IWORK, IFAIL, INFO )
*
*  -- ScaLAPACK routine (version 1.7) --
*     University of Tennessee, Knoxville, Oak Ridge National Laboratory,
*     and University of California, Berkeley.
*     May 1, 1997
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDZ, M, N
      REAL               ORFAC
*     ..
*     .. Array Arguments ..
      INTEGER            IBLOCK( * ), IFAIL( * ), ISPLIT( * ),
     $                   iwork( * )
      REAL               D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  Purpose
*  =======
*
*  SSTEIN2 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)
*          The (n-1) subdiagonal elements of the tridiagonal matrix
*          T, in elements 1 to N-1.  E(N) need not be set.
*
*  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. )
*
*  ORFAC   (input) REAL
*          ORFAC specifies which eigenvectors should be
*          orthogonalized. Eigenvectors that correspond to eigenvalues
*          which are within ORFAC*||T|| of each other are to be
*          orthogonalized.
*
*  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, ODM1
      parameter( zero = 0.0e+0, one = 1.0e+0, ten = 1.0e+1,
     $                   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               EPS, EPS1, NRM, ONENRM, ORTOL, PERTOL, SCL,
     $                   sep, stpcrt, tol, xj, xjm, ztr
*     ..
*     .. 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( orfac.LT.zero ) THEN
         info = -8
      ELSE IF( ldz.LT.max( 1, n ) ) THEN
         info = -10
      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( 'SSTEIN2', -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 = 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 = orfac*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
                  ztr = -sdot( blksiz, work( indrv1+1 ), 1, z( b1, i ),
     $                  1 )
                  CALL saxpy( blksiz, ztr, 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 SSTEIN2
*
      END