subroutine dstein	(	integer	N,
		double precision, dimension( * )	D,
		double precision, dimension( * )	E,
		integer	M,
		double precision, dimension( * )	W,
		integer, dimension( * )	IBLOCK,
		integer, dimension( * )	ISPLIT,
		double precision, dimension( ldz, * )	Z,
		integer	LDZ,
		double precision, dimension( * )	WORK,
		integer, dimension( * )	IWORK,
		integer, dimension( * )	IFAIL,
		integer	INFO
	)

DSTEIN

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Purpose:

 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).

Parameters

[in]	N	N is INTEGER The order of the matrix. N >= 0.
[in]	D	D is DOUBLE PRECISION array, dimension (N) The n diagonal elements of the tridiagonal matrix T.
[in]	E	E is DOUBLE PRECISION array, dimension (N-1) The (n-1) subdiagonal elements of the tridiagonal matrix T, in elements 1 to N-1.
[in]	M	M is INTEGER The number of eigenvectors to be found. 0 <= M <= N.
[in]	W	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. )
[in]	IBLOCK	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. )
[in]	ISPLIT	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. )
[out]	Z	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.
[in]	LDZ	LDZ is INTEGER The leading dimension of the array Z. LDZ >= max(1,N).
[out]	WORK	WORK is DOUBLE PRECISION array, dimension (5*N)
[out]	IWORK	IWORK is INTEGER array, dimension (N)
[out]	IFAIL	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.
[out]	INFO	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.

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.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date

November 2015

Definition at line 176 of file dstein.f.

*
*  -- 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
*

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