subroutine slaed9	(	integer	K,
		integer	KSTART,
		integer	KSTOP,
		integer	N,
		real, dimension( * )	D,
		real, dimension( ldq, * )	Q,
		integer	LDQ,
		real	RHO,
		real, dimension( * )	DLAMDA,
		real, dimension( * )	W,
		real, dimension( lds, * )	S,
		integer	LDS,
		integer	INFO
	)

SLAED9 used by sstedc. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is dense.

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Purpose:

 SLAED9 finds the roots of the secular equation, as defined by the
 values in D, Z, and RHO, between KSTART and KSTOP.  It makes the
 appropriate calls to SLAED4 and then stores the new matrix of
 eigenvectors for use in calculating the next level of Z vectors.

Parameters

[in]	K	K is INTEGER The number of terms in the rational function to be solved by SLAED4. K >= 0.
[in]	KSTART	KSTART is INTEGER
[in]	KSTOP	KSTOP is INTEGER The updated eigenvalues Lambda(I), KSTART <= I <= KSTOP are to be computed. 1 <= KSTART <= KSTOP <= K.
[in]	N	N is INTEGER The number of rows and columns in the Q matrix. N >= K (delation may result in N > K).
[out]	D	D is REAL array, dimension (N) D(I) contains the updated eigenvalues for KSTART <= I <= KSTOP.
[out]	Q	Q is REAL array, dimension (LDQ,N)
[in]	LDQ	LDQ is INTEGER The leading dimension of the array Q. LDQ >= max( 1, N ).
[in]	RHO	RHO is REAL The value of the parameter in the rank one update equation. RHO >= 0 required.
[in]	DLAMDA	DLAMDA is REAL array, dimension (K) The first K elements of this array contain the old roots of the deflated updating problem. These are the poles of the secular equation.
[in]	W	W is REAL array, dimension (K) The first K elements of this array contain the components of the deflation-adjusted updating vector.
[out]	S	S is REAL array, dimension (LDS, K) Will contain the eigenvectors of the repaired matrix which will be stored for subsequent Z vector calculation and multiplied by the previously accumulated eigenvectors to update the system.
[in]	LDS	LDS is INTEGER The leading dimension of S. LDS >= max( 1, K ).
[out]	INFO	INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, an eigenvalue did not converge

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date

September 2012

Contributors:

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

Definition at line 158 of file slaed9.f.

*
*  -- LAPACK computational routine (version 3.4.2) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     September 2012
*
*     .. Scalar Arguments ..
      INTEGER            info, k, kstart, kstop, ldq, lds, n
      REAL               rho
*     ..
*     .. Array Arguments ..
      REAL               d( * ), dlamda( * ), q( ldq, * ), s( lds, * ),
     $                   w( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      INTEGER            i, j
      REAL               temp
*     ..
*     .. External Functions ..
      REAL               slamc3, snrm2
      EXTERNAL           slamc3, snrm2
*     ..
*     .. External Subroutines ..
      EXTERNAL           scopy, slaed4, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, sign, sqrt
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      info = 0
*
      IF( k.LT.0 ) THEN
         info = -1
      ELSE IF( kstart.LT.1 .OR. kstart.GT.max( 1, k ) ) THEN
         info = -2
      ELSE IF( max( 1, kstop ).LT.kstart .OR. kstop.GT.max( 1, k ) )
     $          THEN
         info = -3
      ELSE IF( n.LT.k ) THEN
         info = -4
      ELSE IF( ldq.LT.max( 1, k ) ) THEN
         info = -7
      ELSE IF( lds.LT.max( 1, k ) ) THEN
         info = -12
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'SLAED9', -info )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( k.EQ.0 )
     $   RETURN
*
*     Modify values DLAMDA(i) to make sure all DLAMDA(i)-DLAMDA(j) can
*     be computed with high relative accuracy (barring over/underflow).
*     This is a problem on machines without a guard digit in
*     add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2).
*     The following code replaces DLAMDA(I) by 2*DLAMDA(I)-DLAMDA(I),
*     which on any of these machines zeros out the bottommost
*     bit of DLAMDA(I) if it is 1; this makes the subsequent
*     subtractions DLAMDA(I)-DLAMDA(J) unproblematic when cancellation
*     occurs. On binary machines with a guard digit (almost all
*     machines) it does not change DLAMDA(I) at all. On hexadecimal
*     and decimal machines with a guard digit, it slightly
*     changes the bottommost bits of DLAMDA(I). It does not account
*     for hexadecimal or decimal machines without guard digits
*     (we know of none). We use a subroutine call to compute
*     2*DLAMBDA(I) to prevent optimizing compilers from eliminating
*     this code.
*
      DO 10 i = 1, n
         dlamda( i ) = slamc3( dlamda( i ), dlamda( i ) ) - dlamda( i )
   10 CONTINUE
*
      DO 20 j = kstart, kstop
         CALL slaed4( k, j, dlamda, w, q( 1, j ), rho, d( j ), info )
*
*        If the zero finder fails, the computation is terminated.
*
         IF( info.NE.0 )
     $      GO TO 120
   20 CONTINUE
*
      IF( k.EQ.1 .OR. k.EQ.2 ) THEN
         DO 40 i = 1, k
            DO 30 j = 1, k
               s( j, i ) = q( j, i )
   30       CONTINUE
   40    CONTINUE
         GO TO 120
      END IF
*
*     Compute updated W.
*
      CALL scopy( k, w, 1, s, 1 )
*
*     Initialize W(I) = Q(I,I)
*
      CALL scopy( k, q, ldq+1, w, 1 )
      DO 70 j = 1, k
         DO 50 i = 1, j - 1
            w( i ) = w( i )*( q( i, j ) / ( dlamda( i )-dlamda( j ) ) )
   50    CONTINUE
         DO 60 i = j + 1, k
            w( i ) = w( i )*( q( i, j ) / ( dlamda( i )-dlamda( j ) ) )
   60    CONTINUE
   70 CONTINUE
      DO 80 i = 1, k
         w( i ) = sign( sqrt( -w( i ) ), s( i, 1 ) )
   80 CONTINUE
*
*     Compute eigenvectors of the modified rank-1 modification.
*
      DO 110 j = 1, k
         DO 90 i = 1, k
            q( i, j ) = w( i ) / q( i, j )
   90    CONTINUE
         temp = snrm2( k, q( 1, j ), 1 )
         DO 100 i = 1, k
            s( i, j ) = q( i, j ) / temp
  100    CONTINUE
  110 CONTINUE
*
  120 CONTINUE
      RETURN
*
*     End of SLAED9
*

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