*> \brief \b SLAED3 used by sstedc. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is tridiagonal.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slaed3.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SLAED3( K, N, N1, D, Q, LDQ, RHO, DLAMDA, Q2, INDX,
* CTOT, W, S, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, LDQ, N, N1
* REAL RHO
* ..
* .. Array Arguments ..
* INTEGER CTOT( * ), INDX( * )
* REAL D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ),
* $ S( * ), W( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SLAED3 finds the roots of the secular equation, as defined by the
*> values in D, W, and RHO, between 1 and K. It makes the
*> appropriate calls to SLAED4 and then updates the eigenvectors by
*> multiplying the matrix of eigenvectors of the pair of eigensystems
*> being combined by the matrix of eigenvectors of the K-by-K system
*> which is solved here.
*>
*> This code makes very mild assumptions about floating point
*> arithmetic. It will work on machines with a guard digit in
*> add/subtract, or on those binary machines without guard digits
*> which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
*> It could conceivably fail on hexadecimal or decimal machines
*> without guard digits, but we know of none.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of terms in the rational function to be solved by
*> SLAED4. K >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of rows and columns in the Q matrix.
*> N >= K (deflation may result in N>K).
*> \endverbatim
*>
*> \param[in] N1
*> \verbatim
*> N1 is INTEGER
*> The location of the last eigenvalue in the leading submatrix.
*> min(1,N) <= N1 <= N/2.
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is REAL array, dimension (N)
*> D(I) contains the updated eigenvalues for
*> 1 <= I <= K.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*> Q is REAL array, dimension (LDQ,N)
*> Initially the first K columns are used as workspace.
*> On output the columns 1 to K contain
*> the updated eigenvectors.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*> RHO is REAL
*> The value of the parameter in the rank one update equation.
*> RHO >= 0 required.
*> \endverbatim
*>
*> \param[in,out] DLAMDA
*> \verbatim
*> 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. May be changed on output by
*> having lowest order bit set to zero on Cray X-MP, Cray Y-MP,
*> Cray-2, or Cray C-90, as described above.
*> \endverbatim
*>
*> \param[in] Q2
*> \verbatim
*> Q2 is REAL array, dimension (LDQ2*N)
*> The first K columns of this matrix contain the non-deflated
*> eigenvectors for the split problem.
*> \endverbatim
*>
*> \param[in] INDX
*> \verbatim
*> INDX is INTEGER array, dimension (N)
*> The permutation used to arrange the columns of the deflated
*> Q matrix into three groups (see SLAED2).
*> The rows of the eigenvectors found by SLAED4 must be likewise
*> permuted before the matrix multiply can take place.
*> \endverbatim
*>
*> \param[in] CTOT
*> \verbatim
*> CTOT is INTEGER array, dimension (4)
*> A count of the total number of the various types of columns
*> in Q, as described in INDX. The fourth column type is any
*> column which has been deflated.
*> \endverbatim
*>
*> \param[in,out] W
*> \verbatim
*> W is REAL array, dimension (K)
*> The first K elements of this array contain the components
*> of the deflation-adjusted updating vector. Destroyed on
*> output.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is REAL array, dimension (N1 + 1)*K
*> Will contain the eigenvectors of the repaired matrix which
*> will be multiplied by the previously accumulated eigenvectors
*> to update the system.
*> \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 = 1, an eigenvalue did not converge
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2017
*
*> \ingroup auxOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA \n
*> Modified by Francoise Tisseur, University of Tennessee
*>
* =====================================================================
SUBROUTINE SLAED3( K, N, N1, D, Q, LDQ, RHO, DLAMDA, Q2, INDX,
$ CTOT, W, S, INFO )
*
* -- LAPACK computational routine (version 3.7.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* June 2017
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDQ, N, N1
REAL RHO
* ..
* .. Array Arguments ..
INTEGER CTOT( * ), INDX( * )
REAL D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ),
$ S( * ), W( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, ZERO
PARAMETER ( ONE = 1.0E0, ZERO = 0.0E0 )
* ..
* .. Local Scalars ..
INTEGER I, II, IQ2, J, N12, N2, N23
REAL TEMP
* ..
* .. External Functions ..
REAL SLAMC3, SNRM2
EXTERNAL SLAMC3, SNRM2
* ..
* .. External Subroutines ..
EXTERNAL SCOPY, SGEMM, SLACPY, SLAED4, SLASET, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SIGN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( K.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.K ) THEN
INFO = -2
ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SLAED3', -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, K
DLAMDA( I ) = SLAMC3( DLAMDA( I ), DLAMDA( I ) ) - DLAMDA( I )
10 CONTINUE
*
DO 20 J = 1, K
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 )
$ GO TO 110
IF( K.EQ.2 ) THEN
DO 30 J = 1, K
W( 1 ) = Q( 1, J )
W( 2 ) = Q( 2, J )
II = INDX( 1 )
Q( 1, J ) = W( II )
II = INDX( 2 )
Q( 2, J ) = W( II )
30 CONTINUE
GO TO 110
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 60 J = 1, K
DO 40 I = 1, J - 1
W( I ) = W( I )*( Q( I, J ) / ( DLAMDA( I )-DLAMDA( J ) ) )
40 CONTINUE
DO 50 I = J + 1, K
W( I ) = W( I )*( Q( I, J ) / ( DLAMDA( I )-DLAMDA( J ) ) )
50 CONTINUE
60 CONTINUE
DO 70 I = 1, K
W( I ) = SIGN( SQRT( -W( I ) ), S( I ) )
70 CONTINUE
*
* Compute eigenvectors of the modified rank-1 modification.
*
DO 100 J = 1, K
DO 80 I = 1, K
S( I ) = W( I ) / Q( I, J )
80 CONTINUE
TEMP = SNRM2( K, S, 1 )
DO 90 I = 1, K
II = INDX( I )
Q( I, J ) = S( II ) / TEMP
90 CONTINUE
100 CONTINUE
*
* Compute the updated eigenvectors.
*
110 CONTINUE
*
N2 = N - N1
N12 = CTOT( 1 ) + CTOT( 2 )
N23 = CTOT( 2 ) + CTOT( 3 )
*
CALL SLACPY( 'A', N23, K, Q( CTOT( 1 )+1, 1 ), LDQ, S, N23 )
IQ2 = N1*N12 + 1
IF( N23.NE.0 ) THEN
CALL SGEMM( 'N', 'N', N2, K, N23, ONE, Q2( IQ2 ), N2, S, N23,
$ ZERO, Q( N1+1, 1 ), LDQ )
ELSE
CALL SLASET( 'A', N2, K, ZERO, ZERO, Q( N1+1, 1 ), LDQ )
END IF
*
CALL SLACPY( 'A', N12, K, Q, LDQ, S, N12 )
IF( N12.NE.0 ) THEN
CALL SGEMM( 'N', 'N', N1, K, N12, ONE, Q2, N1, S, N12, ZERO, Q,
$ LDQ )
ELSE
CALL SLASET( 'A', N1, K, ZERO, ZERO, Q( 1, 1 ), LDQ )
END IF
*
*
120 CONTINUE
RETURN
*
* End of SLAED3
*
END