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SUBROUTINE SLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDDIFR,
$ DSIGMA, WORK, INFO )
*
* -- LAPACK auxiliary routine (version 3.3.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2010
*
* .. Scalar Arguments ..
INTEGER ICOMPQ, INFO, K, LDDIFR
* ..
* .. Array Arguments ..
REAL D( * ), DIFL( * ), DIFR( LDDIFR, * ),
$ DSIGMA( * ), VF( * ), VL( * ), WORK( * ),
$ Z( * )
* ..
*
* Purpose
* =======
*
* SLASD8 finds the square roots of the roots of the secular equation,
* as defined by the values in DSIGMA and Z. It makes the appropriate
* calls to SLASD4, and stores, for each element in D, the distance
* to its two nearest poles (elements in DSIGMA). It also updates
* the arrays VF and VL, the first and last components of all the
* right singular vectors of the original bidiagonal matrix.
*
* SLASD8 is called from SLASD6.
*
* Arguments
* =========
*
* ICOMPQ (input) INTEGER
* Specifies whether singular vectors are to be computed in
* factored form in the calling routine:
* = 0: Compute singular values only.
* = 1: Compute singular vectors in factored form as well.
*
* K (input) INTEGER
* The number of terms in the rational function to be solved
* by SLASD4. K >= 1.
*
* D (output) REAL array, dimension ( K )
* On output, D contains the updated singular values.
*
* Z (input/output) REAL array, dimension ( K )
* On entry, the first K elements of this array contain the
* components of the deflation-adjusted updating row vector.
* On exit, Z is updated.
*
* VF (input/output) REAL array, dimension ( K )
* On entry, VF contains information passed through DBEDE8.
* On exit, VF contains the first K components of the first
* components of all right singular vectors of the bidiagonal
* matrix.
*
* VL (input/output) REAL array, dimension ( K )
* On entry, VL contains information passed through DBEDE8.
* On exit, VL contains the first K components of the last
* components of all right singular vectors of the bidiagonal
* matrix.
*
* DIFL (output) REAL array, dimension ( K )
* On exit, DIFL(I) = D(I) - DSIGMA(I).
*
* DIFR (output) REAL array,
* dimension ( LDDIFR, 2 ) if ICOMPQ = 1 and
* dimension ( K ) if ICOMPQ = 0.
* On exit, DIFR(I,1) = D(I) - DSIGMA(I+1), DIFR(K,1) is not
* defined and will not be referenced.
*
* If ICOMPQ = 1, DIFR(1:K,2) is an array containing the
* normalizing factors for the right singular vector matrix.
*
* LDDIFR (input) INTEGER
* The leading dimension of DIFR, must be at least K.
*
* DSIGMA (input/output) REAL array, dimension ( K )
* On entry, the first K elements of this array contain the old
* roots of the deflated updating problem. These are the poles
* of the secular equation.
* On exit, the elements of DSIGMA may be very slightly altered
* in value.
*
* WORK (workspace) REAL array, dimension at least 3 * K
*
* INFO (output) INTEGER
* = 0: successful exit.
* < 0: if INFO = -i, the i-th argument had an illegal value.
* > 0: if INFO = 1, a singular value did not converge
*
* Further Details
* ===============
*
* Based on contributions by
* Ming Gu and Huan Ren, Computer Science Division, University of
* California at Berkeley, USA
*
* =====================================================================
*
* .. Parameters ..
REAL ONE
PARAMETER ( ONE = 1.0E+0 )
* ..
* .. Local Scalars ..
INTEGER I, IWK1, IWK2, IWK2I, IWK3, IWK3I, J
REAL DIFLJ, DIFRJ, DJ, DSIGJ, DSIGJP, RHO, TEMP
* ..
* .. External Subroutines ..
EXTERNAL SCOPY, SLASCL, SLASD4, SLASET, XERBLA
* ..
* .. External Functions ..
REAL SDOT, SLAMC3, SNRM2
EXTERNAL SDOT, SLAMC3, SNRM2
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SIGN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
INFO = -1
ELSE IF( K.LT.1 ) THEN
INFO = -2
ELSE IF( LDDIFR.LT.K ) THEN
INFO = -9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SLASD8', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( K.EQ.1 ) THEN
D( 1 ) = ABS( Z( 1 ) )
DIFL( 1 ) = D( 1 )
IF( ICOMPQ.EQ.1 ) THEN
DIFL( 2 ) = ONE
DIFR( 1, 2 ) = ONE
END IF
RETURN
END IF
*
* Modify values DSIGMA(i) to make sure all DSIGMA(i)-DSIGMA(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 DSIGMA(I) by 2*DSIGMA(I)-DSIGMA(I),
* which on any of these machines zeros out the bottommost
* bit of DSIGMA(I) if it is 1; this makes the subsequent
* subtractions DSIGMA(I)-DSIGMA(J) unproblematic when cancellation
* occurs. On binary machines with a guard digit (almost all
* machines) it does not change DSIGMA(I) at all. On hexadecimal
* and decimal machines with a guard digit, it slightly
* changes the bottommost bits of DSIGMA(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
DSIGMA( I ) = SLAMC3( DSIGMA( I ), DSIGMA( I ) ) - DSIGMA( I )
10 CONTINUE
*
* Book keeping.
*
IWK1 = 1
IWK2 = IWK1 + K
IWK3 = IWK2 + K
IWK2I = IWK2 - 1
IWK3I = IWK3 - 1
*
* Normalize Z.
*
RHO = SNRM2( K, Z, 1 )
CALL SLASCL( 'G', 0, 0, RHO, ONE, K, 1, Z, K, INFO )
RHO = RHO*RHO
*
* Initialize WORK(IWK3).
*
CALL SLASET( 'A', K, 1, ONE, ONE, WORK( IWK3 ), K )
*
* Compute the updated singular values, the arrays DIFL, DIFR,
* and the updated Z.
*
DO 40 J = 1, K
CALL SLASD4( K, J, DSIGMA, Z, WORK( IWK1 ), RHO, D( J ),
$ WORK( IWK2 ), INFO )
*
* If the root finder fails, the computation is terminated.
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SLASD4', -INFO )
RETURN
END IF
WORK( IWK3I+J ) = WORK( IWK3I+J )*WORK( J )*WORK( IWK2I+J )
DIFL( J ) = -WORK( J )
DIFR( J, 1 ) = -WORK( J+1 )
DO 20 I = 1, J - 1
WORK( IWK3I+I ) = WORK( IWK3I+I )*WORK( I )*
$ WORK( IWK2I+I ) / ( DSIGMA( I )-
$ DSIGMA( J ) ) / ( DSIGMA( I )+
$ DSIGMA( J ) )
20 CONTINUE
DO 30 I = J + 1, K
WORK( IWK3I+I ) = WORK( IWK3I+I )*WORK( I )*
$ WORK( IWK2I+I ) / ( DSIGMA( I )-
$ DSIGMA( J ) ) / ( DSIGMA( I )+
$ DSIGMA( J ) )
30 CONTINUE
40 CONTINUE
*
* Compute updated Z.
*
DO 50 I = 1, K
Z( I ) = SIGN( SQRT( ABS( WORK( IWK3I+I ) ) ), Z( I ) )
50 CONTINUE
*
* Update VF and VL.
*
DO 80 J = 1, K
DIFLJ = DIFL( J )
DJ = D( J )
DSIGJ = -DSIGMA( J )
IF( J.LT.K ) THEN
DIFRJ = -DIFR( J, 1 )
DSIGJP = -DSIGMA( J+1 )
END IF
WORK( J ) = -Z( J ) / DIFLJ / ( DSIGMA( J )+DJ )
DO 60 I = 1, J - 1
WORK( I ) = Z( I ) / ( SLAMC3( DSIGMA( I ), DSIGJ )-DIFLJ )
$ / ( DSIGMA( I )+DJ )
60 CONTINUE
DO 70 I = J + 1, K
WORK( I ) = Z( I ) / ( SLAMC3( DSIGMA( I ), DSIGJP )+DIFRJ )
$ / ( DSIGMA( I )+DJ )
70 CONTINUE
TEMP = SNRM2( K, WORK, 1 )
WORK( IWK2I+J ) = SDOT( K, WORK, 1, VF, 1 ) / TEMP
WORK( IWK3I+J ) = SDOT( K, WORK, 1, VL, 1 ) / TEMP
IF( ICOMPQ.EQ.1 ) THEN
DIFR( J, 2 ) = TEMP
END IF
80 CONTINUE
*
CALL SCOPY( K, WORK( IWK2 ), 1, VF, 1 )
CALL SCOPY( K, WORK( IWK3 ), 1, VL, 1 )
*
RETURN
*
* End of SLASD8
*
END
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