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SUBROUTINE DLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDDIFR, |
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$ DSIGMA, WORK, INFO ) |
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* |
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* -- LAPACK auxiliary routine (version 3.2.2) -- |
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* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
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* June 2010 |
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* |
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* .. Scalar Arguments .. |
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INTEGER ICOMPQ, INFO, K, LDDIFR |
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* .. |
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* .. Array Arguments .. |
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DOUBLE PRECISION D( * ), DIFL( * ), DIFR( LDDIFR, * ), |
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$ DSIGMA( * ), VF( * ), VL( * ), WORK( * ), |
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$ Z( * ) |
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* .. |
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* |
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* Purpose |
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* ======= |
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* |
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* DLASD8 finds the square roots of the roots of the secular equation, |
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* as defined by the values in DSIGMA and Z. It makes the appropriate |
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* calls to DLASD4, and stores, for each element in D, the distance |
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* to its two nearest poles (elements in DSIGMA). It also updates |
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* the arrays VF and VL, the first and last components of all the |
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* right singular vectors of the original bidiagonal matrix. |
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* |
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* DLASD8 is called from DLASD6. |
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* |
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* Arguments |
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* ========= |
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* |
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* ICOMPQ (input) INTEGER |
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* Specifies whether singular vectors are to be computed in |
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* factored form in the calling routine: |
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* = 0: Compute singular values only. |
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* = 1: Compute singular vectors in factored form as well. |
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* |
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* K (input) INTEGER |
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* The number of terms in the rational function to be solved |
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* by DLASD4. K >= 1. |
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* |
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* D (output) DOUBLE PRECISION array, dimension ( K ) |
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* On output, D contains the updated singular values. |
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* |
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* Z (input/output) DOUBLE PRECISION array, dimension ( K ) |
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* On entry, the first K elements of this array contain the |
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* components of the deflation-adjusted updating row vector. |
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* On exit, Z is updated. |
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* |
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* VF (input/output) DOUBLE PRECISION array, dimension ( K ) |
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* On entry, VF contains information passed through DBEDE8. |
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* On exit, VF contains the first K components of the first |
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* components of all right singular vectors of the bidiagonal |
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* matrix. |
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* |
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* VL (input/output) DOUBLE PRECISION array, dimension ( K ) |
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* On entry, VL contains information passed through DBEDE8. |
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* On exit, VL contains the first K components of the last |
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* components of all right singular vectors of the bidiagonal |
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* matrix. |
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* |
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* DIFL (output) DOUBLE PRECISION array, dimension ( K ) |
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* On exit, DIFL(I) = D(I) - DSIGMA(I). |
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* |
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* DIFR (output) DOUBLE PRECISION array, |
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* dimension ( LDDIFR, 2 ) if ICOMPQ = 1 and |
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* dimension ( K ) if ICOMPQ = 0. |
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* On exit, DIFR(I,1) = D(I) - DSIGMA(I+1), DIFR(K,1) is not |
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* defined and will not be referenced. |
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* |
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* If ICOMPQ = 1, DIFR(1:K,2) is an array containing the |
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* normalizing factors for the right singular vector matrix. |
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* |
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* LDDIFR (input) INTEGER |
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* The leading dimension of DIFR, must be at least K. |
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* |
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* DSIGMA (input/output) DOUBLE PRECISION array, dimension ( K ) |
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* On entry, the first K elements of this array contain the old |
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* roots of the deflated updating problem. These are the poles |
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* of the secular equation. |
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* On exit, the elements of DSIGMA may be very slightly altered |
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* in value. |
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* |
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* WORK (workspace) DOUBLE PRECISION array, dimension at least 3 * K |
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* |
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* INFO (output) INTEGER |
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* = 0: successful exit. |
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* < 0: if INFO = -i, the i-th argument had an illegal value. |
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* > 0: if INFO = 1, a singular value did not converge |
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* |
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* Further Details |
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* =============== |
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* |
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* Based on contributions by |
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* Ming Gu and Huan Ren, Computer Science Division, University of |
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* California at Berkeley, USA |
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* |
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* ===================================================================== |
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* |
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* .. Parameters .. |
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DOUBLE PRECISION ONE |
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PARAMETER ( ONE = 1.0D+0 ) |
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* .. |
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* .. Local Scalars .. |
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INTEGER I, IWK1, IWK2, IWK2I, IWK3, IWK3I, J |
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DOUBLE PRECISION DIFLJ, DIFRJ, DJ, DSIGJ, DSIGJP, RHO, TEMP |
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* .. |
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* .. External Subroutines .. |
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EXTERNAL DCOPY, DLASCL, DLASD4, DLASET, XERBLA |
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* .. |
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* .. External Functions .. |
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DOUBLE PRECISION DDOT, DLAMC3, DNRM2 |
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EXTERNAL DDOT, DLAMC3, DNRM2 |
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* .. |
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* .. Intrinsic Functions .. |
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INTRINSIC ABS, SIGN, SQRT |
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* .. |
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* .. Executable Statements .. |
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* |
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* Test the input parameters. |
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* |
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INFO = 0 |
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* |
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IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN |
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INFO = -1 |
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ELSE IF( K.LT.1 ) THEN |
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INFO = -2 |
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ELSE IF( LDDIFR.LT.K ) THEN |
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INFO = -9 |
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END IF |
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IF( INFO.NE.0 ) THEN |
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CALL XERBLA( 'DLASD8', -INFO ) |
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RETURN |
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END IF |
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* |
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* Quick return if possible |
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* |
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IF( K.EQ.1 ) THEN |
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D( 1 ) = ABS( Z( 1 ) ) |
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DIFL( 1 ) = D( 1 ) |
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IF( ICOMPQ.EQ.1 ) THEN |
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DIFL( 2 ) = ONE |
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DIFR( 1, 2 ) = ONE |
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END IF |
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RETURN |
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END IF |
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* |
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* Modify values DSIGMA(i) to make sure all DSIGMA(i)-DSIGMA(j) can |
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* be computed with high relative accuracy (barring over/underflow). |
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* This is a problem on machines without a guard digit in |
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* add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2). |
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* The following code replaces DSIGMA(I) by 2*DSIGMA(I)-DSIGMA(I), |
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* which on any of these machines zeros out the bottommost |
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* bit of DSIGMA(I) if it is 1; this makes the subsequent |
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* subtractions DSIGMA(I)-DSIGMA(J) unproblematic when cancellation |
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* occurs. On binary machines with a guard digit (almost all |
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* machines) it does not change DSIGMA(I) at all. On hexadecimal |
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* and decimal machines with a guard digit, it slightly |
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* changes the bottommost bits of DSIGMA(I). It does not account |
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* for hexadecimal or decimal machines without guard digits |
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* (we know of none). We use a subroutine call to compute |
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* 2*DLAMBDA(I) to prevent optimizing compilers from eliminating |
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* this code. |
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* |
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DO 10 I = 1, K |
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DSIGMA( I ) = DLAMC3( DSIGMA( I ), DSIGMA( I ) ) - DSIGMA( I ) |
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10 CONTINUE |
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* |
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* Book keeping. |
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* |
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IWK1 = 1 |
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IWK2 = IWK1 + K |
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IWK3 = IWK2 + K |
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IWK2I = IWK2 - 1 |
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IWK3I = IWK3 - 1 |
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* |
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* Normalize Z. |
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* |
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RHO = DNRM2( K, Z, 1 ) |
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CALL DLASCL( 'G', 0, 0, RHO, ONE, K, 1, Z, K, INFO ) |
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RHO = RHO*RHO |
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* |
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* Initialize WORK(IWK3). |
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* |
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CALL DLASET( 'A', K, 1, ONE, ONE, WORK( IWK3 ), K ) |
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* |
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* Compute the updated singular values, the arrays DIFL, DIFR, |
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* and the updated Z. |
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* |
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DO 40 J = 1, K |
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CALL DLASD4( K, J, DSIGMA, Z, WORK( IWK1 ), RHO, D( J ), |
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$ WORK( IWK2 ), INFO ) |
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* |
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* If the root finder fails, the computation is terminated. |
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* |
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IF( INFO.NE.0 ) THEN |
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RETURN |
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END IF |
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WORK( IWK3I+J ) = WORK( IWK3I+J )*WORK( J )*WORK( IWK2I+J ) |
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DIFL( J ) = -WORK( J ) |
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DIFR( J, 1 ) = -WORK( J+1 ) |
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DO 20 I = 1, J - 1 |
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WORK( IWK3I+I ) = WORK( IWK3I+I )*WORK( I )* |
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$ WORK( IWK2I+I ) / ( DSIGMA( I )- |
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$ DSIGMA( J ) ) / ( DSIGMA( I )+ |
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$ DSIGMA( J ) ) |
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20 CONTINUE |
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DO 30 I = J + 1, K |
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WORK( IWK3I+I ) = WORK( IWK3I+I )*WORK( I )* |
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$ WORK( IWK2I+I ) / ( DSIGMA( I )- |
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$ DSIGMA( J ) ) / ( DSIGMA( I )+ |
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$ DSIGMA( J ) ) |
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30 CONTINUE |
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40 CONTINUE |
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* |
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* Compute updated Z. |
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* |
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DO 50 I = 1, K |
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Z( I ) = SIGN( SQRT( ABS( WORK( IWK3I+I ) ) ), Z( I ) ) |
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50 CONTINUE |
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* |
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* Update VF and VL. |
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* |
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DO 80 J = 1, K |
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DIFLJ = DIFL( J ) |
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DJ = D( J ) |
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DSIGJ = -DSIGMA( J ) |
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IF( J.LT.K ) THEN |
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DIFRJ = -DIFR( J, 1 ) |
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DSIGJP = -DSIGMA( J+1 ) |
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END IF |
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WORK( J ) = -Z( J ) / DIFLJ / ( DSIGMA( J )+DJ ) |
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DO 60 I = 1, J - 1 |
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WORK( I ) = Z( I ) / ( DLAMC3( DSIGMA( I ), DSIGJ )-DIFLJ ) |
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$ / ( DSIGMA( I )+DJ ) |
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60 CONTINUE |
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DO 70 I = J + 1, K |
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WORK( I ) = Z( I ) / ( DLAMC3( DSIGMA( I ), DSIGJP )+DIFRJ ) |
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$ / ( DSIGMA( I )+DJ ) |
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70 CONTINUE |
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TEMP = DNRM2( K, WORK, 1 ) |
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WORK( IWK2I+J ) = DDOT( K, WORK, 1, VF, 1 ) / TEMP |
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WORK( IWK3I+J ) = DDOT( K, WORK, 1, VL, 1 ) / TEMP |
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IF( ICOMPQ.EQ.1 ) THEN |
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DIFR( J, 2 ) = TEMP |
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END IF |
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80 CONTINUE |
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* |
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CALL DCOPY( K, WORK( IWK2 ), 1, VF, 1 ) |
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CALL DCOPY( K, WORK( IWK3 ), 1, VL, 1 ) |
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* |
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RETURN |
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* |
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* End of DLASD8 |
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* |
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END |
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