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SUBROUTINE DLASD8( 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 .. DOUBLE PRECISION D( * ), DIFL( * ), DIFR( LDDIFR, * ), $ DSIGMA( * ), VF( * ), VL( * ), WORK( * ), $ Z( * ) * .. * * Purpose * ======= * * DLASD8 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 DLASD4, 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. * * DLASD8 is called from DLASD6. * * 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 DLASD4. K >= 1. * * D (output) DOUBLE PRECISION array, dimension ( K ) * On output, D contains the updated singular values. * * Z (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension ( K ) * On exit, DIFL(I) = D(I) - DSIGMA(I). * * DIFR (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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 .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) * .. * .. Local Scalars .. INTEGER I, IWK1, IWK2, IWK2I, IWK3, IWK3I, J DOUBLE PRECISION DIFLJ, DIFRJ, DJ, DSIGJ, DSIGJP, RHO, TEMP * .. * .. External Subroutines .. EXTERNAL DCOPY, DLASCL, DLASD4, DLASET, XERBLA * .. * .. External Functions .. DOUBLE PRECISION DDOT, DLAMC3, DNRM2 EXTERNAL DDOT, DLAMC3, DNRM2 * .. * .. 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( 'DLASD8', -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 ) = DLAMC3( 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 = DNRM2( K, Z, 1 ) CALL DLASCL( 'G', 0, 0, RHO, ONE, K, 1, Z, K, INFO ) RHO = RHO*RHO * * Initialize WORK(IWK3). * CALL DLASET( '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 DLASD4( 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( 'DLASD4', -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 ) / ( DLAMC3( DSIGMA( I ), DSIGJ )-DIFLJ ) $ / ( DSIGMA( I )+DJ ) 60 CONTINUE DO 70 I = J + 1, K WORK( I ) = Z( I ) / ( DLAMC3( DSIGMA( I ), DSIGJP )+DIFRJ ) $ / ( DSIGMA( I )+DJ ) 70 CONTINUE TEMP = DNRM2( K, WORK, 1 ) WORK( IWK2I+J ) = DDOT( K, WORK, 1, VF, 1 ) / TEMP WORK( IWK3I+J ) = DDOT( K, WORK, 1, VL, 1 ) / TEMP IF( ICOMPQ.EQ.1 ) THEN DIFR( J, 2 ) = TEMP END IF 80 CONTINUE * CALL DCOPY( K, WORK( IWK2 ), 1, VF, 1 ) CALL DCOPY( K, WORK( IWK3 ), 1, VL, 1 ) * RETURN * * End of DLASD8 * END |