*> \brief \b SLASQ3 checks for deflation, computes a shift and calls dqds. Used by sbdsqr. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SLASQ3 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SLASQ3( I0, N0, Z, PP, DMIN, SIGMA, DESIG, QMAX, NFAIL, * ITER, NDIV, IEEE, TTYPE, DMIN1, DMIN2, DN, DN1, * DN2, G, TAU ) * * .. Scalar Arguments .. * LOGICAL IEEE * INTEGER I0, ITER, N0, NDIV, NFAIL, PP * REAL DESIG, DMIN, DMIN1, DMIN2, DN, DN1, DN2, G, * $ QMAX, SIGMA, TAU * .. * .. Array Arguments .. * REAL Z( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SLASQ3 checks for deflation, computes a shift (TAU) and calls dqds. *> In case of failure it changes shifts, and tries again until output *> is positive. *> \endverbatim * * Arguments: * ========== * *> \param[in] I0 *> \verbatim *> I0 is INTEGER *> First index. *> \endverbatim *> *> \param[in,out] N0 *> \verbatim *> N0 is INTEGER *> Last index. *> \endverbatim *> *> \param[in,out] Z *> \verbatim *> Z is REAL array, dimension ( 4*N ) *> Z holds the qd array. *> \endverbatim *> *> \param[in,out] PP *> \verbatim *> PP is INTEGER *> PP=0 for ping, PP=1 for pong. *> PP=2 indicates that flipping was applied to the Z array *> and that the initial tests for deflation should not be *> performed. *> \endverbatim *> *> \param[out] DMIN *> \verbatim *> DMIN is REAL *> Minimum value of d. *> \endverbatim *> *> \param[out] SIGMA *> \verbatim *> SIGMA is REAL *> Sum of shifts used in current segment. *> \endverbatim *> *> \param[in,out] DESIG *> \verbatim *> DESIG is REAL *> Lower order part of SIGMA *> \endverbatim *> *> \param[in] QMAX *> \verbatim *> QMAX is REAL *> Maximum value of q. *> \endverbatim *> *> \param[in,out] NFAIL *> \verbatim *> NFAIL is INTEGER *> Increment NFAIL by 1 each time the shift was too big. *> \endverbatim *> *> \param[in,out] ITER *> \verbatim *> ITER is INTEGER *> Increment ITER by 1 for each iteration. *> \endverbatim *> *> \param[in,out] NDIV *> \verbatim *> NDIV is INTEGER *> Increment NDIV by 1 for each division. *> \endverbatim *> *> \param[in] IEEE *> \verbatim *> IEEE is LOGICAL *> Flag for IEEE or non IEEE arithmetic (passed to SLASQ5). *> \endverbatim *> *> \param[in,out] TTYPE *> \verbatim *> TTYPE is INTEGER *> Shift type. *> \endverbatim *> *> \param[in,out] DMIN1 *> \verbatim *> DMIN1 is REAL *> \endverbatim *> *> \param[in,out] DMIN2 *> \verbatim *> DMIN2 is REAL *> \endverbatim *> *> \param[in,out] DN *> \verbatim *> DN is REAL *> \endverbatim *> *> \param[in,out] DN1 *> \verbatim *> DN1 is REAL *> \endverbatim *> *> \param[in,out] DN2 *> \verbatim *> DN2 is REAL *> \endverbatim *> *> \param[in,out] G *> \verbatim *> G is REAL *> \endverbatim *> *> \param[in,out] TAU *> \verbatim *> TAU is REAL *> *> These are passed as arguments in order to save their values *> between calls to SLASQ3. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date November 2015 * *> \ingroup auxOTHERcomputational * * ===================================================================== SUBROUTINE SLASQ3( I0, N0, Z, PP, DMIN, SIGMA, DESIG, QMAX, NFAIL, $ ITER, NDIV, IEEE, TTYPE, DMIN1, DMIN2, DN, DN1, $ DN2, G, TAU ) * * -- LAPACK computational routine (version 3.6.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2015 * * .. Scalar Arguments .. LOGICAL IEEE INTEGER I0, ITER, N0, NDIV, NFAIL, PP REAL DESIG, DMIN, DMIN1, DMIN2, DN, DN1, DN2, G, $ QMAX, SIGMA, TAU * .. * .. Array Arguments .. REAL Z( * ) * .. * * ===================================================================== * * .. Parameters .. REAL CBIAS PARAMETER ( CBIAS = 1.50E0 ) REAL ZERO, QURTR, HALF, ONE, TWO, HUNDRD PARAMETER ( ZERO = 0.0E0, QURTR = 0.250E0, HALF = 0.5E0, $ ONE = 1.0E0, TWO = 2.0E0, HUNDRD = 100.0E0 ) * .. * .. Local Scalars .. INTEGER IPN4, J4, N0IN, NN, TTYPE REAL EPS, S, T, TEMP, TOL, TOL2 * .. * .. External Subroutines .. EXTERNAL SLASQ4, SLASQ5, SLASQ6 * .. * .. External Function .. REAL SLAMCH LOGICAL SISNAN EXTERNAL SISNAN, SLAMCH * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, SQRT * .. * .. Executable Statements .. * N0IN = N0 EPS = SLAMCH( 'Precision' ) TOL = EPS*HUNDRD TOL2 = TOL**2 * * Check for deflation. * 10 CONTINUE * IF( N0.LT.I0 ) $ RETURN IF( N0.EQ.I0 ) $ GO TO 20 NN = 4*N0 + PP IF( N0.EQ.( I0+1 ) ) $ GO TO 40 * * Check whether E(N0-1) is negligible, 1 eigenvalue. * IF( Z( NN-5 ).GT.TOL2*( SIGMA+Z( NN-3 ) ) .AND. $ Z( NN-2*PP-4 ).GT.TOL2*Z( NN-7 ) ) $ GO TO 30 * 20 CONTINUE * Z( 4*N0-3 ) = Z( 4*N0+PP-3 ) + SIGMA N0 = N0 - 1 GO TO 10 * * Check whether E(N0-2) is negligible, 2 eigenvalues. * 30 CONTINUE * IF( Z( NN-9 ).GT.TOL2*SIGMA .AND. $ Z( NN-2*PP-8 ).GT.TOL2*Z( NN-11 ) ) $ GO TO 50 * 40 CONTINUE * IF( Z( NN-3 ).GT.Z( NN-7 ) ) THEN S = Z( NN-3 ) Z( NN-3 ) = Z( NN-7 ) Z( NN-7 ) = S END IF T = HALF*( ( Z( NN-7 )-Z( NN-3 ) )+Z( NN-5 ) ) IF( Z( NN-5 ).GT.Z( NN-3 )*TOL2.AND.T.NE.ZERO ) THEN S = Z( NN-3 )*( Z( NN-5 ) / T ) IF( S.LE.T ) THEN S = Z( NN-3 )*( Z( NN-5 ) / $ ( T*( ONE+SQRT( ONE+S / T ) ) ) ) ELSE S = Z( NN-3 )*( Z( NN-5 ) / ( T+SQRT( T )*SQRT( T+S ) ) ) END IF T = Z( NN-7 ) + ( S+Z( NN-5 ) ) Z( NN-3 ) = Z( NN-3 )*( Z( NN-7 ) / T ) Z( NN-7 ) = T END IF Z( 4*N0-7 ) = Z( NN-7 ) + SIGMA Z( 4*N0-3 ) = Z( NN-3 ) + SIGMA N0 = N0 - 2 GO TO 10 * 50 CONTINUE IF( PP.EQ.2 ) $ PP = 0 * * Reverse the qd-array, if warranted. * IF( DMIN.LE.ZERO .OR. N0.LT.N0IN ) THEN IF( CBIAS*Z( 4*I0+PP-3 ).LT.Z( 4*N0+PP-3 ) ) THEN IPN4 = 4*( I0+N0 ) DO 60 J4 = 4*I0, 2*( I0+N0-1 ), 4 TEMP = Z( J4-3 ) Z( J4-3 ) = Z( IPN4-J4-3 ) Z( IPN4-J4-3 ) = TEMP TEMP = Z( J4-2 ) Z( J4-2 ) = Z( IPN4-J4-2 ) Z( IPN4-J4-2 ) = TEMP TEMP = Z( J4-1 ) Z( J4-1 ) = Z( IPN4-J4-5 ) Z( IPN4-J4-5 ) = TEMP TEMP = Z( J4 ) Z( J4 ) = Z( IPN4-J4-4 ) Z( IPN4-J4-4 ) = TEMP 60 CONTINUE IF( N0-I0.LE.4 ) THEN Z( 4*N0+PP-1 ) = Z( 4*I0+PP-1 ) Z( 4*N0-PP ) = Z( 4*I0-PP ) END IF DMIN2 = MIN( DMIN2, Z( 4*N0+PP-1 ) ) Z( 4*N0+PP-1 ) = MIN( Z( 4*N0+PP-1 ), Z( 4*I0+PP-1 ), $ Z( 4*I0+PP+3 ) ) Z( 4*N0-PP ) = MIN( Z( 4*N0-PP ), Z( 4*I0-PP ), $ Z( 4*I0-PP+4 ) ) QMAX = MAX( QMAX, Z( 4*I0+PP-3 ), Z( 4*I0+PP+1 ) ) DMIN = -ZERO END IF END IF * * Choose a shift. * CALL SLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN, DN1, $ DN2, TAU, TTYPE, G ) * * Call dqds until DMIN > 0. * 70 CONTINUE * CALL SLASQ5( I0, N0, Z, PP, TAU, SIGMA, DMIN, DMIN1, DMIN2, DN, $ DN1, DN2, IEEE, EPS ) * NDIV = NDIV + ( N0-I0+2 ) ITER = ITER + 1 * * Check status. * IF( DMIN.GE.ZERO .AND. DMIN1.GE.ZERO ) THEN * * Success. * GO TO 90 * ELSE IF( DMIN.LT.ZERO .AND. DMIN1.GT.ZERO .AND. $ Z( 4*( N0-1 )-PP ).LT.TOL*( SIGMA+DN1 ) .AND. $ ABS( DN ).LT.TOL*SIGMA ) THEN * * Convergence hidden by negative DN. * Z( 4*( N0-1 )-PP+2 ) = ZERO DMIN = ZERO GO TO 90 ELSE IF( DMIN.LT.ZERO ) THEN * * TAU too big. Select new TAU and try again. * NFAIL = NFAIL + 1 IF( TTYPE.LT.-22 ) THEN * * Failed twice. Play it safe. * TAU = ZERO ELSE IF( DMIN1.GT.ZERO ) THEN * * Late failure. Gives excellent shift. * TAU = ( TAU+DMIN )*( ONE-TWO*EPS ) TTYPE = TTYPE - 11 ELSE * * Early failure. Divide by 4. * TAU = QURTR*TAU TTYPE = TTYPE - 12 END IF GO TO 70 ELSE IF( SISNAN( DMIN ) ) THEN * * NaN. * IF( TAU.EQ.ZERO ) THEN GO TO 80 ELSE TAU = ZERO GO TO 70 END IF ELSE * * Possible underflow. Play it safe. * GO TO 80 END IF * * Risk of underflow. * 80 CONTINUE CALL SLASQ6( I0, N0, Z, PP, DMIN, DMIN1, DMIN2, DN, DN1, DN2 ) NDIV = NDIV + ( N0-I0+2 ) ITER = ITER + 1 TAU = ZERO * 90 CONTINUE IF( TAU.LT.SIGMA ) THEN DESIG = DESIG + TAU T = SIGMA + DESIG DESIG = DESIG - ( T-SIGMA ) ELSE T = SIGMA + TAU DESIG = SIGMA - ( T-TAU ) + DESIG END IF SIGMA = T * RETURN * * End of SLASQ3 * END