#include "ESMF_LapackBlas.inc" !> \brief \b DLASQ4 ! ! =========== DOCUMENTATION =========== ! ! Online html documentation available at ! http://www.netlib.org/lapack/explore-html/ ! !> \htmlonly !> Download DLASQ4 + dependencies !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasq4.f"> !> [TGZ]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasq4.f"> !> [ZIP]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasq4.f"> !> [TXT]</a> !> \endhtmlonly ! ! Definition: ! =========== ! ! SUBROUTINE DLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN, ! DN1, DN2, TAU, TTYPE, G ) ! ! .. Scalar Arguments .. ! INTEGER I0, N0, N0IN, PP, TTYPE ! DOUBLE PRECISION DMIN, DMIN1, DMIN2, DN, DN1, DN2, G, TAU ! .. ! .. Array Arguments .. ! DOUBLE PRECISION Z( * ) ! .. ! ! !> \par Purpose: ! ============= !> !> \verbatim !> !> DLASQ4 computes an approximation TAU to the smallest eigenvalue !> using values of d from the previous transform. !> \endverbatim ! ! Arguments: ! ========== ! !> \param[in] I0 !> \verbatim !> I0 is INTEGER !> First index. !> \endverbatim !> !> \param[in] N0 !> \verbatim !> N0 is INTEGER !> Last index. !> \endverbatim !> !> \param[in] Z !> \verbatim !> Z is DOUBLE PRECISION array, dimension ( 4*N ) !> Z holds the qd array. !> \endverbatim !> !> \param[in] PP !> \verbatim !> PP is INTEGER !> PP=0 for ping, PP=1 for pong. !> \endverbatim !> !> \param[in] N0IN !> \verbatim !> N0IN is INTEGER !> The value of N0 at start of EIGTEST. !> \endverbatim !> !> \param[in] DMIN !> \verbatim !> DMIN is DOUBLE PRECISION !> Minimum value of d. !> \endverbatim !> !> \param[in] DMIN1 !> \verbatim !> DMIN1 is DOUBLE PRECISION !> Minimum value of d, excluding D( N0 ). !> \endverbatim !> !> \param[in] DMIN2 !> \verbatim !> DMIN2 is DOUBLE PRECISION !> Minimum value of d, excluding D( N0 ) and D( N0-1 ). !> \endverbatim !> !> \param[in] DN !> \verbatim !> DN is DOUBLE PRECISION !> d(N) !> \endverbatim !> !> \param[in] DN1 !> \verbatim !> DN1 is DOUBLE PRECISION !> d(N-1) !> \endverbatim !> !> \param[in] DN2 !> \verbatim !> DN2 is DOUBLE PRECISION !> d(N-2) !> \endverbatim !> !> \param[out] TAU !> \verbatim !> TAU is DOUBLE PRECISION !> This is the shift. !> \endverbatim !> !> \param[out] TTYPE !> \verbatim !> TTYPE is INTEGER !> Shift type. !> \endverbatim !> !> \param[in,out] G !> \verbatim !> G is REAL !> G is passed as an argument in order to save its value between !> calls to DLASQ4. !> \endverbatim ! ! Authors: ! ======== ! !> \author Univ. of Tennessee !> \author Univ. of California Berkeley !> \author Univ. of Colorado Denver !> \author NAG Ltd. ! !> \date November 2011 ! !> \ingroup auxOTHERcomputational ! !> \par Further Details: ! ===================== !> !> \verbatim !> !> CNST1 = 9/16 !> \endverbatim !> ! ===================================================================== SUBROUTINE DLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN, & & DN1, DN2, TAU, TTYPE, G ) ! ! -- LAPACK computational routine (version 3.4.0) -- ! -- LAPACK is a software package provided by Univ. of Tennessee, -- ! -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- ! November 2011 ! ! .. Scalar Arguments .. INTEGER I0, N0, N0IN, PP, TTYPE DOUBLE PRECISION DMIN, DMIN1, DMIN2, DN, DN1, DN2, G, TAU ! .. ! .. Array Arguments .. DOUBLE PRECISION Z( * ) ! .. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION CNST1, CNST2, CNST3 PARAMETER ( CNST1 = 0.5630D0, CNST2 = 1.010D0, & & CNST3 = 1.050D0 ) DOUBLE PRECISION QURTR, THIRD, HALF, ZERO, ONE, TWO, HUNDRD PARAMETER ( QURTR = 0.250D0, THIRD = 0.3330D0, & & HALF = 0.50D0, ZERO = 0.0D0, ONE = 1.0D0, & & TWO = 2.0D0, HUNDRD = 100.0D0 ) ! .. ! .. Local Scalars .. INTEGER I4, NN, NP DOUBLE PRECISION A2, B1, B2, GAM, GAP1, GAP2, S ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! ! A negative DMIN forces the shift to take that absolute value ! TTYPE records the type of shift. ! IF( DMIN.LE.ZERO ) THEN TAU = -DMIN TTYPE = -1 RETURN END IF ! NN = 4*N0 + PP IF( N0IN.EQ.N0 ) THEN ! ! No eigenvalues deflated. ! IF( DMIN.EQ.DN .OR. DMIN.EQ.DN1 ) THEN ! B1 = SQRT( Z( NN-3 ) )*SQRT( Z( NN-5 ) ) B2 = SQRT( Z( NN-7 ) )*SQRT( Z( NN-9 ) ) A2 = Z( NN-7 ) + Z( NN-5 ) ! ! Cases 2 and 3. ! IF( DMIN.EQ.DN .AND. DMIN1.EQ.DN1 ) THEN GAP2 = DMIN2 - A2 - DMIN2*QURTR IF( GAP2.GT.ZERO .AND. GAP2.GT.B2 ) THEN GAP1 = A2 - DN - ( B2 / GAP2 )*B2 ELSE GAP1 = A2 - DN - ( B1+B2 ) END IF IF( GAP1.GT.ZERO .AND. GAP1.GT.B1 ) THEN S = MAX( DN-( B1 / GAP1 )*B1, HALF*DMIN ) TTYPE = -2 ELSE S = ZERO IF( DN.GT.B1 ) & & S = DN - B1 IF( A2.GT.( B1+B2 ) ) & & S = MIN( S, A2-( B1+B2 ) ) S = MAX( S, THIRD*DMIN ) TTYPE = -3 END IF ELSE ! ! Case 4. ! TTYPE = -4 S = QURTR*DMIN IF( DMIN.EQ.DN ) THEN GAM = DN A2 = ZERO IF( Z( NN-5 ) .GT. Z( NN-7 ) ) & & RETURN B2 = Z( NN-5 ) / Z( NN-7 ) NP = NN - 9 ELSE NP = NN - 2*PP B2 = Z( NP-2 ) GAM = DN1 IF( Z( NP-4 ) .GT. Z( NP-2 ) ) & & RETURN A2 = Z( NP-4 ) / Z( NP-2 ) IF( Z( NN-9 ) .GT. Z( NN-11 ) ) & & RETURN B2 = Z( NN-9 ) / Z( NN-11 ) NP = NN - 13 END IF ! ! Approximate contribution to norm squared from I < NN-1. ! A2 = A2 + B2 DO 10 I4 = NP, 4*I0 - 1 + PP, -4 IF( B2.EQ.ZERO ) & & GO TO 20 B1 = B2 IF( Z( I4 ) .GT. Z( I4-2 ) ) & & RETURN B2 = B2*( Z( I4 ) / Z( I4-2 ) ) A2 = A2 + B2 IF( HUNDRD*MAX( B2, B1 ).LT.A2 .OR. CNST1.LT.A2 ) & & GO TO 20 10 CONTINUE 20 CONTINUE A2 = CNST3*A2 ! ! Rayleigh quotient residual bound. ! IF( A2.LT.CNST1 ) & & S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 ) END IF ELSE IF( DMIN.EQ.DN2 ) THEN ! ! Case 5. ! TTYPE = -5 S = QURTR*DMIN ! ! Compute contribution to norm squared from I > NN-2. ! NP = NN - 2*PP B1 = Z( NP-2 ) B2 = Z( NP-6 ) GAM = DN2 IF( Z( NP-8 ).GT.B2 .OR. Z( NP-4 ).GT.B1 ) & & RETURN A2 = ( Z( NP-8 ) / B2 )*( ONE+Z( NP-4 ) / B1 ) ! ! Approximate contribution to norm squared from I < NN-2. ! IF( N0-I0.GT.2 ) THEN B2 = Z( NN-13 ) / Z( NN-15 ) A2 = A2 + B2 DO 30 I4 = NN - 17, 4*I0 - 1 + PP, -4 IF( B2.EQ.ZERO ) & & GO TO 40 B1 = B2 IF( Z( I4 ) .GT. Z( I4-2 ) ) & & RETURN B2 = B2*( Z( I4 ) / Z( I4-2 ) ) A2 = A2 + B2 IF( HUNDRD*MAX( B2, B1 ).LT.A2 .OR. CNST1.LT.A2 ) & & GO TO 40 30 CONTINUE 40 CONTINUE A2 = CNST3*A2 END IF ! IF( A2.LT.CNST1 ) & & S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 ) ELSE ! ! Case 6, no information to guide us. ! IF( TTYPE.EQ.-6 ) THEN G = G + THIRD*( ONE-G ) ELSE IF( TTYPE.EQ.-18 ) THEN G = QURTR*THIRD ELSE G = QURTR END IF S = G*DMIN TTYPE = -6 END IF ! ELSE IF( N0IN.EQ.( N0+1 ) ) THEN ! ! One eigenvalue just deflated. Use DMIN1, DN1 for DMIN and DN. ! IF( DMIN1.EQ.DN1 .AND. DMIN2.EQ.DN2 ) THEN ! ! Cases 7 and 8. ! TTYPE = -7 S = THIRD*DMIN1 IF( Z( NN-5 ).GT.Z( NN-7 ) ) & & RETURN B1 = Z( NN-5 ) / Z( NN-7 ) B2 = B1 IF( B2.EQ.ZERO ) & & GO TO 60 DO 50 I4 = 4*N0 - 9 + PP, 4*I0 - 1 + PP, -4 A2 = B1 IF( Z( I4 ).GT.Z( I4-2 ) ) & & RETURN B1 = B1*( Z( I4 ) / Z( I4-2 ) ) B2 = B2 + B1 IF( HUNDRD*MAX( B1, A2 ).LT.B2 ) & & GO TO 60 50 CONTINUE 60 CONTINUE B2 = SQRT( CNST3*B2 ) A2 = DMIN1 / ( ONE+B2**2 ) GAP2 = HALF*DMIN2 - A2 IF( GAP2.GT.ZERO .AND. GAP2.GT.B2*A2 ) THEN S = MAX( S, A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ) ) ELSE S = MAX( S, A2*( ONE-CNST2*B2 ) ) TTYPE = -8 END IF ELSE ! ! Case 9. ! S = QURTR*DMIN1 IF( DMIN1.EQ.DN1 ) & & S = HALF*DMIN1 TTYPE = -9 END IF ! ELSE IF( N0IN.EQ.( N0+2 ) ) THEN ! ! Two eigenvalues deflated. Use DMIN2, DN2 for DMIN and DN. ! ! Cases 10 and 11. ! IF( DMIN2.EQ.DN2 .AND. TWO*Z( NN-5 ).LT.Z( NN-7 ) ) THEN TTYPE = -10 S = THIRD*DMIN2 IF( Z( NN-5 ).GT.Z( NN-7 ) ) & & RETURN B1 = Z( NN-5 ) / Z( NN-7 ) B2 = B1 IF( B2.EQ.ZERO ) & & GO TO 80 DO 70 I4 = 4*N0 - 9 + PP, 4*I0 - 1 + PP, -4 IF( Z( I4 ).GT.Z( I4-2 ) ) & & RETURN B1 = B1*( Z( I4 ) / Z( I4-2 ) ) B2 = B2 + B1 IF( HUNDRD*B1.LT.B2 ) & & GO TO 80 70 CONTINUE 80 CONTINUE B2 = SQRT( CNST3*B2 ) A2 = DMIN2 / ( ONE+B2**2 ) GAP2 = Z( NN-7 ) + Z( NN-9 ) - & & SQRT( Z( NN-11 ) )*SQRT( Z( NN-9 ) ) - A2 IF( GAP2.GT.ZERO .AND. GAP2.GT.B2*A2 ) THEN S = MAX( S, A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ) ) ELSE S = MAX( S, A2*( ONE-CNST2*B2 ) ) END IF ELSE S = QURTR*DMIN2 TTYPE = -11 END IF ELSE IF( N0IN.GT.( N0+2 ) ) THEN ! ! Case 12, more than two eigenvalues deflated. No information. ! S = ZERO TTYPE = -12 END IF ! TAU = S RETURN ! ! End of DLASQ4 ! END