root / src / lapack / double / dlasq4.f @ 10
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SUBROUTINE DLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN, |
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$ DN1, DN2, TAU, TTYPE, G ) |
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* |
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* -- LAPACK routine (version 3.2) -- |
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* |
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* -- Contributed by Osni Marques of the Lawrence Berkeley National -- |
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* -- Laboratory and Beresford Parlett of the Univ. of California at -- |
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* -- Berkeley -- |
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* -- November 2008 -- |
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* |
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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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* |
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* .. Scalar Arguments .. |
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INTEGER I0, N0, N0IN, PP, TTYPE |
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DOUBLE PRECISION DMIN, DMIN1, DMIN2, DN, DN1, DN2, G, TAU |
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* .. |
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* .. Array Arguments .. |
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DOUBLE PRECISION 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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* DLASQ4 computes an approximation TAU to the smallest eigenvalue |
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* using values of d from the previous transform. |
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* |
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* I0 (input) INTEGER |
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* First index. |
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* |
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* N0 (input) INTEGER |
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* Last index. |
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* |
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* Z (input) DOUBLE PRECISION array, dimension ( 4*N ) |
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* Z holds the qd array. |
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* |
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* PP (input) INTEGER |
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* PP=0 for ping, PP=1 for pong. |
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* |
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* NOIN (input) INTEGER |
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* The value of N0 at start of EIGTEST. |
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* |
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* DMIN (input) DOUBLE PRECISION |
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* Minimum value of d. |
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* |
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* DMIN1 (input) DOUBLE PRECISION |
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* Minimum value of d, excluding D( N0 ). |
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* |
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* DMIN2 (input) DOUBLE PRECISION |
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* Minimum value of d, excluding D( N0 ) and D( N0-1 ). |
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* |
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* DN (input) DOUBLE PRECISION |
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* d(N) |
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* |
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* DN1 (input) DOUBLE PRECISION |
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* d(N-1) |
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* |
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* DN2 (input) DOUBLE PRECISION |
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* d(N-2) |
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* |
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* TAU (output) DOUBLE PRECISION |
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* This is the shift. |
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* |
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* TTYPE (output) INTEGER |
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* Shift type. |
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* |
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* G (input/output) REAL |
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* G is passed as an argument in order to save its value between |
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* calls to DLASQ4. |
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* |
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* Further Details |
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* =============== |
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* CNST1 = 9/16 |
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* |
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* ===================================================================== |
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* |
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* .. Parameters .. |
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DOUBLE PRECISION CNST1, CNST2, CNST3 |
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PARAMETER ( CNST1 = 0.5630D0, CNST2 = 1.010D0, |
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$ CNST3 = 1.050D0 ) |
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DOUBLE PRECISION QURTR, THIRD, HALF, ZERO, ONE, TWO, HUNDRD |
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PARAMETER ( QURTR = 0.250D0, THIRD = 0.3330D0, |
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$ HALF = 0.50D0, ZERO = 0.0D0, ONE = 1.0D0, |
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$ TWO = 2.0D0, HUNDRD = 100.0D0 ) |
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* .. |
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* .. Local Scalars .. |
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INTEGER I4, NN, NP |
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DOUBLE PRECISION A2, B1, B2, GAM, GAP1, GAP2, S |
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* .. |
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* .. Intrinsic Functions .. |
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INTRINSIC MAX, MIN, SQRT |
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* .. |
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* .. Executable Statements .. |
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* |
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* A negative DMIN forces the shift to take that absolute value |
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* TTYPE records the type of shift. |
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* |
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IF( DMIN.LE.ZERO ) THEN |
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TAU = -DMIN |
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TTYPE = -1 |
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RETURN |
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END IF |
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* |
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NN = 4*N0 + PP |
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IF( N0IN.EQ.N0 ) THEN |
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* |
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* No eigenvalues deflated. |
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* |
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IF( DMIN.EQ.DN .OR. DMIN.EQ.DN1 ) THEN |
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* |
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B1 = SQRT( Z( NN-3 ) )*SQRT( Z( NN-5 ) ) |
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B2 = SQRT( Z( NN-7 ) )*SQRT( Z( NN-9 ) ) |
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A2 = Z( NN-7 ) + Z( NN-5 ) |
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* |
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* Cases 2 and 3. |
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* |
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IF( DMIN.EQ.DN .AND. DMIN1.EQ.DN1 ) THEN |
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GAP2 = DMIN2 - A2 - DMIN2*QURTR |
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IF( GAP2.GT.ZERO .AND. GAP2.GT.B2 ) THEN |
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GAP1 = A2 - DN - ( B2 / GAP2 )*B2 |
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ELSE |
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GAP1 = A2 - DN - ( B1+B2 ) |
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END IF |
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IF( GAP1.GT.ZERO .AND. GAP1.GT.B1 ) THEN |
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S = MAX( DN-( B1 / GAP1 )*B1, HALF*DMIN ) |
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TTYPE = -2 |
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ELSE |
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S = ZERO |
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IF( DN.GT.B1 ) |
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$ S = DN - B1 |
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IF( A2.GT.( B1+B2 ) ) |
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$ S = MIN( S, A2-( B1+B2 ) ) |
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S = MAX( S, THIRD*DMIN ) |
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TTYPE = -3 |
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END IF |
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ELSE |
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* |
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* Case 4. |
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* |
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TTYPE = -4 |
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S = QURTR*DMIN |
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IF( DMIN.EQ.DN ) THEN |
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GAM = DN |
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A2 = ZERO |
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IF( Z( NN-5 ) .GT. Z( NN-7 ) ) |
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$ RETURN |
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B2 = Z( NN-5 ) / Z( NN-7 ) |
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NP = NN - 9 |
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ELSE |
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NP = NN - 2*PP |
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B2 = Z( NP-2 ) |
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GAM = DN1 |
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IF( Z( NP-4 ) .GT. Z( NP-2 ) ) |
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$ RETURN |
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A2 = Z( NP-4 ) / Z( NP-2 ) |
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IF( Z( NN-9 ) .GT. Z( NN-11 ) ) |
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$ RETURN |
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B2 = Z( NN-9 ) / Z( NN-11 ) |
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NP = NN - 13 |
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END IF |
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* |
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* Approximate contribution to norm squared from I < NN-1. |
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* |
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A2 = A2 + B2 |
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DO 10 I4 = NP, 4*I0 - 1 + PP, -4 |
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IF( B2.EQ.ZERO ) |
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$ GO TO 20 |
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B1 = B2 |
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IF( Z( I4 ) .GT. Z( I4-2 ) ) |
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$ RETURN |
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B2 = B2*( Z( I4 ) / Z( I4-2 ) ) |
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A2 = A2 + B2 |
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IF( HUNDRD*MAX( B2, B1 ).LT.A2 .OR. CNST1.LT.A2 ) |
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$ GO TO 20 |
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10 CONTINUE |
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20 CONTINUE |
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A2 = CNST3*A2 |
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* |
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* Rayleigh quotient residual bound. |
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* |
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IF( A2.LT.CNST1 ) |
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$ S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 ) |
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END IF |
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ELSE IF( DMIN.EQ.DN2 ) THEN |
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* |
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* Case 5. |
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* |
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TTYPE = -5 |
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S = QURTR*DMIN |
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* |
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* Compute contribution to norm squared from I > NN-2. |
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* |
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NP = NN - 2*PP |
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B1 = Z( NP-2 ) |
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B2 = Z( NP-6 ) |
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GAM = DN2 |
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IF( Z( NP-8 ).GT.B2 .OR. Z( NP-4 ).GT.B1 ) |
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$ RETURN |
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A2 = ( Z( NP-8 ) / B2 )*( ONE+Z( NP-4 ) / B1 ) |
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* |
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* Approximate contribution to norm squared from I < NN-2. |
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* |
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IF( N0-I0.GT.2 ) THEN |
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B2 = Z( NN-13 ) / Z( NN-15 ) |
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A2 = A2 + B2 |
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DO 30 I4 = NN - 17, 4*I0 - 1 + PP, -4 |
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IF( B2.EQ.ZERO ) |
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$ GO TO 40 |
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B1 = B2 |
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IF( Z( I4 ) .GT. Z( I4-2 ) ) |
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$ RETURN |
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B2 = B2*( Z( I4 ) / Z( I4-2 ) ) |
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A2 = A2 + B2 |
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IF( HUNDRD*MAX( B2, B1 ).LT.A2 .OR. CNST1.LT.A2 ) |
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$ GO TO 40 |
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30 CONTINUE |
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40 CONTINUE |
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A2 = CNST3*A2 |
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END IF |
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* |
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IF( A2.LT.CNST1 ) |
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$ S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 ) |
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ELSE |
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* |
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* Case 6, no information to guide us. |
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* |
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IF( TTYPE.EQ.-6 ) THEN |
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G = G + THIRD*( ONE-G ) |
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ELSE IF( TTYPE.EQ.-18 ) THEN |
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G = QURTR*THIRD |
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ELSE |
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G = QURTR |
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END IF |
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S = G*DMIN |
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TTYPE = -6 |
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END IF |
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* |
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ELSE IF( N0IN.EQ.( N0+1 ) ) THEN |
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* |
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* One eigenvalue just deflated. Use DMIN1, DN1 for DMIN and DN. |
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* |
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IF( DMIN1.EQ.DN1 .AND. DMIN2.EQ.DN2 ) THEN |
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* |
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* Cases 7 and 8. |
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* |
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TTYPE = -7 |
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S = THIRD*DMIN1 |
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IF( Z( NN-5 ).GT.Z( NN-7 ) ) |
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$ RETURN |
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B1 = Z( NN-5 ) / Z( NN-7 ) |
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B2 = B1 |
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IF( B2.EQ.ZERO ) |
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$ GO TO 60 |
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DO 50 I4 = 4*N0 - 9 + PP, 4*I0 - 1 + PP, -4 |
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A2 = B1 |
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IF( Z( I4 ).GT.Z( I4-2 ) ) |
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$ RETURN |
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B1 = B1*( Z( I4 ) / Z( I4-2 ) ) |
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B2 = B2 + B1 |
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IF( HUNDRD*MAX( B1, A2 ).LT.B2 ) |
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$ GO TO 60 |
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50 CONTINUE |
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60 CONTINUE |
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B2 = SQRT( CNST3*B2 ) |
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A2 = DMIN1 / ( ONE+B2**2 ) |
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GAP2 = HALF*DMIN2 - A2 |
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IF( GAP2.GT.ZERO .AND. GAP2.GT.B2*A2 ) THEN |
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S = MAX( S, A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ) ) |
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ELSE |
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S = MAX( S, A2*( ONE-CNST2*B2 ) ) |
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TTYPE = -8 |
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END IF |
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ELSE |
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* |
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* Case 9. |
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* |
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S = QURTR*DMIN1 |
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IF( DMIN1.EQ.DN1 ) |
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$ S = HALF*DMIN1 |
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TTYPE = -9 |
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END IF |
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* |
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ELSE IF( N0IN.EQ.( N0+2 ) ) THEN |
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* |
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* Two eigenvalues deflated. Use DMIN2, DN2 for DMIN and DN. |
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* |
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* Cases 10 and 11. |
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* |
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IF( DMIN2.EQ.DN2 .AND. TWO*Z( NN-5 ).LT.Z( NN-7 ) ) THEN |
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TTYPE = -10 |
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S = THIRD*DMIN2 |
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IF( Z( NN-5 ).GT.Z( NN-7 ) ) |
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$ RETURN |
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B1 = Z( NN-5 ) / Z( NN-7 ) |
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B2 = B1 |
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IF( B2.EQ.ZERO ) |
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$ GO TO 80 |
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DO 70 I4 = 4*N0 - 9 + PP, 4*I0 - 1 + PP, -4 |
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IF( Z( I4 ).GT.Z( I4-2 ) ) |
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$ RETURN |
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B1 = B1*( Z( I4 ) / Z( I4-2 ) ) |
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B2 = B2 + B1 |
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IF( HUNDRD*B1.LT.B2 ) |
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$ GO TO 80 |
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70 CONTINUE |
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80 CONTINUE |
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B2 = SQRT( CNST3*B2 ) |
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A2 = DMIN2 / ( ONE+B2**2 ) |
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GAP2 = Z( NN-7 ) + Z( NN-9 ) - |
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$ SQRT( Z( NN-11 ) )*SQRT( Z( NN-9 ) ) - A2 |
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IF( GAP2.GT.ZERO .AND. GAP2.GT.B2*A2 ) THEN |
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S = MAX( S, A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ) ) |
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ELSE |
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S = MAX( S, A2*( ONE-CNST2*B2 ) ) |
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END IF |
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ELSE |
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S = QURTR*DMIN2 |
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TTYPE = -11 |
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END IF |
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ELSE IF( N0IN.GT.( N0+2 ) ) THEN |
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* |
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* Case 12, more than two eigenvalues deflated. No information. |
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* |
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S = ZERO |
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TTYPE = -12 |
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END IF |
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* |
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TAU = S |
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RETURN |
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* |
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* End of DLASQ4 |
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* |
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END |