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RAYBENDS.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME RAYBENDS
* Problem:
* ********
* A ray bending problem. A ray across a inhomogeneous 2D medium is
* represented by a beta-spline whose knots can be chosen. The problem
* is then to optimize the positions of these knots in order to obtain a
* ray path corresponding to the minimum travel time from source to receiver,
* according to Fermat principle.
* In this version, 10 points are used in every interval of the curve
* defining the ray in order to compute more accurate travel times.
* Source: a test example in
* T.J. Moser, G. Nolet and R. Snieder,
* "Ray bending revisited",
* Bulletin of the Seism. Society of America 21(1).
* SIF input: Ph Toint, Dec 1991.
* classification OXR2-MY-V-0
* number of spline knots ( >= 4 )
* ( n = 2 * ( NK - 1 ) )
*IE NK 4 $-PARAMETER n = 6
*IE NK 11 $-PARAMETER n = 20
*IE NK 26 $-PARAMETER n = 50 original value
*IE NK 32 $-PARAMETER n = 62
*IE NK 64 $-PARAMETER n = 126
*IE NK 512 $-PARAMETER n = 1022
IE NK 1024 $-PARAMETER n = 2046
* source position
RE XSRC 0.0
RE ZSRC 0.0
* receiver position
RE XRCV 100.0
RE ZRCV 100.0
* derived from the number of knots
IA NK-1 NK -1
IA NK-2 NK -2
IA NK+1 NK 1
* useful constants
IE 0 0
IE 1 1
IE 2 2
IE 3 3
VARIABLES
* the unknowns are the two coordinates of the spline knots
DO I 0 NK
X X(I)
X Z(I)
OD I
GROUPS
* The objective is the travel time along the currently defined ray
* One group is defined for the time spent on each spline segment.
DO I 0 NK+1
XN TIME(I)
XN TIME(I) 'SCALE' 2.0
OD I
BOUNDS
FR RAYBENDS 'DEFAULT'
* The extreme knots coincide with the source and receiver
ZX RAYBENDS X(0) XSRC
ZX RAYBENDS Z(0) ZSRC
ZX RAYBENDS X(NK) XRCV
ZX RAYBENDS Z(NK) ZRCV
START POINT
* The initial knots are chosen as equidistant points on the straight
* line joining the source and receiver.
R- XRANGE XRCV XSRC
R- ZRANGE ZRCV ZSRC
RI RKNOTS NK
DO I 0 NK
RI REALI I
R/ FRAC REALI RKNOTS
R* XINCR FRAC XRANGE
R* ZINCR FRAC ZRANGE
R+ XC XSRC XINCR
R+ ZC ZSRC ZINCR
ZV RAYBENDS X(I) XC
ZV RAYBENDS Z(I) ZC
OD I
ELEMENT TYPE
EV TT X1 X2
EV TT X3 X4
EV TT Z1 Z2
EV TT Z3 Z4
IV TT ZZ1 ZZ2
IV TT ZZ3 ZZ4
IV TT X2MX1 X3MX1
IV TT X4MX1
* The range is difficult here. One knows that the element is invariant
* wrt to translations in all 4 Xs!!!!
ELEMENT USES
XT 'DEFAULT' TT
ZV T(0) X1 X(0)
ZV T(0) X2 X(0)
ZV T(0) X3 X(0)
ZV T(0) X4 X(1)
ZV T(0) Z1 Z(0)
ZV T(0) Z2 Z(0)
ZV T(0) Z3 Z(0)
ZV T(0) Z4 Z(1)
ZV T(1) X1 X(0)
ZV T(1) X2 X(0)
ZV T(1) X3 X(1)
ZV T(1) X4 X(2)
ZV T(1) Z1 Z(0)
ZV T(1) Z2 Z(0)
ZV T(1) Z3 Z(1)
ZV T(1) Z4 Z(2)
DO I 2 NK-1
IA I-1 I -1
IA I-2 I -2
IA I+1 I 1
ZV T(I) X1 X(I-2)
ZV T(I) X2 X(I-1)
ZV T(I) X3 X(I)
ZV T(I) X4 X(I+1)
ZV T(I) Z1 Z(I-2)
ZV T(I) Z2 Z(I-1)
ZV T(I) Z3 Z(I)
ZV T(I) Z4 Z(I+1)
OD I
ZV T(NK) X1 X(NK-2)
ZV T(NK) X2 X(NK-1)
ZV T(NK) X3 X(NK)
ZV T(NK) X4 X(NK)
ZV T(NK) Z1 Z(NK-2)
ZV T(NK) Z2 Z(NK-1)
ZV T(NK) Z3 Z(NK)
ZV T(NK) Z4 Z(NK)
ZV T(NK+1) X1 X(NK-1)
ZV T(NK+1) X2 X(NK)
ZV T(NK+1) X3 X(NK)
ZV T(NK+1) X4 X(NK)
ZV T(NK+1) Z1 Z(NK-1)
ZV T(NK+1) Z2 Z(NK)
ZV T(NK+1) Z3 Z(NK)
ZV T(NK+1) Z4 Z(NK)
GROUP USES
DO I 0 NK+1
XE TIME(I) T(I)
OD I
OBJECT BOUND
* Analytical solution to the continuous problem
*LO RAYBENDS 96.2424
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS RAYBENDS
TEMPORARIES
R FVAL
R GVAL
R HVAL
F FVAL
F GVAL
F HVAL
INDIVIDUALS
T TT
R X2MX1 X1 -1.0 X2 1.0
R X3MX1 X1 -1.0 X3 1.0
R X4MX1 X1 -1.0 X4 1.0
R ZZ1 Z1 1.0
R ZZ2 Z2 1.0
R ZZ3 Z3 1.0
R ZZ4 Z4 1.0
F FVAL( X2MX1, X3MX1, X4MX1,
F+ ZZ1, ZZ2, ZZ3, ZZ4 )
G X2MX1 GVAL( 1, X2MX1, X3MX1, X4MX1,
G+ ZZ1, ZZ2, ZZ3, ZZ4 )
G X3MX1 GVAL( 2, X2MX1, X3MX1, X4MX1,
G+ ZZ1, ZZ2, ZZ3, ZZ4 )
G X4MX1 GVAL( 3, X2MX1, X3MX1, X4MX1,
G+ ZZ1, ZZ2, ZZ3, ZZ4 )
G ZZ1 GVAL( 4, X2MX1, X3MX1, X4MX1,
G+ ZZ1, ZZ2, ZZ3, ZZ4 )
G ZZ2 GVAL( 5, X2MX1, X3MX1, X4MX1,
G+ ZZ1, ZZ2, ZZ3, ZZ4 )
G ZZ3 GVAL( 6, X2MX1, X3MX1, X4MX1,
G+ ZZ1, ZZ2, ZZ3, ZZ4 )
G ZZ4 GVAL( 7, X2MX1, X3MX1, X4MX1,
G+ ZZ1, ZZ2, ZZ3, ZZ4 )
H X2MX1 X2MX1 HVAL( 1, 1, X2MX1, X3MX1, X4MX1,
H+ ZZ1, ZZ2, ZZ3, ZZ4 )
H X2MX1 X3MX1 HVAL( 1, 2, X2MX1, X3MX1, X4MX1,
H+ ZZ1, ZZ2, ZZ3, ZZ4 )
H X2MX1 X4MX1 HVAL( 1, 3, X2MX1, X3MX1, X4MX1,
H+ ZZ1, ZZ2, ZZ3, ZZ4 )
H X2MX1 ZZ1 HVAL( 1, 4, X2MX1, X3MX1, X4MX1,
H+ ZZ1, ZZ2, ZZ3, ZZ4 )
H X2MX1 ZZ2 HVAL( 1, 5, X2MX1, X3MX1, X4MX1,
H+ ZZ1, ZZ2, ZZ3, ZZ4 )
H X2MX1 ZZ3 HVAL( 1, 6, X2MX1, X3MX1, X4MX1,
H+ ZZ1, ZZ2, ZZ3, ZZ4 )
H X2MX1 ZZ4 HVAL( 1, 7, X2MX1, X3MX1, X4MX1,
H+ ZZ1, ZZ2, ZZ3, ZZ4 )
H X3MX1 X3MX1 HVAL( 2, 2, X2MX1, X3MX1, X4MX1,
H+ ZZ1, ZZ2, ZZ3, ZZ4 )
H X3MX1 X4MX1 HVAL( 2, 3, X2MX1, X3MX1, X4MX1,
H+ ZZ1, ZZ2, ZZ3, ZZ4 )
H X3MX1 ZZ1 HVAL( 2, 4, X2MX1, X3MX1, X4MX1,
H+ ZZ1, ZZ2, ZZ3, ZZ4 )
H X3MX1 ZZ2 HVAL( 2, 5, X2MX1, X3MX1, X4MX1,
H+ ZZ1, ZZ2, ZZ3, ZZ4 )
H X3MX1 ZZ3 HVAL( 2, 6, X2MX1, X3MX1, X4MX1,
H+ ZZ1, ZZ2, ZZ3, ZZ4 )
H X3MX1 ZZ4 HVAL( 2, 7, X2MX1, X3MX1, X4MX1,
H+ ZZ1, ZZ2, ZZ3, ZZ4 )
H X4MX1 X4MX1 HVAL( 3, 3, X2MX1, X3MX1, X4MX1,
H+ ZZ1, ZZ2, ZZ3, ZZ4 )
H X4MX1 ZZ1 HVAL( 3, 4, X2MX1, X3MX1, X4MX1,
H+ ZZ1, ZZ2, ZZ3, ZZ4 )
H X4MX1 ZZ2 HVAL( 3, 5, X2MX1, X3MX1, X4MX1,
H+ ZZ1, ZZ2, ZZ3, ZZ4 )
H X4MX1 ZZ3 HVAL( 3, 6, X2MX1, X3MX1, X4MX1,
H+ ZZ1, ZZ2, ZZ3, ZZ4 )
H X4MX1 ZZ4 HVAL( 3, 7, X2MX1, X3MX1, X4MX1,
H+ ZZ1, ZZ2, ZZ3, ZZ4 )
H ZZ1 ZZ1 HVAL( 4, 4, X2MX1, X3MX1, X4MX1,
H+ ZZ1, ZZ2, ZZ3, ZZ4 )
H ZZ1 ZZ2 HVAL( 4, 5, X2MX1, X3MX1, X4MX1,
H+ ZZ1, ZZ2, ZZ3, ZZ4 )
H ZZ1 ZZ3 HVAL( 4, 6, X2MX1, X3MX1, X4MX1,
H+ ZZ1, ZZ2, ZZ3, ZZ4 )
H ZZ1 ZZ4 HVAL( 4, 7, X2MX1, X3MX1, X4MX1,
H+ ZZ1, ZZ2, ZZ3, ZZ4 )
H ZZ2 ZZ2 HVAL( 5, 5, X2MX1, X3MX1, X4MX1,
H+ ZZ1, ZZ2, ZZ3, ZZ4 )
H ZZ2 ZZ3 HVAL( 5, 6, X2MX1, X3MX1, X4MX1,
H+ ZZ1, ZZ2, ZZ3, ZZ4 )
H ZZ2 ZZ4 HVAL( 5, 7, X2MX1, X3MX1, X4MX1,
H+ ZZ1, ZZ2, ZZ3, ZZ4 )
H ZZ3 ZZ3 HVAL( 6, 6, X2MX1, X3MX1, X4MX1,
H+ ZZ1, ZZ2, ZZ3, ZZ4 )
H ZZ3 ZZ4 HVAL( 6, 7, X2MX1, X3MX1, X4MX1,
H+ ZZ1, ZZ2, ZZ3, ZZ4 )
H ZZ4 ZZ4 HVAL( 7, 7, X2MX1, X3MX1, X4MX1,
H+ ZZ1, ZZ2, ZZ3, ZZ4 )
ENDATA
C
DOUBLE PRECISION FUNCTION FVAL( X2MX1, X3MX1, X4MX1,
+ ZZ1, ZZ2, ZZ3, ZZ4 )
DOUBLE PRECISION X2MX1, X3MX1, X4MX1, ZZ1, ZZ2, ZZ3, ZZ4
C
C Compute the objective's value
C
DOUBLE PRECISION F, G(7), H(7,7)
C
CALL EVAL( 0, F, G, H, X2MX1, X3MX1, X4MX1, ZZ1, ZZ2, ZZ3, ZZ4)
FVAL = F
RETURN
END
C
C
C
DOUBLE PRECISION FUNCTION GVAL( I, X2MX1, X3MX1, X4MX1,
+ ZZ1, ZZ2, ZZ3, ZZ4 )
DOUBLE PRECISION X2MX1, X3MX1, X4MX1, ZZ1, ZZ2, ZZ3, ZZ4
INTEGER I
C
C Compute the gradient values
C
DOUBLE PRECISION F, G(7), H(7,7)
C
CALL EVAL( 1, F, G, H, X2MX1, X3MX1, X4MX1, ZZ1, ZZ2, ZZ3, ZZ4)
GVAL = G( I )
RETURN
END
C
C
C
DOUBLE PRECISION FUNCTION HVAL( I, J, X2MX1, X3MX1, X4MX1,
+ ZZ1, ZZ2, ZZ3, ZZ4 )
DOUBLE PRECISION X2MX1, X3MX1, X4MX1, ZZ1, ZZ2, ZZ3, ZZ4
INTEGER I, J
C
C Compute the Hessian values
C
DOUBLE PRECISION F, G(7), H(7,7)
C
CALL EVAL( 2, F, G, H, X2MX1, X3MX1, X4MX1, ZZ1, ZZ2, ZZ3, ZZ4)
IF ( I .LE. J ) THEN
HVAL = H( I, J )
ELSE
HVAL = H( J, I )
ENDIF
RETURN
END
C
C
C
SUBROUTINE EVAL( DER, F, G, H, X2MX1, X3MX1, X4MX1,
+ ZZ1, ZZ2, ZZ3, ZZ4 )
DOUBLE PRECISION X2MX1, X3MX1, X4MX1, ZZ1, ZZ2, ZZ3, ZZ4
DOUBLE PRECISION F, G(7), H(7,7)
INTEGER DER
C
C number of points within a spline interval
C
INTEGER M
PARAMETER ( M = 10 )
C
C
DOUBLE PRECISION C(M+1), DX, Z(M+1), T(M+1), ZDZ1(M+1)
DOUBLE PRECISION ZDZ2(M+1), ZDZ3(M+1), ZDZ4(M+1), DXD2, DXD3
DOUBLE PRECISION DXD4, V, VDZ1, VDZ2, VDZ3, VDZ4, VDZ1Z1
DOUBLE PRECISION VDZ1Z2, VDZ1Z3, VDZ1Z4, VDZ2Z3, VDZ2Z4, VDZ3Z4
DOUBLE PRECISION VDZ2Z2, VDZ3Z3, VDZ4Z4, DZ, DZDZ1, DZDZ2, DZDZ3
DOUBLE PRECISION DZDZ4, CC, CN, CSQ, CNSQ, CCB, CNCB, R, RD2
DOUBLE PRECISION RD3, RD4, RDZ1, RDZ2, RDZ3, RDZ4, FACT1, FACT2
DOUBLE PRECISION FACT3, RD2D2, RD2D3, RD2D4, RD2DZ1, RD2DZ2
DOUBLE PRECISION RD2DZ3, RD2DZ4, RD3D3, RD3D4, RD3DZ1, RD3DZ2
DOUBLE PRECISION RD3DZ3, RD3DZ4, RD4D4, RD4DZ1, RD4DZ2, RD4DZ3
DOUBLE PRECISION RD4DZ4, RDZ1Z1, RDZ1Z2, RDZ1Z3, RDZ1Z4, RDZ2Z2
DOUBLE PRECISION RDZ2Z3, RDZ2Z4, RDZ3Z3, RDZ3Z4, RDZ4Z4
DOUBLE PRECISION Q
EXTERNAL Q
INTEGER K, L
INTRINSIC SQRT
DO 10 K = 1, M+1
T(K) = 0.1D0 * ( K - 1.0D0 )
ZDZ1(K) = Q ( 1, T(K) )
ZDZ2(K) = Q ( 2, T(K) )
ZDZ3(K) = Q ( 3, T(K) )
ZDZ4(K) = Q ( 4, T(K) )
Z(K) = ZZ1 * ZDZ1( K ) + ZZ2 * ZDZ2( K )
+ + ZZ3 * ZDZ3( K ) + ZZ4 * ZDZ4( K )
C(K) = 1.0D0 + 0.01 * Z(K)
10 CONTINUE
F = 0.0D0
IF ( DER .GE. 1 ) THEN
DO 40 K = 1, 7
G(K) = 0.0D0
IF ( DER .EQ. 2 ) THEN
DO 50 L = 1, 7
H( K, L ) = 0.0D0
50 CONTINUE
ENDIF
40 CONTINUE
ENDIF
DO 30 K = 1, M
DXD2 = Q( 2, T( K + 1 ) ) - Q( 2, T( K ) )
DXD3 = Q( 3, T( K + 1 ) ) - Q( 3, T( K ) )
DXD4 = Q( 4, T( K + 1 ) ) - Q( 4, T( K ) )
DX = X2MX1 * DXD2 + X3MX1 * DXD3 + X4MX1 * DXD4
CC = C( K )
CN = C( K + 1 )
V = 1.0D0 / CC + 1.0D0 / CN
IF ( DER . GE. 1 ) THEN
CSQ = - 0.01 / ( CC * CC )
CNSQ = - 0.01 / ( CN * CN )
VDZ1 = ZDZ1( K + 1 ) * CNSQ + ZDZ1( K ) * CSQ
VDZ2 = ZDZ2( K + 1 ) * CNSQ + ZDZ2( K ) * CSQ
VDZ3 = ZDZ3( K + 1 ) * CNSQ + ZDZ3( K ) * CSQ
VDZ4 = ZDZ4( K + 1 ) * CNSQ + ZDZ4( K ) * CSQ
IF ( DER .EQ. 2 ) THEN
CCB = 0.0002 / CC**3
CNCB = 0.0002 / CN**3
VDZ1Z1 = ZDZ1( K + 1 ) * ZDZ1( K + 1 ) * CNCB
+ + ZDZ1( K ) * ZDZ1( K ) * CCB
VDZ1Z2 = ZDZ1( K + 1 ) * ZDZ2( K + 1 ) * CNCB
+ + ZDZ1( K ) * ZDZ2( K ) * CCB
VDZ1Z3 = ZDZ1( K + 1 ) * ZDZ3( K + 1 ) * CNCB
+ + ZDZ1( K ) * ZDZ3( K ) * CCB
VDZ1Z4 = ZDZ1( K + 1 ) * ZDZ4( K + 1 ) * CNCB
+ + ZDZ1( K ) * ZDZ4( K ) * CCB
VDZ2Z2 = ZDZ2( K + 1 ) * ZDZ2( K + 1 ) * CNCB
+ + ZDZ2( K ) * ZDZ2( K ) * CCB
VDZ2Z3 = ZDZ2( K + 1 ) * ZDZ3( K + 1 ) * CNCB
+ + ZDZ2( K ) * ZDZ3( K ) * CCB
VDZ2Z4 = ZDZ2( K + 1 ) * ZDZ4( K + 1 ) * CNCB
+ + ZDZ2( K ) * ZDZ4( K ) * CCB
VDZ3Z3 = ZDZ3( K + 1 ) * ZDZ3( K + 1 ) * CNCB
+ + ZDZ3( K ) * ZDZ3( K ) * CCB
VDZ3Z4 = ZDZ3( K + 1 ) * ZDZ4( K + 1 ) * CNCB
+ + ZDZ3( K ) * ZDZ4( K ) * CCB
VDZ4Z4 = ZDZ4( K + 1 ) * ZDZ4( K + 1 ) * CNCB
+ + ZDZ4( K ) * ZDZ4( K ) * CCB
ENDIF
ENDIF
DZ = Z( K + 1) - Z( K )
IF ( DER .GE. 1 ) THEN
DZDZ1 = ZDZ1( K + 1 ) - ZDZ1( K )
DZDZ2 = ZDZ2( K + 1 ) - ZDZ2( K )
DZDZ3 = ZDZ3( K + 1 ) - ZDZ3( K )
DZDZ4 = ZDZ4( K + 1 ) - ZDZ4( K )
ENDIF
R = SQRT( DX * DX + DZ * DZ )
IF ( DER .GE. 1 ) THEN
RD2 = DX * DXD2 / R
RD3 = DX * DXD3 / R
RD4 = DX * DXD4 / R
RDZ1 = DZ * DZDZ1 / R
RDZ2 = DZ * DZDZ2 / R
RDZ3 = DZ * DZDZ3 / R
RDZ4 = DZ * DZDZ4 / R
IF ( DER .EQ. 2 ) THEN
FACT1 = ( 1.0D0 - DX**2 / ( R * R ) ) / R
FACT2 = - DX * DZ / R**3
FACT3 = ( 1.0D0 - DZ**2 / ( R * R ) ) / R
RD2D2 = FACT1 * DXD2 * DXD2
RD2D3 = FACT1 * DXD2 * DXD3
RD2D4 = FACT1 * DXD2 * DXD4
RD2DZ1 = FACT2 * DXD2 * DZDZ1
RD2DZ2 = FACT2 * DXD2 * DZDZ2
RD2DZ3 = FACT2 * DXD2 * DZDZ3
RD2DZ4 = FACT2 * DXD2 * DZDZ4
RD3D3 = FACT1 * DXD3 * DXD3
RD3D4 = FACT1 * DXD3 * DXD4
RD3DZ1 = FACT2 * DXD3 * DZDZ1
RD3DZ2 = FACT2 * DXD3 * DZDZ2
RD3DZ3 = FACT2 * DXD3 * DZDZ3
RD3DZ4 = FACT2 * DXD3 * DZDZ4
RD4D4 = FACT1 * DXD4 * DXD4
RD4DZ1 = FACT2 * DXD4 * DZDZ1
RD4DZ2 = FACT2 * DXD4 * DZDZ2
RD4DZ3 = FACT2 * DXD4 * DZDZ3
RD4DZ4 = FACT2 * DXD4 * DZDZ4
RDZ1Z1 = FACT3 * DZDZ1 * DZDZ1
RDZ1Z2 = FACT3 * DZDZ1 * DZDZ2
RDZ1Z3 = FACT3 * DZDZ1 * DZDZ3
RDZ1Z4 = FACT3 * DZDZ1 * DZDZ4
RDZ2Z2 = FACT3 * DZDZ2 * DZDZ2
RDZ2Z3 = FACT3 * DZDZ2 * DZDZ3
RDZ2Z4 = FACT3 * DZDZ2 * DZDZ4
RDZ3Z3 = FACT3 * DZDZ3 * DZDZ3
RDZ3Z4 = FACT3 * DZDZ3 * DZDZ4
RDZ4Z4 = FACT3 * DZDZ4 * DZDZ4
ENDIF
ENDIF
F = F + V * R
IF ( DER .GE. 1 ) THEN
G(1) = G(1) + V * RD2
G(2) = G(2) + V * RD3
G(3) = G(3) + V * RD4
G(4) = G(4) + V * RDZ1 + VDZ1 * R
G(5) = G(5) + V * RDZ2 + VDZ2 * R
G(6) = G(6) + V * RDZ3 + VDZ3 * R
G(7) = G(7) + V * RDZ4 + VDZ4 * R
IF ( DER .EQ. 2 ) THEN
H(1,1) = H(1,1) + V * RD2D2
H(1,2) = H(1,2) + V * RD2D3
H(1,3) = H(1,3) + V * RD2D4
H(1,4) = H(1,4) + VDZ1 * RD2 + V * RD2DZ1
H(1,5) = H(1,5) + VDZ2 * RD2 + V * RD2DZ2
H(1,6) = H(1,6) + VDZ3 * RD2 + V * RD2DZ3
H(1,7) = H(1,7) + VDZ4 * RD2 + V * RD2DZ4
H(2,2) = H(2,2) + V * RD3D3
H(2,3) = H(2,3) + V * RD3D4
H(2,4) = H(2,4) + VDZ1 * RD3 + V * RD3DZ1
H(2,5) = H(2,5) + VDZ2 * RD3 + V * RD3DZ2
H(2,6) = H(2,6) + VDZ3 * RD3 + V * RD3DZ3
H(2,7) = H(2,7) + VDZ4 * RD3 + V * RD3DZ4
H(3,3) = H(3,3) + V * RD4D4
H(3,4) = H(3,4) + VDZ1 * RD4 + V * RD4DZ1
H(3,5) = H(3,5) + VDZ2 * RD4 + V * RD4DZ2
H(3,6) = H(3,6) + VDZ3 * RD4 + V * RD4DZ3
H(3,7) = H(3,7) + VDZ4 * RD4 + V * RD4DZ4
H(4,4) = H(4,4) + V * RDZ1Z1 + VDZ1Z1 * R
+ + 2.0D0 * VDZ1 * RDZ1
H(4,5) = H(4,5) + V * RDZ1Z2 + VDZ1 * RDZ2 + VDZ2 * RDZ1
+ + R * VDZ1Z2
H(4,6) = H(4,6) + V * RDZ1Z3 + VDZ1 * RDZ3 + VDZ3 * RDZ1
+ + R * VDZ1Z3
H(4,7) = H(4,7) + V * RDZ1Z4 + VDZ1 * RDZ4 + VDZ4 * RDZ1
+ + R * VDZ1Z4
H(5,5) = H(5,5) + V * RDZ2Z2 + VDZ2Z2 * R
+ + 2.0D0 * VDZ2 * RDZ2
H(5,6) = H(5,6) + V * RDZ2Z3 + VDZ2 * RDZ3 + VDZ3 * RDZ2
+ + R * VDZ2Z3
H(5,7) = H(5,7) + V * RDZ2Z4 + VDZ2 * RDZ4 + VDZ4 * RDZ2
+ + R * VDZ2Z4
H(6,6) = H(6,6) + V * RDZ3Z3 + VDZ3Z3 * R
+ + 2.0D0 * VDZ3 * RDZ3
H(6,7) = H(6,7) + V * RDZ3Z4 + VDZ3 * RDZ4 + VDZ4 * RDZ3
+ + R * VDZ3Z4
H(7,7) = H(7,7) + V * RDZ4Z4 + VDZ4Z4 * R
+ + 2.0D0 * VDZ4 * RDZ4
ENDIF
ENDIF
30 CONTINUE
END
C
C
C
DOUBLE PRECISION FUNCTION Q( I, X )
DOUBLE PRECISION X
INTEGER I
C
C Compute the beta-spline coefficients
C
IF ( I .EQ. 1 ) THEN
Q = 0.166666667 - 0.5 * X + 0.5 * X * X - 0.1666667 * X**3
ELSE IF ( I .EQ. 2) THEN
Q = 0.666666667 - X * X + 0.5 * X**3
ELSE IF ( I. EQ. 3 ) THEN
Q = 0.166666667 + 0.5 * X + 0.5 * X * X - 0.5 * X**3
ELSE
Q = 0.166666667 * X**3
ENDIF
RETURN
END