C ALGORITHM 792, COLLECTED ALGORITHMS FROM ACM.
C THIS WORK PUBLISHED IN TRANSACTIONS ON MATHEMATICAL SOFTWARE,
C VOL. 25,NO. 1, March, 1999, P. 78-- 94.
#! /bin/sh
# This is a shell archive, meaning:
# 1. Remove everything above the #! /bin/sh line.
# 2. Save the resulting text in a file.
# 3. Execute the file with /bin/sh (not csh) to create the files:
# Doc/
# Doc/PackingList
# Fortran77/
# Fortran77/Drivers/
# Fortran77/Drivers/Sp/
# Fortran77/Drivers/Sp/data
# Fortran77/Drivers/Sp/driver.f
# Fortran77/Drivers/Sp/res
# Fortran77/Drivers/Sp/res1_2
# Fortran77/Drivers/Sp/src_790.f
# Fortran77/Src/
# Fortran77/Src/Sp/
# Fortran77/Src/Sp/src.f
# This archive created: Tue Jul 27 14:09:54 1999
export PATH; PATH=/bin:$PATH
if test ! -d 'Doc'
then
mkdir 'Doc'
fi
cd 'Doc'
if test -f 'PackingList'
then
echo shar: will not over-write existing file "'PackingList'"
else
cat << SHAR_EOF > 'PackingList'
Algorithm:792
Language:Fortran77
Precision:Sp
Src:src.f
Driver_0:driver.f src_790.f @Src
DriverLib_0:port
Data_0:stdin=data
Res_0:stdout=res
SHAR_EOF
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cd ..
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fi
cd 'Fortran77'
if test ! -d 'Drivers'
then
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fi
cd 'Drivers'
if test ! -d 'Sp'
then
mkdir 'Sp'
fi
cd 'Sp'
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else
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1
1
10
17
30
6
2
1
10
17
30
6
3
1
10
17
24
3
4
1
10
17
30
6
5
1
10
17
30
5
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C
C CS2TST
C 11/20/98
C
C This program computes interpolation errors using the
C scattered data package CSHEP2D for each of ten test
C functions and a 33 by 33 uniform grid of interpolation
C points in the unit square.
C
C This program uses Subroutines TESTDT and TSTFN1 from
C ACM Algorithm SURVEY to generate a node set and and the
C test function values.
C
INTEGER NMAX, NRMAX, NI
PARAMETER (NMAX=100, NRMAX=10, NI=33)
C
C Array storage:
C
DOUBLE PRECISION X(NMAX), Y(NMAX), W(NMAX), RW(NMAX),
. A(9,NMAX), P(NI), FT(NI,NI)
INTEGER LCELL(NRMAX,NRMAX), LNEXT(NMAX)
C
DOUBLE PRECISION DEL, DUM, DX, DY, ERMAX, ERMEAN, PW,
. RMAX, SSA, SSE, SSM, SUM, XMIN, YMIN
DOUBLE PRECISION CS2VAL
INTEGER I, IER, J, K, KF, KFF, KFL, KS, LOUT,
. N, NC, NFUN, NP, NR, NSET, NW, NWMAX
C
C Data:
C
C LOUT = Logical unit number for output.
C NSET = Number of node sets.
C NFUN = Number of test functions.
C
DATA LOUT/2/, NSET/5/, NFUN/10/
OPEN (LOUT,FILE='res1_2')
C
C Input format:
C
100 FORMAT (I2)
C
C Get a user-specified node set number KS.
C
1 WRITE (*,110) NSET
110 FORMAT (///13X,'CS2TST: CSHEP2D Test Program'//
. 5X,'Specify a node set number in the range 1',
. ' to ',I2,':'/)
READ (*,100,ERR=1) KS
IF (KS .LT. 1 .OR. KS .GT. NSET) GO TO 1
C
C Copy N and the nodal coordinates for node set KS.
C
CALL TESTDT (KS, N,X,Y)
IF (N .LT. 10 .OR. N .GT. NMAX) GO TO 20
C
C Allow the user to specify a range of function numbers.
C
WRITE (*,120) NFUN
120 FORMAT (//5X,'Specify the first test function ',
. '(1 to ',I2,'):'/)
READ (*,100,ERR=1) KFF
IF (KFF .LT. 1 .OR. KFF .GT. NFUN) GO TO 1
C
WRITE (*,130) KFF, NFUN
130 FORMAT (//5X,'Specify the last test function (',
. I2,' to ',I2,'):'/)
READ (*,100,ERR=1) KFL
IF (KFL .LT. KFF .OR. KFL .GT. NFUN) GO TO 1
NFUN = KFL-KFF+1
C
C Input NC, NW, and NR from the console.
C
NWMAX = MIN(40,N-1)
2 WRITE (*,140) N
140 FORMAT (//5X,'N =',I4//5X,
. 'Specify the number of nodes NC for the ',
. 'least squares fits.'/5X,'NC = 17 is ',
. 'recommended. NC GE 9.'//
. 10X,'NC = '/)
READ (*,100,ERR=2) NC
IF (NC .LT. 9 .OR. NC .GT. NWMAX) GO TO 2
C
3 WRITE (*,150)
150 FORMAT (///5X,'Specify the number of nodes NW for ',
. 'the weights. NW = 30 is'/5X,'recommended. ',
. ' 1 LE NW LE MIN(40,N-1).'//10X,'NW = '/)
READ (*,100,ERR=2) NW
IF (1 .GT. NW .OR. NW .GT. NWMAX) GO TO 3
C
4 WRITE (*,160) NRMAX
160 FORMAT (///5X,'Specify the number of rows and column',
. 's NR in the uniform grid'/5X,'of cells used',
. ' to locate nearest neighbors. NR = Sqrt(N/',
. '3) is'/5X,'recommended. 1 LE NR LE ',I2,
. '.'//10X,'NR = '/)
READ (*,100,ERR=3) NR
IF (NR .LT. 1 .OR. NR .GT. NRMAX) GO TO 4
C
C Set up uniform grid points.
C
DEL = 1./DBLE(NI-1)
DO 5 I = 1,NI
P(I) = DBLE(I-1)*DEL
5 CONTINUE
C
C Initialize the average SSE/SSM value to zero.
C
SSA = 0.
C
C Print a heading and loop on test functions.
C
WRITE (LOUT,200) KS, N, NI, NC, NW, NR
DO 11 KF = KFF,KFL
C
C Compute true function values at the nodes.
C
DO 6 K = 1,N
CALL TSTFN1 (KF,X(K),Y(K),0, W(K),DUM,DUM)
6 CONTINUE
C
C Compute true function values FT on the uniform grid, and
C accumulate the sum of values SUM and sum of squared
C values SSM.
C
SUM = 0.
SSM = 0.
DO 8 I = 1,NI
DO 7 J = 1,NI
CALL TSTFN1 (KF,P(I),P(J),0, FT(I,J),DUM,DUM)
SUM = SUM + FT(I,J)
SSM = SSM + FT(I,J)**2
7 CONTINUE
8 CONTINUE
C
C Compute the sum of squared deviations from the mean SSM.
C
SSM = SSM - SUM*SUM/DBLE(NI*NI)
C
C Compute parameters A and RW defining the interpolant.
C
CALL CSHEP2 (N,X,Y,W,NC,NW,NR, LCELL,LNEXT,XMIN,
. YMIN,DX,DY,RMAX,RW,A,IER)
IF (IER .NE. 0) GO TO 21
C
C Compute interpolation errors.
C
ERMEAN = 0.
ERMAX = 0.
SSE = 0.
DO 10 I = 1,NI
DO 9 J = 1,NI
PW = CS2VAL (P(I),P(J),N,X,Y,W,NR,LCELL,LNEXT,
. XMIN,YMIN,DX,DY,RMAX,RW,A) -
. FT(I,J)
ERMEAN = ERMEAN + ABS(PW)
ERMAX = MAX(ERMAX,ABS(PW))
SSE = SSE + PW*PW
9 CONTINUE
10 CONTINUE
NP = NI*NI
ERMEAN = ERMEAN/DBLE(NP)
SSE = SSE/SSM
SSA = SSA + SSE
WRITE (LOUT,210) KF, ERMAX, ERMEAN, SSE
11 CONTINUE
C
C Print the average SSE/SSM value (averaged over the test
C functions).
C
WRITE (LOUT,220) SSA/DBLE(NFUN)
STOP
C
C N is outside its valid range.
C
20 WRITE (LOUT,500) N, NMAX
STOP
C
C Error in CSHEP2.
C
21 IF (IER .EQ. 2) WRITE (LOUT,510)
IF (IER .EQ. 3) WRITE (LOUT,520)
STOP
C
C Print formats:
C
200 FORMAT (30X,'CS2TST Output:'//
. 1X,16X,'CSHEP2D Interpolation Errors for ',
. 'N nodes and an'/
. 1X,6X,'NI by NI Uniform Grid of Interpolation',
. 'ion Points in the Unit Square'//1X,
. 6X,'Node set ',I2,4X,'N =',I4,4X,'NI = ',I2,
. 4X,'NC = ',I2,4X,'NW = ',I2,4X,'NR = ',I2///
. 1X,16X,'Function',4X,'Max Error',4X,
. 'Mean Error',4X,'SSE/SSM'/)
210 FORMAT (1X,19X,I2,9X,F7.4,6X,F8.5,2X,F9.6)
220 FORMAT (//1X,11X,'Average SSE/SSM over the test ',
. 'functions = ',F9.6)
C
C Error message formats:
C
500 FORMAT (///1X,10X,'*** Error in data -- N = ',I4,
. ', Maximum value =',I4,' ***')
510 FORMAT (///1X,14X,'*** Error in CSHEP2 -- duplicate ',
. 'nodes encountered ***')
520 FORMAT (///1X,14X,'*** Error in CSHEP2 -- all nodes ',
. 'are collinear ***')
END
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CS2TST: CSHEP2D Test Program
Specify a node set number in the range 1 to 5:
Specify the first test function (1 to 10):
Specify the last test function ( 1 to 10):
N = 100
Specify the number of nodes NC for the least squares fits.
NC = 17 is recommended. NC GE 9.
NC =
Specify the number of nodes NW for the weights. NW = 30 is
recommended. 1 LE NW LE MIN(40,N-1).
NW =
Specify the number of rows and columns NR in the uniform grid
of cells used to locate nearest neighbors. NR = Sqrt(N/3) is
recommended. 1 LE NR LE 10.
NR =
SHAR_EOF
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CS2TST Output:
CSHEP2D Interpolation Errors for N nodes and an
NI by NI Uniform Grid of Interpolationion Points in the Unit Square
Node set 1 N = 100 NI = 33 NC = 17 NW = 30 NR = 6
Function Max Error Mean Error SSE/SSM
1 0.0451 0.00396 0.000560
2 0.0367 0.00222 0.002090
3 0.0100 0.00050 0.000182
4 0.0012 0.00016 0.000010
5 0.0109 0.00076 0.000345
6 0.0020 0.00010 0.000010
7 0.8037 0.07071 0.011088
8 0.2982 0.02706 0.008277
9 12.7220 0.76454 0.003507
10 0.2634 0.01735 0.022579
Average SSE/SSM over the test functions = 0.004865
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SUBROUTINE CSHEP2 (N,X,Y,F,NC,NW,NR, LCELL,LNEXT,XMIN,
. YMIN,DX,DY,RMAX,RW,A,IER)
INTEGER N, NC, NW, NR, LCELL(NR,NR), LNEXT(N), IER
DOUBLE PRECISION X(N), Y(N), F(N), XMIN, YMIN, DX,
. DY, RMAX, RW(N), A(9,N)
C
C***********************************************************
C
C From CSHEP2D
C Robert J. Renka
C Dept. of Computer Science
C Univ. of North Texas
C renka@cs.unt.edu
C 02/13/97
C
C This subroutine computes a set of parameters defining a
C C2 (twice continuously differentiable) bivariate function
C C(X,Y) which interpolates data values F at a set of N
C arbitrarily distributed points (X,Y) in the plane (nodes).
C The interpolant C may be evaluated at an arbitrary point
C by function CS2VAL, and its first partial derivatives are
C computed by Subroutine CS2GRD.
C
C The interpolation scheme is a modified Cubic Shepard
C method:
C
C C = [W(1)*C(1)+W(2)*C(2)+..+W(N)*C(N)]/[W(1)+W(2)+..+W(N)]
C
C for bivariate functions W(k) and C(k). The nodal func-
C tions are given by
C
C C(k)(x,y) = A(1,k)*(x-X(k))**3 +
C A(2,k)*(x-X(k))**2*(y-Y(k)) +
C A(3,k)*(x-X(k))*(y-Y(k))**2 +
C A(4,k)*(y-Y(k))**3 + A(5,k)*(x-X(k))**2 +
C A(6,k)*(x-X(k))*(y-Y(k)) + A(7,k)*(y-Y(k))**2
C + A(8,k)*(x-X(k)) + A(9,k)*(y-Y(k)) + F(k) .
C
C Thus, C(k) is a cubic function which interpolates the data
C value at node k. Its coefficients A(,k) are obtained by a
C weighted least squares fit to the closest NC data points
C with weights similar to W(k). Note that the radius of
C influence for the least squares fit is fixed for each k,
C but varies with k.
C
C The weights are taken to be
C
C W(k)(x,y) = ( (R(k)-D(k))+ / R(k)*D(k) )**3 ,
C
C where (R(k)-D(k))+ = 0 if R(k) < D(k), and D(k)(x,y) is
C the Euclidean distance between (x,y) and (X(k),Y(k)). The
C radius of influence R(k) varies with k and is chosen so
C that NW nodes are within the radius. Note that W(k) is
C not defined at node (X(k),Y(k)), but C(x,y) has limit F(k)
C as (x,y) approaches (X(k),Y(k)).
C
C On input:
C
C N = Number of nodes and data values. N .GE. 10.
C
C X,Y = Arrays of length N containing the Cartesian
C coordinates of the nodes.
C
C F = Array of length N containing the data values
C in one-to-one correspondence with the nodes.
C
C NC = Number of data points to be used in the least
C squares fit for coefficients defining the nodal
C functions C(k). Values found to be optimal for
C test data sets ranged from 11 to 25. A recom-
C mended value for general data sets is NC = 17.
C For nodes lying on (or close to) a rectangular
C grid, the recommended value is NC = 11. In any
C case, NC must be in the range 9 to Min(40,N-1).
C
C NW = Number of nodes within (and defining) the radii
C of influence R(k) which enter into the weights
C W(k). For N sufficiently large, a recommended
C value is NW = 30. In general, NW should be
C about 1.5*NC. 1 .LE. NW .LE. Min(40,N-1).
C
C NR = Number of rows and columns in the cell grid de-
C fined in Subroutine STORE2. A rectangle con-
C taining the nodes is partitioned into cells in
C order to increase search efficiency. NR =
C Sqrt(N/3) is recommended. NR .GE. 1.
C
C The above parameters are not altered by this routine.
C
C LCELL = Array of length .GE. NR**2.
C
C LNEXT = Array of length .GE. N.
C
C RW = Array of length .GE. N.
C
C A = Array of length .GE. 9N.
C
C On output:
C
C LCELL = NR by NR array of nodal indexes associated
C with cells. Refer to Subroutine STORE2.
C
C LNEXT = Array of length N containing next-node
C indexes. Refer to Subroutine STORE2.
C
C XMIN,YMIN,DX,DY = Minimum nodal coordinates and cell
C dimensions. Refer to Subroutine
C STORE2.
C
C RMAX = Largest element in RW -- maximum radius R(k).
C
C RW = Array containing the the radii R(k) which enter
C into the weights W(k).
C
C A = 9 by N array containing the coefficients for
C cubic nodal function C(k) in column k.
C
C Note that the output parameters described above are not
C defined unless IER = 0.
C
C IER = Error indicator:
C IER = 0 if no errors were encountered.
C IER = 1 if N, NC, NW, or NR is outside its
C valid range.
C IER = 2 if duplicate nodes were encountered.
C IER = 3 if all nodes are collinear.
C
C Modules required by CSHEP2: GETNP2, GIVENS, ROTATE,
C SETUP2, STORE2
C
C Intrinsic functions called by CSHEP2: ABS, DBLE, MAX,
C MIN, SQRT
C
C***********************************************************
C
INTEGER LMX
PARAMETER (LMX=40)
INTEGER I, IERR, IP1, IRM1, IROW, J, JP1, K, LMAX,
. LNP, NEQ, NN, NNC, NNR, NNW, NP, NPTS(LMX),
. NCWMAX
DOUBLE PRECISION B(10,10), C, DDX, DDY, DMIN, DTOL,
. FK, RC, RS, RSMX, RSOLD, RTOL, RWS,
. S, SF, SFC, SFS, STF, SUM, T, XK,
. XMN, YK, YMN
C
DATA RTOL/1.D-5/, DTOL/.01/
C
C Local parameters:
C
C B = Transpose of the augmented regression matrix
C C = First component of the plane rotation used to
C zero the lower triangle of B**T -- computed
C by Subroutine GIVENS
C DDX,DDY = Local variables for DX and DY
C DMIN = Minimum of the magnitudes of the diagonal
C elements of the regression matrix after
C zeros are introduced below the diagonal
C DTOL = Tolerance for detecting an ill-conditioned
C system. The system is accepted when
C DMIN*RC .GE. DTOL.
C FK = Data value at mode K -- F(K)
C I = Index for A, B, and NPTS
C IERR = Error flag for the call to Subroutine STORE2
C IP1 = I+1
C IRM1 = IROW-1
C IROW = Row index for B
C J = Index for A and B
C JP1 = J+1
C K = Nodal function index and column index for A
C LMAX = Maximum number of NPTS elements
C LMX = Maximum value of LMAX
C LNP = Current length of NPTS
C NEQ = Number of equations in the least squares fit
C NN,NNC,NNR = Local copies of N, NC, and NR
C NNW = Local copy of NW
C NP = NPTS element
C NPTS = Array containing the indexes of a sequence of
C nodes to be used in the least squares fit
C or to compute RW. The nodes are ordered
C by distance from K, and the last element
C (usually indexed by LNP) is used only to
C determine RC, or RW(K) if NW > NC.
C NCWMAX = Max(NC,NW)
C RC = Radius of influence which enters into the
C weights for C(K) (see Subroutine SETUP2)
C RS = Squared distance between K and NPTS(LNP) --
C used to compute RC and RW(K)
C RSMX = Maximum squared RW element encountered
C RSOLD = Squared distance between K and NPTS(LNP-1) --
C used to compute a relative change in RS
C between succeeding NPTS elements
C RTOL = Tolerance for detecting a sufficiently large
C relative change in RS. If the change is
C not greater than RTOL, the nodes are
C treated as being the same distance from K
C RWS = Current squared value of RW(K)
C S = Second component of the plane rotation deter-
C mined by subroutine GIVENS
C SF = Scale factor for the linear terms (columns 8
C and 9) in the least squares fit -- inverse
C of the root-mean-square distance between K
C and the nodes (other than K) in the least
C squares fit
C SFS = Scale factor for the quadratic terms (columns
C 5, 6, and 7) in the least squares fit --
C SF*SF
C SFC = Scale factor for the cubic terms (first 4
C columns) in the least squares fit -- SF**3
C STF = Marquardt stabilization factor used to damp
C out the first 4 solution components (third
C partials of the cubic) when the system is
C ill-conditioned. As STF increases, the
C fitting function approaches a quadratic
C polynomial.
C SUM = Sum of squared Euclidean distances between
C node K and the nodes used in the least
C squares fit (unless additional nodes are
C added for stability)
C T = Temporary variable for accumulating a scalar
C product in the back solve
C XK,YK = Coordinates of node K -- X(K), Y(K)
C XMN,YMN = Local variables for XMIN and YMIN
C
NN = N
NNC = NC
NNW = NW
NNR = NR
NCWMAX = MAX(NNC,NNW)
LMAX = MIN(LMX,NN-1)
IF (NNC .LT. 9 .OR. NNW .LT. 1 .OR. NCWMAX .GT.
. LMAX .OR. NNR .LT. 1) GO TO 21
C
C Create the cell data structure, and initialize RSMX.
C
CALL STORE2 (NN,X,Y,NNR, LCELL,LNEXT,XMN,YMN,DDX,DDY,
. IERR)
IF (IERR .NE. 0) GO TO 23
RSMX = 0.
C
C Outer loop on node K:
C
DO 16 K = 1,NN
XK = X(K)
YK = Y(K)
FK = F(K)
C
C Mark node K to exclude it from the search for nearest
C neighbors.
C
LNEXT(K) = -LNEXT(K)
C
C Initialize for loop on NPTS.
C
RS = 0.
SUM = 0.
RWS = 0.
RC = 0.
LNP = 0
C
C Compute NPTS, LNP, RWS, NEQ, RC, and SFS.
C
1 SUM = SUM + RS
IF (LNP .EQ. LMAX) GO TO 2
LNP = LNP + 1
RSOLD = RS
CALL GETNP2 (XK,YK,X,Y,NNR,LCELL,LNEXT,XMN,YMN,
. DDX,DDY, NP,RS)
IF (RS .EQ. 0.) GO TO 22
NPTS(LNP) = NP
IF ( (RS-RSOLD)/RS .LT. RTOL ) GO TO 1
IF (RWS .EQ. 0. .AND. LNP .GT. NNW) RWS = RS
IF (RC .EQ. 0. .AND. LNP .GT. NNC) THEN
C
C RC = 0 (not yet computed) and LNP > NC. RC = Sqrt(RS)
C is sufficiently large to (strictly) include NC nodes.
C The least squares fit will include NEQ = LNP - 1
C equations for 9 .LE. NC .LE. NEQ .LT. LMAX .LE. N-1.
C
NEQ = LNP - 1
RC = SQRT(RS)
SFS = DBLE(NEQ)/SUM
ENDIF
C
C Bottom of loop -- test for termination.
C
IF (LNP .GT. NCWMAX) GO TO 3
GO TO 1
C
C All LMAX nodes are included in NPTS. RWS and/or RC**2 is
C (arbitrarily) taken to be 10 percent larger than the
C distance RS to the last node included.
C
2 IF (RWS .EQ. 0.) RWS = 1.1*RS
IF (RC .EQ. 0.) THEN
NEQ = LMAX
RC = SQRT(1.1*RS)
SFS = DBLE(NEQ)/SUM
ENDIF
C
C Store RW(K), update RSMX if necessary, and compute SF
C and SFC.
C
3 RW(K) = SQRT(RWS)
IF (RWS .GT. RSMX) RSMX = RWS
SF = SQRT(SFS)
SFC = SF*SFS
C
C A Q-R decomposition is used to solve the least squares
C system. The transpose of the augmented regression
C matrix is stored in B with columns (rows of B) defined
C as follows: 1-4 are the cubic terms, 5-7 are the quad-
C ratic terms, 8 and 9 are the linear terms, and the last
C column is the right hand side.
C
C Set up the equations and zero out the lower triangle with
C Givens rotations.
C
I = 0
4 I = I + 1
NP = NPTS(I)
IROW = MIN(I,10)
CALL SETUP2 (XK,YK,FK,X(NP),Y(NP),F(NP),SF,SFS,
. SFC,RC, B(1,IROW))
IF (I .EQ. 1) GO TO 4
IRM1 = IROW-1
DO 5 J = 1,IRM1
JP1 = J + 1
CALL GIVENS (B(J,J),B(J,IROW),C,S)
CALL ROTATE (10-J,C,S,B(JP1,J),B(JP1,IROW))
5 CONTINUE
IF (I .LT. NEQ) GO TO 4
C
C Test the system for ill-conditioning.
C
DMIN = MIN( ABS(B(1,1)),ABS(B(2,2)),ABS(B(3,3)),
. ABS(B(4,4)),ABS(B(5,5)),ABS(B(6,6)),
. ABS(B(7,7)),ABS(B(8,8)),ABS(B(9,9)) )
IF (DMIN*RC .GE. DTOL) GO TO 11
IF (NEQ .EQ. LMAX) GO TO 7
C
C Increase RC and add another equation to the system to
C improve the conditioning. The number of NPTS elements
C is also increased if necessary.
C
6 RSOLD = RS
NEQ = NEQ + 1
IF (NEQ .EQ. LMAX) THEN
RC = SQRT(1.1*RS)
GO TO 4
ENDIF
IF (NEQ .LT. LNP) THEN
C
C NEQ < LNP.
C
NP = NPTS(NEQ+1)
RS = (X(NP)-XK)**2 + (Y(NP)-YK)**2
IF ( (RS-RSOLD)/RS .LT. RTOL ) GO TO 6
RC = SQRT(RS)
GO TO 4
ENDIF
C
C NEQ = LNP. Add an element to NPTS.
C
LNP = LNP + 1
CALL GETNP2 (XK,YK,X,Y,NNR,LCELL,LNEXT,XMN,YMN,
. DDX,DDY, NP,RS)
IF (NP .EQ. 0) GO TO 22
NPTS(LNP) = NP
IF ( (RS-RSOLD)/RS .LT. RTOL ) GO TO 6
RC = SQRT(RS)
GO TO 4
C
C Stabilize the system by damping third partials -- add
C multiples of the first four unit vectors to the first
C four equations.
C
7 STF = 1.0/RC
DO 10 I = 1,4
B(I,10) = STF
IP1 = I + 1
DO 8 J = IP1,10
B(J,10) = 0.
8 CONTINUE
DO 9 J = I,9
JP1 = J + 1
CALL GIVENS (B(J,J),B(J,10),C,S)
CALL ROTATE (10-J,C,S,B(JP1,J),B(JP1,10))
9 CONTINUE
10 CONTINUE
C
C Test the damped system for ill-conditioning.
C
DMIN = MIN( ABS(B(5,5)),ABS(B(6,6)),ABS(B(7,7)),
. ABS(B(8,8)),ABS(B(9,9)) )
IF (DMIN*RC .LT. DTOL) GO TO 23
C
C Solve the 9 by 9 triangular system for the coefficients.
C
11 DO 13 I = 9,1,-1
T = 0.
IF (I .NE. 9) THEN
IP1 = I + 1
DO 12 J = IP1,9
T = T + B(J,I)*A(J,K)
12 CONTINUE
ENDIF
A(I,K) = (B(10,I)-T)/B(I,I)
13 CONTINUE
C
C Scale the coefficients to adjust for the column scaling.
C
DO 14 I = 1,4
A(I,K) = A(I,K)*SFC
14 CONTINUE
A(5,K) = A(5,K)*SFS
A(6,K) = A(6,K)*SFS
A(7,K) = A(7,K)*SFS
A(8,K) = A(8,K)*SF
A(9,K) = A(9,K)*SF
C
C Unmark K and the elements of NPTS.
C
LNEXT(K) = -LNEXT(K)
DO 15 I = 1,LNP
NP = NPTS(I)
LNEXT(NP) = -LNEXT(NP)
15 CONTINUE
16 CONTINUE
C
C No errors encountered.
C
XMIN = XMN
YMIN = YMN
DX = DDX
DY = DDY
RMAX = SQRT(RSMX)
IER = 0
RETURN
C
C N, NC, NW, or NR is outside its valid range.
C
21 IER = 1
RETURN
C
C Duplicate nodes were encountered by GETNP2.
C
22 IER = 2
RETURN
C
C No unique solution due to collinear nodes.
C
23 XMIN = XMN
YMIN = YMN
DX = DDX
DY = DDY
IER = 3
RETURN
END
DOUBLE PRECISION FUNCTION CS2VAL (PX,PY,N,X,Y,F,NR,
. LCELL,LNEXT,XMIN,YMIN,DX,DY,RMAX,RW,A)
INTEGER N, NR, LCELL(NR,NR), LNEXT(N)
DOUBLE PRECISION PX, PY, X(N), Y(N), F(N), XMIN, YMIN,
. DX, DY, RMAX, RW(N), A(9,N)
C
C***********************************************************
C
C From CSHEP2D
C Robert J. Renka
C Dept. of Computer Science
C Univ. of North Texas
C renka@cs.unt.edu
C 02/03/97
C
C This function returns the value C(PX,PY), where C is the
C weighted sum of cubic nodal functions defined in Subrou-
C tine CSHEP2. CS2GRD may be called to compute a gradient
C of C along with the value, and/or to test for errors.
C CS2HES may be called to compute a value, first partial
C derivatives, and second partial derivatives at a point.
C
C On input:
C
C PX,PY = Cartesian coordinates of the point P at
C which C is to be evaluated.
C
C N = Number of nodes and data values defining C.
C N .GE. 10.
C
C X,Y,F = Arrays of length N containing the nodes and
C data values interpolated by C.
C
C NR = Number of rows and columns in the cell grid.
C Refer to Subroutine STORE2. NR .GE. 1.
C
C LCELL = NR by NR array of nodal indexes associated
C with cells. Refer to Subroutine STORE2.
C
C LNEXT = Array of length N containing next-node
C indexes. Refer to Subroutine STORE2.
C
C XMIN,YMIN,DX,DY = Minimum nodal coordinates and cell
C dimensions. DX and DY must be
C positive. Refer to Subroutine
C STORE2.
C
C RMAX = Largest element in RW -- maximum radius R(k).
C
C RW = Array containing the the radii R(k) which enter
C into the weights W(k) defining C.
C
C A = 9 by N array containing the coefficients for
C cubic nodal function C(k) in column k.
C
C Input parameters are not altered by this function. The
C parameters other than PX and PY should be input unaltered
C from their values on output from CSHEP2. This function
C should not be called if a nonzero error flag was returned
C by CSHEP2.
C
C On output:
C
C CS2VAL = Function value C(PX,PY) unless N, NR, DX,
C DY, or RMAX is invalid, in which case no
C value is returned.
C
C Modules required by CS2VAL: NONE
C
C Intrinsic functions called by CS2VAL: INT, SQRT
C
C***********************************************************
C
INTEGER I, IMAX, IMIN, J, JMAX, JMIN, K, KP
DOUBLE PRECISION D, DELX, DELY, R, SW, SWC, W, XP, YP
C
C Local parameters:
C
C D = Distance between P and node K
C DELX = XP - X(K)
C DELY = YP - Y(K)
C I = Cell row index in the range IMIN to IMAX
C IMIN,IMAX = Range of cell row indexes of the cells
C intersected by a disk of radius RMAX
C centered at P
C J = Cell column index in the range JMIN to JMAX
C JMIN,JMAX = Range of cell column indexes of the cells
C intersected by a disk of radius RMAX
C centered at P
C K = Index of a node in cell (I,J)
C KP = Previous value of K in the sequence of nodes
C in cell (I,J)
C R = Radius of influence for node K
C SW = Sum of weights W(K)
C SWC = Sum of weighted nodal function values at P
C W = Weight W(K) value at P: ((R-D)+/(R*D))**3,
C where (R-D)+ = 0 if R < D
C XP,YP = Local copies of PX and PY -- coordinates of P
C
XP = PX
YP = PY
IF (N .LT. 10 .OR. NR .LT. 1 .OR. DX .LE. 0. .OR.
. DY .LE. 0. .OR. RMAX .LT. 0.) RETURN
C
C Set IMIN, IMAX, JMIN, and JMAX to cell indexes defining
C the range of the search for nodes whose radii include
C P. The cells which must be searched are those inter-
C sected by (or contained in) a circle of radius RMAX
C centered at P.
C
IMIN = INT((XP-XMIN-RMAX)/DX) + 1
IMAX = INT((XP-XMIN+RMAX)/DX) + 1
IF (IMIN .LT. 1) IMIN = 1
IF (IMAX .GT. NR) IMAX = NR
JMIN = INT((YP-YMIN-RMAX)/DY) + 1
JMAX = INT((YP-YMIN+RMAX)/DY) + 1
IF (JMIN .LT. 1) JMIN = 1
IF (JMAX .GT. NR) JMAX = NR
C
C The following is a test for no cells within the circle
C of radius RMAX.
C
IF (IMIN .GT. IMAX .OR. JMIN .GT. JMAX) GO TO 6
C
C Accumulate weight values in SW and weighted nodal function
C values in SWC. The weights are W(K) = ((R-D)+/(R*D))**3
C for R = RW(K) and D = distance between P and node K.
C
SW = 0.
SWC = 0.
C
C Outer loop on cells (I,J).
C
DO 4 J = JMIN,JMAX
DO 3 I = IMIN,IMAX
K = LCELL(I,J)
IF (K .EQ. 0) GO TO 3
C
C Inner loop on nodes K.
C
1 DELX = XP - X(K)
DELY = YP - Y(K)
D = SQRT(DELX*DELX + DELY*DELY)
R = RW(K)
IF (D .GE. R) GO TO 2
IF (D .EQ. 0.) GO TO 5
W = (1.0/D - 1.0/R)**3
SW = SW + W
SWC = SWC + W*( ( (A(1,K)*DELX+A(2,K)*DELY+
. A(5,K))*DELX + (A(3,K)*DELY+
. A(6,K))*DELY + A(8,K) )*DELX +
. ( (A(4,K)*DELY+A(7,K))*DELY +
. A(9,K) )*DELY + F(K) )
C
C Bottom of loop on nodes in cell (I,J).
C
2 KP = K
K = LNEXT(KP)
IF (K .NE. KP) GO TO 1
3 CONTINUE
4 CONTINUE
C
C SW = 0 iff P is not within the radius R(K) for any node K.
C
IF (SW .EQ. 0.) GO TO 6
CS2VAL = SWC/SW
RETURN
C
C (PX,PY) = (X(K),Y(K)).
C
5 CS2VAL = F(K)
RETURN
C
C All weights are 0 at P.
C
6 CS2VAL = 0.
RETURN
END
SUBROUTINE CS2GRD (PX,PY,N,X,Y,F,NR,LCELL,LNEXT,XMIN,
. YMIN,DX,DY,RMAX,RW,A, C,CX,CY,IER)
INTEGER N, NR, LCELL(NR,NR), LNEXT(N), IER
DOUBLE PRECISION PX, PY, X(N), Y(N), F(N), XMIN, YMIN,
. DX, DY, RMAX, RW(N), A(9,N), C, CX,
. CY
C
C***********************************************************
C
C From CSHEP2D
C Robert J. Renka
C Dept. of Computer Science
C Univ. of North Texas
C renka@cs.unt.edu
C 02/03/97
C
C This subroutine computes the value and gradient at P =
C (PX,PY) of the interpolatory function C defined in Sub-
C routine CSHEP2. C is a weighted sum of cubic nodal
C functions.
C
C On input:
C
C PX,PY = Cartesian coordinates of the point P at
C which C and its partial derivatives are
C to be evaluated.
C
C N = Number of nodes and data values defining C.
C N .GE. 10.
C
C X,Y,F = Arrays of length N containing the nodes and
C data values interpolated by C.
C
C NR = Number of rows and columns in the cell grid.
C Refer to Subroutine STORE2. NR .GE. 1.
C
C LCELL = NR by NR array of nodal indexes associated
C with cells. Refer to Subroutine STORE2.
C
C LNEXT = Array of length N containing next-node
C indexes. Refer to Subroutine STORE2.
C
C XMIN,YMIN,DX,DY = Minimum nodal coordinates and cell
C dimensions. DX and DY must be
C positive. Refer to Subroutine
C STORE2.
C
C RMAX = Largest element in RW -- maximum radius R(k).
C
C RW = Array of length N containing the the radii R(k)
C which enter into the weights W(k) defining C.
C
C A = 9 by N array containing the coefficients for
C cubic nodal function C(k) in column k.
C
C Input parameters are not altered by this subroutine.
C The parameters other than PX and PY should be input
C unaltered from their values on output from CSHEP2. This
C subroutine should not be called if a nonzero error flag
C was returned by CSHEP2.
C
C On output:
C
C C = Value of C at (PX,PY) unless IER .EQ. 1, in
C which case no values are returned.
C
C CX,CY = First partial derivatives of C at (PX,PY)
C unless IER .EQ. 1.
C
C IER = Error indicator:
C IER = 0 if no errors were encountered.
C IER = 1 if N, NR, DX, DY or RMAX is invalid.
C IER = 2 if no errors were encountered but
C (PX,PY) is not within the radius R(k)
C for any node k (and thus C=CX=CY=0).
C
C Modules required by CS2GRD: None
C
C Intrinsic functions called by CS2GRD: INT, SQRT
C
C***********************************************************
C
INTEGER I, IMAX, IMIN, J, JMAX, JMIN, K, KP
DOUBLE PRECISION CK, CKX, CKY, D, DELX, DELY, R, SW,
. SWC, SWCX, SWCY, SWS, SWX, SWY, T, W,
. WX, WY, XP, YP
C
C Local parameters:
C
C CK = Value of cubic nodal function C(K) at P
C CKX,CKY = Partial derivatives of C(K) with respect to X
C and Y, respectively
C D = Distance between P and node K
C DELX = XP - X(K)
C DELY = YP - Y(K)
C I = Cell row index in the range IMIN to IMAX
C IMIN,IMAX = Range of cell row indexes of the cells
C intersected by a disk of radius RMAX
C centered at P
C J = Cell column index in the range JMIN to JMAX
C JMIN,JMAX = Range of cell column indexes of the cells
C intersected by a disk of radius RMAX
C centered at P
C K = Index of a node in cell (I,J)
C KP = Previous value of K in the sequence of nodes
C in cell (I,J)
C R = Radius of influence for node K
C SW = Sum of weights W(K)
C SWC = Sum of weighted nodal function values at P
C SWCX,SWCY = Partial derivatives of SWC with respect to X
C and Y, respectively
C SWS = SW**2
C SWX,SWY = Partial derivatives of SW with respect to X
C and Y, respectively
C T = Temporary variable
C W = Weight W(K) value at P: ((R-D)+/(R*D))**3,
C where (R-D)+ = 0 if R < D
C WX,WY = Partial derivatives of W with respect to X
C and Y, respectively
C XP,YP = Local copies of PX and PY -- coordinates of P
C
XP = PX
YP = PY
IF (N .LT. 10 .OR. NR .LT. 1 .OR. DX .LE. 0. .OR.
. DY .LE. 0. .OR. RMAX .LT. 0.) GO TO 6
C
C Set IMIN, IMAX, JMIN, and JMAX to cell indexes defining
C the range of the search for nodes whose radii include
C P. The cells which must be searched are those inter-
C sected by (or contained in) a circle of radius RMAX
C centered at P.
C
IMIN = INT((XP-XMIN-RMAX)/DX) + 1
IMAX = INT((XP-XMIN+RMAX)/DX) + 1
IF (IMIN .LT. 1) IMIN = 1
IF (IMAX .GT. NR) IMAX = NR
JMIN = INT((YP-YMIN-RMAX)/DY) + 1
JMAX = INT((YP-YMIN+RMAX)/DY) + 1
IF (JMIN .LT. 1) JMIN = 1
IF (JMAX .GT. NR) JMAX = NR
C
C The following is a test for no cells within the circle
C of radius RMAX.
C
IF (IMIN .GT. IMAX .OR. JMIN .GT. JMAX) GO TO 7
C
C C = SWC/SW = Sum(W(K)*C(K))/Sum(W(K)), where the sum is
C from K = 1 to N, C(K) is the cubic nodal function value,
C and W(K) = ((R-D)+/(R*D))**3 for radius R(K) and dist-
C ance D(K). Thus
C
C CX = (SWCX*SW - SWC*SWX)/SW**2 and
C CY = (SWCY*SW - SWC*SWY)/SW**2
C
C where SWCX and SWX are partial derivatives with respect
C to X of SWC and SW, respectively. SWCY and SWY are de-
C fined similarly.
C
SW = 0.
SWX = 0.
SWY = 0.
SWC = 0.
SWCX = 0.
SWCY = 0.
C
C Outer loop on cells (I,J).
C
DO 4 J = JMIN,JMAX
DO 3 I = IMIN,IMAX
K = LCELL(I,J)
IF (K .EQ. 0) GO TO 3
C
C Inner loop on nodes K.
C
1 DELX = XP - X(K)
DELY = YP - Y(K)
D = SQRT(DELX*DELX + DELY*DELY)
R = RW(K)
IF (D .GE. R) GO TO 2
IF (D .EQ. 0.) GO TO 5
T = (1.0/D - 1.0/R)
W = T**3
T = -3.0*T*T/(D**3)
WX = DELX*T
WY = DELY*T
T = A(2,K)*DELX + A(3,K)*DELY + A(6,K)
CKY = ( 3.0*A(4,K)*DELY + A(3,K)*DELX +
. 2.0*A(7,K) )*DELY + T*DELX + A(9,K)
T = T*DELY + A(8,K)
CKX = ( 3.0*A(1,K)*DELX + A(2,K)*DELY +
. 2.0*A(5,K) )*DELX + T
CK = ( (A(1,K)*DELX+A(5,K))*DELX + T )*DELX +
. ( (A(4,K)*DELY+A(7,K))*DELY + A(9,K) )*DELY +
. F(K)
SW = SW + W
SWX = SWX + WX
SWY = SWY + WY
SWC = SWC + W*CK
SWCX = SWCX + WX*CK + W*CKX
SWCY = SWCY + WY*CK + W*CKY
C
C Bottom of loop on nodes in cell (I,J).
C
2 KP = K
K = LNEXT(KP)
IF (K .NE. KP) GO TO 1
3 CONTINUE
4 CONTINUE
C
C SW = 0 iff P is not within the radius R(K) for any node K.
C
IF (SW .EQ. 0.) GO TO 7
C = SWC/SW
SWS = SW*SW
CX = (SWCX*SW - SWC*SWX)/SWS
CY = (SWCY*SW - SWC*SWY)/SWS
IER = 0
RETURN
C
C (PX,PY) = (X(K),Y(K)).
C
5 C = F(K)
CX = A(8,K)
CY = A(9,K)
IER = 0
RETURN
C
C Invalid input parameter.
C
6 IER = 1
RETURN
C
C No cells contain a point within RMAX of P, or
C SW = 0 and thus D .GE. RW(K) for all K.
C
7 C = 0.
CX = 0.
CY = 0.
IER = 2
RETURN
END
SUBROUTINE CS2HES (PX,PY,N,X,Y,F,NR,LCELL,LNEXT,XMIN,
. YMIN,DX,DY,RMAX,RW,A, C,CX,CY,CXX,
. CXY,CYY,IER)
INTEGER N, NR, LCELL(NR,NR), LNEXT(N), IER
DOUBLE PRECISION PX, PY, X(N), Y(N), F(N), XMIN, YMIN,
. DX, DY, RMAX, RW(N), A(9,N), C, CX,
. CY, CXX, CXY, CYY
C
C***********************************************************
C
C From CSHEP2D
C Robert J. Renka
C Dept. of Computer Science
C Univ. of North Texas
C renka@cs.unt.edu
C 02/03/97
C
C This subroutine computes the value, gradient, and
C Hessian at P = (PX,PY) of the interpolatory function C
C defined in Subroutine CSHEP2. C is a weighted sum of
C cubic nodal functions.
C
C On input:
C
C PX,PY = Cartesian coordinates of the point P at
C which C and its partial derivatives are
C to be evaluated.
C
C N = Number of nodes and data values defining C.
C N .GE. 10.
C
C X,Y,F = Arrays of length N containing the nodes and
C data values interpolated by C.
C
C NR = Number of rows and columns in the cell grid.
C Refer to Subroutine STORE2. NR .GE. 1.
C
C LCELL = NR by NR array of nodal indexes associated
C with cells. Refer to Subroutine STORE2.
C
C LNEXT = Array of length N containing next-node
C indexes. Refer to Subroutine STORE2.
C
C XMIN,YMIN,DX,DY = Minimum nodal coordinates and cell
C dimensions. DX and DY must be
C positive. Refer to Subroutine
C STORE2.
C
C RMAX = Largest element in RW -- maximum radius R(k).
C
C RW = Array of length N containing the the radii R(k)
C which enter into the weights W(k) defining C.
C
C A = 9 by N array containing the coefficients for
C cubic nodal function C(k) in column k.
C
C Input parameters are not altered by this subroutine.
C The parameters other than PX and PY should be input
C unaltered from their values on output from CSHEP2. This
C subroutine should not be called if a nonzero error flag
C was returned by CSHEP2.
C
C On output:
C
C C = Value of C at (PX,PY) unless IER .EQ. 1, in
C which case no values are returned.
C
C CX,CY = First partial derivatives of C at (PX,PY)
C unless IER .EQ. 1.
C
C CXX,CXY,CYY = Second partial derivatives of C at
C (PX,PY) unless IER .EQ. 1.
C
C IER = Error indicator:
C IER = 0 if no errors were encountered.
C IER = 1 if N, NR, DX, DY or RMAX is invalid.
C IER = 2 if no errors were encountered but
C (PX,PY) is not within the radius R(k)
C for any node k (and thus C = 0).
C
C Modules required by CS2HES: None
C
C Intrinsic functions called by CS2HES: INT, SQRT
C
C***********************************************************
C
INTEGER I, IMAX, IMIN, J, JMAX, JMIN, K, KP
DOUBLE PRECISION CK, CKX, CKXX, CKXY, CKY, CKYY, D,
. DELX, DELY, DXSQ, DYSQ, R, SW, SWC,
. SWCX, SWCXX, SWCXY, SWCY, SWCYY, SWS,
. SWX, SWXX, SWXY, SWY, SWYY, T1, T2,
. T3, T4, W, WX, WXX, WXY, WY, WYY, XP,
. YP
C
C Local parameters:
C
C CK = Value of cubic nodal function C(K) at P
C CKX,CKY = Partial derivatives of C(K) with respect to X
C and Y, respectively
C CKXX,CKXY,CKYY = Second partial derivatives of CK
C D = Distance between P and node K
C DELX = XP - X(K)
C DELY = YP - Y(K)
C DXSQ,DYSQ = DELX**2, DELY**2
C I = Cell row index in the range IMIN to IMAX
C IMIN,IMAX = Range of cell row indexes of the cells
C intersected by a disk of radius RMAX
C centered at P
C J = Cell column index in the range JMIN to JMAX
C JMIN,JMAX = Range of cell column indexes of the cells
C intersected by a disk of radius RMAX
C centered at P
C K = Index of a node in cell (I,J)
C KP = Previous value of K in the sequence of nodes
C in cell (I,J)
C R = Radius of influence for node K
C SW = Sum of weights W(K)
C SWC = Sum of weighted nodal function values at P
C SWCX,SWCY = Partial derivatives of SWC with respect to X
C and Y, respectively
C SWCXX,SWCXY,SWCYY = Second partial derivatives of SWC
C SWS = SW**2
C SWX,SWY = Partial derivatives of SW with respect to X
C and Y, respectively
C SWXX,SWXY,SWYY = Second partial derivatives of SW
C T1,T2,T3,T4 = Temporary variables
C W = Weight W(K) value at P: ((R-D)+/(R*D))**3,
C where (R-D)+ = 0 if R < D
C WX,WY = Partial derivatives of W with respect to X
C and Y, respectively
C WXX,WXY,WYY = Second partial derivatives of W
C XP,YP = Local copies of PX and PY -- coordinates of P
C
XP = PX
YP = PY
IF (N .LT. 10 .OR. NR .LT. 1 .OR. DX .LE. 0. .OR.
. DY .LE. 0. .OR. RMAX .LT. 0.) GO TO 6
C
C Set IMIN, IMAX, JMIN, and JMAX to cell indexes defining
C the range of the search for nodes whose radii include
C P. The cells which must be searched are those inter-
C sected by (or contained in) a circle of radius RMAX
C centered at P.
C
IMIN = INT((XP-XMIN-RMAX)/DX) + 1
IMAX = INT((XP-XMIN+RMAX)/DX) + 1
IF (IMIN .LT. 1) IMIN = 1
IF (IMAX .GT. NR) IMAX = NR
JMIN = INT((YP-YMIN-RMAX)/DY) + 1
JMAX = INT((YP-YMIN+RMAX)/DY) + 1
IF (JMIN .LT. 1) JMIN = 1
IF (JMAX .GT. NR) JMAX = NR
C
C The following is a test for no cells within the circle
C of radius RMAX.
C
IF (IMIN .GT. IMAX .OR. JMIN .GT. JMAX) GO TO 7
C
C C = SWC/SW = Sum(W(K)*C(K))/Sum(W(K)), where the sum is
C from K = 1 to N, C(K) is the cubic nodal function value,
C and W(K) = ((R-D)+/(R*D))**3 for radius R(K) and dist-
C ance D(K). Thus
C
C CX = (SWCX*SW - SWC*SWX)/SW**2 and
C CY = (SWCY*SW - SWC*SWY)/SW**2
C
C where SWCX and SWX are partial derivatives with respect
C to x of SWC and SW, respectively. SWCY and SWY are de-
C fined similarly. The second partials are
C
C CXX = ( SW*(SWCXX - 2*SWX*CX) - SWC*SWXX )/SW**2
C CXY = ( SW*(SWCXY-SWX*CY-SWY*CX) - SWC*SWXY )/SW**2
C CYY = ( SW*(SWCYY - 2*SWY*CY) - SWC*SWYY )/SW**2
C
C where SWCXX and SWXX are second partials with respect
C to x, SWCXY and SWXY are mixed partials, and SWCYY and
C SWYY are second partials with respect to y.
C
SW = 0.
SWX = 0.
SWY = 0.
SWXX = 0.
SWXY = 0.
SWYY = 0.
SWC = 0.
SWCX = 0.
SWCY = 0.
SWCXX = 0.
SWCXY = 0.
SWCYY = 0.
C
C Outer loop on cells (I,J).
C
DO 4 J = JMIN,JMAX
DO 3 I = IMIN,IMAX
K = LCELL(I,J)
IF (K .EQ. 0) GO TO 3
C
C Inner loop on nodes K.
C
1 DELX = XP - X(K)
DELY = YP - Y(K)
DXSQ = DELX*DELX
DYSQ = DELY*DELY
D = SQRT(DXSQ + DYSQ)
R = RW(K)
IF (D .GE. R) GO TO 2
IF (D .EQ. 0.) GO TO 5
T1 = (1.0/D - 1.0/R)
W = T1**3
T2 = -3.0*T1*T1/(D**3)
WX = DELX*T2
WY = DELY*T2
T1 = 3.0*T1*(2.0+3.0*D*T1)/(D**6)
WXX = T1*DXSQ + T2
WXY = T1*DELX*DELY
WYY = T1*DYSQ + T2
T1 = A(1,K)*DELX + A(2,K)*DELY + A(5,K)
T2 = T1 + T1 + A(1,K)*DELX
T3 = A(4,K)*DELY + A(3,K)*DELX + A(7,K)
T4 = T3 + T3 + A(4,K)*DELY
CK = (T1*DELX + A(6,K)*DELY + A(8,K))*DELX +
. (T3*DELY + A(9,K))*DELY + F(K)
CKX = T2*DELX + (A(3,K)*DELY+A(6,K))*DELY + A(8,K)
CKY = T4*DELY + (A(2,K)*DELX+A(6,K))*DELX + A(9,K)
CKXX = T2 + 3.0*A(1,K)*DELX
CKXY = 2.0*(A(2,K)*DELX + A(3,K)*DELY) + A(6,K)
CKYY = T4 + 3.0*A(4,K)*DELY
SW = SW + W
SWX = SWX + WX
SWY = SWY + WY
SWXX = SWXX + WXX
SWXY = SWXY + WXY
SWYY = SWYY + WYY
SWC = SWC + W*CK
SWCX = SWCX + WX*CK + W*CKX
SWCY = SWCY + WY*CK + W*CKY
SWCXX = SWCXX + W*CKXX + 2.0*WX*CKX + CK*WXX
SWCXY = SWCXY + W*CKXY + WX*CKY + WY*CKX + CK*WXY
SWCYY = SWCYY + W*CKYY + 2.0*WY*CKY + CK*WYY
C
C Bottom of loop on nodes in cell (I,J).
C
2 KP = K
K = LNEXT(KP)
IF (K .NE. KP) GO TO 1
3 CONTINUE
4 CONTINUE
C
C SW = 0 iff P is not within the radius R(K) for any node K.
C
IF (SW .EQ. 0.) GO TO 7
C = SWC/SW
SWS = SW*SW
CX = (SWCX*SW - SWC*SWX)/SWS
CY = (SWCY*SW - SWC*SWY)/SWS
CXX = (SW*(SWCXX-2.0*SWX*CX) - SWC*SWXX)/SWS
CXY = (SW*(SWCXY-SWY*CX-SWX*CY) - SWC*SWXY)/SWS
CYY = (SW*(SWCYY-2.0*SWY*CY) - SWC*SWYY)/SWS
IER = 0
RETURN
C
C (PX,PY) = (X(K),Y(K)).
C
5 C = F(K)
CX = A(8,K)
CY = A(9,K)
CXX = 2.0*A(5,K)
CXY = A(6,K)
CYY = 2.0*A(7,K)
IER = 0
RETURN
C
C Invalid input parameter.
C
6 IER = 1
RETURN
C
C No cells contain a point within RMAX of P, or
C SW = 0 and thus D .GE. RW(K) for all K.
C
7 C = 0.
CX = 0.
CY = 0.
CXX = 0.
CXY = 0.
CYY = 0.
IER = 2
RETURN
END
SUBROUTINE GETNP2 (PX,PY,X,Y,NR,LCELL,LNEXT,XMIN,YMIN,
. DX,DY, NP,DSQ)
INTEGER NR, LCELL(NR,NR), LNEXT(*), NP
DOUBLE PRECISION PX, PY, X(*), Y(*), XMIN, YMIN, DX,
. DY, DSQ
C
C***********************************************************
C
C From CSHEP2D
C Robert J. Renka
C Dept. of Computer Science
C Univ. of North Texas
C renka@cs.unt.edu
C 02/03/97
C
C Given a set of N nodes and the data structure defined in
C Subroutine STORE2, this subroutine uses the cell method to
C find the closest unmarked node NP to a specified point P.
C NP is then marked by setting LNEXT(NP) to -LNEXT(NP). (A
C node is marked if and only if the corresponding LNEXT ele-
C ment is negative. The absolute values of LNEXT elements,
C however, must be preserved.) Thus, the closest M nodes to
C P may be determined by a sequence of M calls to this rou-
C tine. Note that if the nearest neighbor to node K is to
C be determined (PX = X(K) and PY = Y(K)), then K should be
C marked before the call to this routine.
C
C The search is begun in the cell containing (or closest
C to) P and proceeds outward in rectangular layers until all
C cells which contain points within distance R of P have
C been searched, where R is the distance from P to the first
C unmarked node encountered (infinite if no unmarked nodes
C are present).
C
C This code is essentially unaltered from the subroutine
C of the same name in QSHEP2D.
C
C On input:
C
C PX,PY = Cartesian coordinates of the point P whose
C nearest unmarked neighbor is to be found.
C
C X,Y = Arrays of length N, for N .GE. 2, containing
C the Cartesian coordinates of the nodes.
C
C NR = Number of rows and columns in the cell grid.
C Refer to Subroutine STORE2. NR .GE. 1.
C
C LCELL = NR by NR array of nodal indexes associated
C with cells. Refer to Subroutine STORE2.
C
C LNEXT = Array of length N containing next-node
C indexes (or their negatives). Refer to
C Subroutine STORE2.
C
C XMIN,YMIN,DX,DY = Minimum nodal coordinates and cell
C dimensions. DX and DY must be
C positive. Refer to Subroutine
C STORE2.
C
C Input parameters other than LNEXT are not altered by
C this routine. With the exception of (PX,PY) and the signs
C of LNEXT elements, these parameters should be unaltered
C from their values on output from Subroutine STORE2.
C
C On output:
C
C NP = Index (for X and Y) of the nearest unmarked
C node to P, or 0 if all nodes are marked or NR
C .LT. 1 or DX .LE. 0 or DY .LE. 0. LNEXT(NP)
C .LT. 0 IF NP .NE. 0.
C
C DSQ = Squared Euclidean distance between P and node
C NP, or 0 if NP = 0.
C
C Modules required by GETNP2: None
C
C Intrinsic functions called by GETNP2: ABS, INT, SQRT
C
C***********************************************************
C
INTEGER I, I0, I1, I2, IMAX, IMIN, J, J0, J1, J2,
. JMAX, JMIN, L, LMIN, LN
LOGICAL FIRST
DOUBLE PRECISION DELX, DELY, R, RSMIN, RSQ, XP, YP
C
C Local parameters:
C
C DELX,DELY = PX-XMIN, PY-YMIN
C FIRST = Logical variable with value TRUE iff the
C first unmarked node has yet to be
C encountered
C I,J = Cell indexes in the range [I1,I2] X [J1,J2]
C I0,J0 = Indexes of the cell containing or closest
C to P
C I1,I2,J1,J2 = Range of cell indexes defining the layer
C whose intersection with the range
C [IMIN,IMAX] X [JMIN,JMAX] is currently
C being searched
C IMIN,IMAX = Cell row indexes defining the range of the
C search
C JMIN,JMAX = Cell column indexes defining the range of
C the search
C L,LN = Indexes of nodes in cell (I,J)
C LMIN = Current candidate for NP
C R = Distance from P to node LMIN
C RSMIN = Squared distance from P to node LMIN
C RSQ = Squared distance from P to node L
C XP,YP = Local copy of PX,PY -- coordinates of P
C
XP = PX
YP = PY
C
C Test for invalid input parameters.
C
IF (NR .LT. 1 .OR. DX .LE. 0. .OR. DY .LE. 0.)
. GO TO 9
C
C Initialize parameters.
C
FIRST = .TRUE.
IMIN = 1
IMAX = NR
JMIN = 1
JMAX = NR
DELX = XP - XMIN
DELY = YP - YMIN
I0 = INT(DELX/DX) + 1
IF (I0 .LT. 1) I0 = 1
IF (I0 .GT. NR) I0 = NR
J0 = INT(DELY/DY) + 1
IF (J0 .LT. 1) J0 = 1
IF (J0 .GT. NR) J0 = NR
I1 = I0
I2 = I0
J1 = J0
J2 = J0
C
C Outer loop on layers, inner loop on layer cells, excluding
C those outside the range [IMIN,IMAX] X [JMIN,JMAX].
C
1 DO 6 J = J1,J2
IF (J .GT. JMAX) GO TO 7
IF (J .LT. JMIN) GO TO 6
DO 5 I = I1,I2
IF (I .GT. IMAX) GO TO 6
IF (I .LT. IMIN) GO TO 5
IF (J .NE. J1 .AND. J .NE. J2 .AND. I .NE. I1
. .AND. I .NE. I2) GO TO 5
C
C Search cell (I,J) for unmarked nodes L.
C
L = LCELL(I,J)
IF (L .EQ. 0) GO TO 5
C
C Loop on nodes in cell (I,J).
C
2 LN = LNEXT(L)
IF (LN .LT. 0) GO TO 4
C
C Node L is not marked.
C
RSQ = (X(L)-XP)**2 + (Y(L)-YP)**2
IF (.NOT. FIRST) GO TO 3
C
C Node L is the first unmarked neighbor of P encountered.
C Initialize LMIN to the current candidate for NP, and
C RSMIN to the squared distance from P to LMIN. IMIN,
C IMAX, JMIN, and JMAX are updated to define the smal-
C lest rectangle containing a circle of radius R =
C Sqrt(RSMIN) centered at P, and contained in [1,NR] X
C [1,NR] (except that, if P is outside the rectangle
C defined by the nodes, it is possible that IMIN > NR,
C IMAX < 1, JMIN > NR, or JMAX < 1). FIRST is reset to
C FALSE.
C
LMIN = L
RSMIN = RSQ
R = SQRT(RSMIN)
IMIN = INT((DELX-R)/DX) + 1
IF (IMIN .LT. 1) IMIN = 1
IMAX = INT((DELX+R)/DX) + 1
IF (IMAX .GT. NR) IMAX = NR
JMIN = INT((DELY-R)/DY) + 1
IF (JMIN .LT. 1) JMIN = 1
JMAX = INT((DELY+R)/DY) + 1
IF (JMAX .GT. NR) JMAX = NR
FIRST = .FALSE.
GO TO 4
C
C Test for node L closer than LMIN to P.
C
3 IF (RSQ .GE. RSMIN) GO TO 4
C
C Update LMIN and RSMIN.
C
LMIN = L
RSMIN = RSQ
C
C Test for termination of loop on nodes in cell (I,J).
C
4 IF (ABS(LN) .EQ. L) GO TO 5
L = ABS(LN)
GO TO 2
5 CONTINUE
6 CONTINUE
C
C Test for termination of loop on cell layers.
C
7 IF (I1 .LE. IMIN .AND. I2 .GE. IMAX .AND.
. J1 .LE. JMIN .AND. J2 .GE. JMAX) GO TO 8
I1 = I1 - 1
I2 = I2 + 1
J1 = J1 - 1
J2 = J2 + 1
GO TO 1
C
C Unless no unmarked nodes were encountered, LMIN is the
C closest unmarked node to P.
C
8 IF (FIRST) GO TO 9
NP = LMIN
DSQ = RSMIN
LNEXT(LMIN) = -LNEXT(LMIN)
RETURN
C
C Error: NR, DX, or DY is invalid or all nodes are marked.
C
9 NP = 0
DSQ = 0.
RETURN
END
SUBROUTINE GIVENS ( A,B, C,S)
DOUBLE PRECISION A, B, C, S
C
C***********************************************************
C
C From SRFPACK
C Robert J. Renka
C Dept. of Computer Science
C Univ. of North Texas
C renka@cs.unt.edu
C 09/01/88
C
C This subroutine constructs the Givens plane rotation,
C
C ( C S)
C G = ( ) , where C*C + S*S = 1,
C (-S C)
C
C which zeros the second component of the vector (A,B)**T
C (transposed). Subroutine ROTATE may be called to apply
C the transformation to a 2 by N matrix.
C
C This routine is identical to subroutine SROTG from the
C LINPACK BLAS (Basic Linear Algebra Subroutines).
C
C On input:
C
C A,B = Components of the vector defining the rota-
C tion. These are overwritten by values R
C and Z (described below) which define C and S.
C
C On output:
C
C A = Signed Euclidean norm R of the input vector:
C R = +/-SQRT(A*A + B*B)
C
C B = Value Z such that:
C C = SQRT(1-Z*Z) and S=Z if ABS(Z) .LE. 1, and
C C = 1/Z and S = SQRT(1-C*C) if ABS(Z) > 1.
C
C C = +/-(A/R) or 1 if R = 0.
C
C S = +/-(B/R) or 0 if R = 0.
C
C Modules required by GIVENS: None
C
C Intrinsic functions called by GIVENS: ABS, SQRT
C
C***********************************************************
C
DOUBLE PRECISION AA, BB, R, U, V
C
C Local parameters:
C
C AA,BB = Local copies of A and B
C R = C*A + S*B = +/-SQRT(A*A+B*B)
C U,V = Variables used to scale A and B for computing R
C
AA = A
BB = B
IF (ABS(AA) .LE. ABS(BB)) GO TO 1
C
C ABS(A) > ABS(B).
C
U = AA + AA
V = BB/U
R = SQRT(.25 + V*V) * U
C = AA/R
S = V * (C + C)
C
C Note that R has the sign of A, C > 0, and S has
C SIGN(A)*SIGN(B).
C
B = S
A = R
RETURN
C
C ABS(A) .LE. ABS(B).
C
1 IF (BB .EQ. 0.) GO TO 2
U = BB + BB
V = AA/U
C
C Store R in A.
C
A = SQRT(.25 + V*V) * U
S = BB/A
C = V * (S + S)
C
C Note that R has the sign of B, S > 0, and C has
C SIGN(A)*SIGN(B).
C
B = 1.
IF (C .NE. 0.) B = 1./C
RETURN
C
C A = B = 0.
C
2 C = 1.
S = 0.
RETURN
END
SUBROUTINE ROTATE (N,C,S, X,Y )
INTEGER N
DOUBLE PRECISION C, S, X(N), Y(N)
C
C***********************************************************
C
C From SRFPACK
C Robert J. Renka
C Dept. of Computer Science
C Univ. of North Texas
C renka@cs.unt.edu
C 09/01/88
C
C ( C S)
C This subroutine applies the Givens rotation ( ) to
C (-S C)
C (X(1) ... X(N))
C the 2 by N matrix ( ) .
C (Y(1) ... Y(N))
C
C This routine is identical to subroutine SROT from the
C LINPACK BLAS (Basic Linear Algebra Subroutines).
C
C On input:
C
C N = Number of columns to be rotated.
C
C C,S = Elements of the Givens rotation. Refer to
C subroutine GIVENS.
C
C The above parameters are not altered by this routine.
C
C X,Y = Arrays of length .GE. N containing the compo-
C nents of the vectors to be rotated.
C
C On output:
C
C X,Y = Arrays containing the rotated vectors (not
C altered if N < 1).
C
C Modules required by ROTATE: None
C
C***********************************************************
C
INTEGER I
DOUBLE PRECISION XI, YI
C
DO 1 I = 1,N
XI = X(I)
YI = Y(I)
X(I) = C*XI + S*YI
Y(I) = -S*XI + C*YI
1 CONTINUE
RETURN
END
SUBROUTINE SETUP2 (XK,YK,ZK,XI,YI,ZI,S1,S2,S3,R, ROW)
DOUBLE PRECISION XK, YK, ZK, XI, YI, ZI, S1, S2, S3,
. R, ROW(10)
C
C***********************************************************
C
C From CSHEP2D
C Robert J. Renka
C Dept. of Computer Science
C Univ. of North Texas
C renka@cs.unt.edu
C 02/03/97
C
C This subroutine sets up the I-th row of an augmented re-
C gression matrix for a weighted least squares fit of a
C cubic function f(x,y) to a set of data values z, where
C f(XK,YK) = ZK. The first four columns (cubic terms) are
C scaled by S3, the next three columns (quadratic terms)
C are scaled by S2, and the eighth and ninth columns (lin-
C ear terms) are scaled by S1.
C
C On input:
C
C XK,YK = Coordinates of node K.
C
C ZK = Data value at node K to be interpolated by f.
C
C XI,YI,ZI = Coordinates and data value at node I.
C
C S1,S2,S3 = Scale factors.
C
C R = Radius of influence about node K defining the
C weight.
C
C The above parameters are not altered by this routine.
C
C ROW = Array of length 10.
C
C On output:
C
C ROW = Array containing a row of the augmented re-
C gression matrix.
C
C Modules required by SETUP2: None
C
C Intrinsic function called by SETUP2: SQRT
C
C***********************************************************
C
INTEGER I
DOUBLE PRECISION D, DX, DXSQ, DY, DYSQ, W, W1, W2, W3
C
C Local parameters:
C
C D = Distance between nodes K and I
C DX = XI - XK
C DXSQ = DX*DX
C DY = YI - YK
C DYSQ = DY*DY
C I = DO-loop index
C W = Weight associated with the row: (R-D)/(R*D)
C (0 if D = 0 or D > R)
C W1 = S1*W
C W2 = S2*W
C W3 = W3*W
C
DX = XI - XK
DY = YI - YK
DXSQ = DX*DX
DYSQ = DY*DY
D = SQRT(DXSQ + DYSQ)
IF (D .LE. 0. .OR. D .GE. R) GO TO 1
W = (R-D)/R/D
W1 = S1*W
W2 = S2*W
W3 = S3*W
ROW(1) = DXSQ*DX*W3
ROW(2) = DXSQ*DY*W3
ROW(3) = DX*DYSQ*W3
ROW(4) = DYSQ*DY*W3
ROW(5) = DXSQ*W2
ROW(6) = DX*DY*W2
ROW(7) = DYSQ*W2
ROW(8) = DX*W1
ROW(9) = DY*W1
ROW(10) = (ZI - ZK)*W
RETURN
C
C Nodes K and I coincide or node I is outside of the radius
C of influence. Set ROW to the zero vector.
C
1 DO 2 I = 1,10
ROW(I) = 0.
2 CONTINUE
RETURN
END
SUBROUTINE STORE2 (N,X,Y,NR, LCELL,LNEXT,XMIN,YMIN,DX,
. DY,IER)
INTEGER N, NR, LCELL(NR,NR), LNEXT(N), IER
DOUBLE PRECISION X(N), Y(N), XMIN, YMIN, DX, DY
C
C***********************************************************
C
C From CSHEP2D
C Robert J. Renka
C Dept. of Computer Science
C Univ. of North Texas
C renka@cs.unt.edu
C 03/28/97
C
C Given a set of N arbitrarily distributed nodes in the
C plane, this subroutine creates a data structure for a
C cell-based method of solving closest-point problems. The
C smallest rectangle containing the nodes is partitioned
C into an NR by NR uniform grid of cells, and nodes are as-
C sociated with cells. In particular, the data structure
C stores the indexes of the nodes contained in each cell.
C For a uniform random distribution of nodes, the nearest
C node to an arbitrary point can be determined in constant
C expected time.
C
C This code is essentially unaltered from the subroutine
C of the same name in QSHEP2D.
C
C On input:
C
C N = Number of nodes. N .GE. 2.
C
C X,Y = Arrays of length N containing the Cartesian
C coordinates of the nodes.
C
C NR = Number of rows and columns in the grid. The
C cell density (average number of nodes per cell)
C is D = N/(NR**2). A recommended value, based
C on empirical evidence, is D = 3 -- NR =
C Sqrt(N/3). NR .GE. 1.
C
C The above parameters are not altered by this routine.
C
C LCELL = Array of length .GE. NR**2.
C
C LNEXT = Array of length .GE. N.
C
C On output:
C
C LCELL = NR by NR cell array such that LCELL(I,J)
C contains the index (for X and Y) of the
C first node (node with smallest index) in
C cell (I,J), or LCELL(I,J) = 0 if no nodes
C are contained in the cell. The upper right
C corner of cell (I,J) has coordinates (XMIN+
C I*DX,YMIN+J*DY). LCELL is not defined if
C IER .NE. 0.
C
C LNEXT = Array of next-node indexes such that
C LNEXT(K) contains the index of the next node
C in the cell which contains node K, or
C LNEXT(K) = K if K is the last node in the
C cell for K = 1,...,N. (The nodes contained
C in a cell are ordered by their indexes.)
C If, for example, cell (I,J) contains nodes
C 2, 3, and 5 (and no others), then LCELL(I,J)
C = 2, LNEXT(2) = 3, LNEXT(3) = 5, and
C LNEXT(5) = 5. LNEXT is not defined if
C IER .NE. 0.
C
C XMIN,YMIN = Cartesian coordinates of the lower left
C corner of the rectangle defined by the
C nodes (smallest nodal coordinates) un-
C less IER = 1. The upper right corner is
C (XMAX,YMAX) for XMAX = XMIN + NR*DX and
C YMAX = YMIN + NR*DY.
C
C DX,DY = Dimensions of the cells unless IER = 1. DX
C = (XMAX-XMIN)/NR and DY = (YMAX-YMIN)/NR,
C where XMIN, XMAX, YMIN, and YMAX are the
C extrema of X and Y.
C
C IER = Error indicator:
C IER = 0 if no errors were encountered.
C IER = 1 if N < 2 or NR < 1.
C IER = 2 if DX = 0 or DY = 0.
C
C Modules required by STORE2: None
C
C Intrinsic functions called by STORE2: DBLE, INT
C
C***********************************************************
C
INTEGER I, J, K, L, NN, NNR
DOUBLE PRECISION DELX, DELY, XMN, XMX, YMN, YMX
C
C Local parameters:
C
C DELX,DELY = Components of the cell dimensions -- local
C copies of DX,DY
C I,J = Cell indexes
C K = Nodal index
C L = Index of a node in cell (I,J)
C NN = Local copy of N
C NNR = Local copy of NR
C XMN,XMX = Range of nodal X coordinates
C YMN,YMX = Range of nodal Y coordinates
C
NN = N
NNR = NR
IF (NN .LT. 2 .OR. NNR .LT. 1) GO TO 5
C
C Compute the dimensions of the rectangle containing the
C nodes.
C
XMN = X(1)
XMX = XMN
YMN = Y(1)
YMX = YMN
DO 1 K = 2,NN
IF (X(K) .LT. XMN) XMN = X(K)
IF (X(K) .GT. XMX) XMX = X(K)
IF (Y(K) .LT. YMN) YMN = Y(K)
IF (Y(K) .GT. YMX) YMX = Y(K)
1 CONTINUE
XMIN = XMN
YMIN = YMN
C
C Compute cell dimensions and test for zero area.
C
DELX = (XMX-XMN)/DBLE(NNR)
DELY = (YMX-YMN)/DBLE(NNR)
DX = DELX
DY = DELY
IF (DELX .EQ. 0. .OR. DELY .EQ. 0.) GO TO 6
C
C Initialize LCELL.
C
DO 3 J = 1,NNR
DO 2 I = 1,NNR
LCELL(I,J) = 0
2 CONTINUE
3 CONTINUE
C
C Loop on nodes, storing indexes in LCELL and LNEXT.
C
DO 4 K = NN,1,-1
I = INT((X(K)-XMN)/DELX) + 1
IF (I .GT. NNR) I = NNR
J = INT((Y(K)-YMN)/DELY) + 1
IF (J .GT. NNR) J = NNR
L = LCELL(I,J)
LNEXT(K) = L
IF (L .EQ. 0) LNEXT(K) = K
LCELL(I,J) = K
4 CONTINUE
C
C No errors encountered.
C
IER = 0
RETURN
C
C Invalid input parameter.
C
5 IER = 1
RETURN
C
C DX = 0 or DY = 0.
C
6 IER = 2
RETURN
END
SHAR_EOF
fi # end of overwriting check
cd ..
cd ..
if test ! -d 'Src'
then
mkdir 'Src'
fi
cd 'Src'
if test ! -d 'Sp'
then
mkdir 'Sp'
fi
cd 'Sp'
if test -f 'src.f'
then
echo shar: will not over-write existing file "'src.f'"
else
cat << SHAR_EOF > 'src.f'
SUBROUTINE TESTDT (K, N,X,Y)
DOUBLE PRECISION X(100), Y(100)
INTEGER K, N
C
C***********************************************************
C
C Robert J. Renka
C Dept. of Computer Science
C Univ. of North Texas
C renka@cs.unt.edu
C 03/28/97
C
C This subroutine returns one of five sets of nodes used
C for testing scattered data fitting methods. All five sets
C approximately cover the unit square [0,1] X [0,1]: the
C convex hulls of sets 1 and 3 extend slightly outside the
C square but do not completely cover it, those of sets 2 and
C 5 coincide with the unit square, and the convex hull of
C set 4 is a large subset of the unit square.
C
C On input:
C
C K = Integer in the range 1 to 5 which determines the
C choice of data set as follows:
C
C K = 1 - Franke's 100-node set
C K = 2 - Franke's 33-node set
C K = 3 - Lawson's 25-node set
C K = 4 - Random 100-node set
C K = 5 - Gridded 81-node set
C
C X,Y = Arrays of length at least N(K), where
C N(1) = 100, N(2) = 33, N(3) = 25,
C N(4) = 100, and N(5) = 81.
C
C Input parameters are not altered by this routine.
C
C On output:
C
C N = Number of nodes in set K, or 0 if K is outside
C its valid range.
C
C X,Y = Nodal coordinates of node set K.
C
C Subprograms required by TESTDT: None
C
C***********************************************************
C
DOUBLE PRECISION X1(100), Y1(100), X2(33), Y2(33),
. X3(25), Y3(25), X4(100), Y4(100),
. X5(81), Y5(81)
INTEGER I
C
C Node set 1: Franke's 100-node set.
C
DATA (X1(I),Y1(I), I = 1,20)/
. 0.0227035, -0.0310206, 0.0539888, 0.1586742,
. 0.0217008, 0.2576924, 0.0175129, 0.3414014,
. 0.0019029, 0.4943596, -0.0509685, 0.5782854,
. 0.0395408, 0.6993418, -0.0487061, 0.7470194,
. 0.0315828, 0.9107649, -0.0418785, 0.9962890,
. 0.1324189, 0.0501330, 0.1090271, 0.0918555,
. 0.1254439, 0.2592973, 0.0934540, 0.3381592,
. 0.0767578, 0.4171125, 0.1451874, 0.5615563,
. 0.0626494, 0.6552235, 0.1452734, 0.7524066,
. 0.0958668, 0.9146523, 0.0695559, 0.9632421/
DATA (X1(I),Y1(I), I = 21,40)/
. 0.2645602, 0.0292939, 0.2391645, 0.0602303,
. 0.2088990, 0.2668783, 0.2767329, 0.3696044,
. 0.1714726, 0.4801738, 0.2266781, 0.5940595,
. 0.1909212, 0.6878797, 0.1867647, 0.8185576,
. 0.2304634, 0.9046507, 0.2426219, 0.9805412,
. 0.3663168, 0.0396955, 0.3857662, 0.0684484,
. 0.3832392, 0.2389548, 0.3179087, 0.3124129,
. 0.3466321, 0.4902989, 0.3776591, 0.5199303,
. 0.3873159, 0.6445227, 0.3812917, 0.8203789,
. 0.3795364, 0.8938079, 0.2803515, 0.9711719/
DATA (X1(I),Y1(I), I = 41,60)/
. 0.4149771, -0.0284618, 0.4277679, 0.1560965,
. 0.4200010, 0.2262471, 0.4663631, 0.3175094,
. 0.4855658, 0.3891417, 0.4092026, 0.5084949,
. 0.4792578, 0.6324247, 0.4812279, 0.7511007,
. 0.3977761, 0.8489712, 0.4027321, 0.9978728,
. 0.5848691, -0.0271948, 0.5730076, 0.1272430,
. 0.6063893, 0.2709269, 0.5013894, 0.3477728,
. 0.5741311, 0.4259422, 0.6106955, 0.6084711,
. 0.5990105, 0.6733781, 0.5380621, 0.7235242,
. 0.6096967, 0.9242411, 0.5026188, 1.0308762/
DATA (X1(I),Y1(I), I = 61,80)/
. 0.6616928, 0.0255959, 0.6427836, 0.0707835,
. 0.6396475, 0.2008336, 0.6703963, 0.3259843,
. 0.7001181, 0.4890704, 0.6333590, 0.5096324,
. 0.6908947, 0.6697880, 0.6895638, 0.7759569,
. 0.6718889, 0.9366096, 0.6837675, 1.0064516,
. 0.7736939, 0.0285374, 0.7635332, 0.1021403,
. 0.7410424, 0.1936581, 0.8258981, 0.3235775,
. 0.7306034, 0.4714228, 0.8086609, 0.6091595,
. 0.8214531, 0.6685053, 0.7290640, 0.8022808,
. 0.8076643, 0.8476790, 0.8170951, 1.0512371/
DATA (X1(I),Y1(I), I = 81,100)/
. 0.8424572, 0.0380499, 0.8684053, 0.0902048,
. 0.8366923, 0.2083092, 0.9418461, 0.3318491,
. 0.8478122, 0.4335632, 0.8599583, 0.5910139,
. 0.9175700, 0.6307383, 0.8596328, 0.8144841,
. 0.9279871, 0.9042310, 0.8512805, 0.9696030,
. 1.0449820, -0.0120900, 0.9670631, 0.1334114,
. 0.9857884, 0.2695844, 0.9676313, 0.3795281,
. 1.0129299, 0.4396054, 0.9657040, 0.5044425,
. 1.0019855, 0.6941519, 1.0359297, 0.7459923,
. 1.0414677, 0.8682081, 0.9471506, 0.9801409/
C
C Node set 2: Franke's 33-node set.
C
DATA (X2(I),Y2(I), I = 1,33)/
. 0.05, 0.45, 0.00, 0.50,
. 0.00, 1.00, 0.00, 0.00,
. 0.10, 0.15, 0.10, 0.75,
. 0.15, 0.30, 0.20, 0.10,
. 0.25, 0.20, 0.30, 0.35,
. 0.35, 0.85, 0.50, 0.00,
. 0.50, 1.00, 0.55, 0.95,
. 0.60, 0.25, 0.60, 0.65,
. 0.60, 0.85, 0.65, 0.70,
. 0.70, 0.20, 0.70, 0.65,
. 0.70, 0.90, 0.75, 0.10,
. 0.75, 0.35, 0.75, 0.85,
. 0.80, 0.40, 0.80, 0.65,
. 0.85, 0.25, 0.90, 0.35,
. 0.90, 0.80, 0.95, 0.90,
. 1.00, 0.00, 1.00, 0.50,
. 1.00, 1.00/
C
C Node set 3: Lawson's 25-node set.
C
DATA (X3(I),Y3(I), I = 1,25)/
. 0.13750, 0.97500, 0.91250, 0.98750,
. 0.71250, 0.76250, 0.22500, 0.83750,
. -0.05000, 0.41250, 0.47500, 0.63750,
. 0.05000, -0.05000, 0.45000, 1.03750,
. 1.08750, 0.55000, 0.53750, 0.80000,
. -0.03750, 0.75000, 0.18750, 0.57500,
. 0.71250, 0.55000, 0.85000, 0.43750,
. 0.70000, 0.31250, 0.27500, 0.42500,
. 0.45000, 0.28750, 0.81250, 0.18750,
. 0.45000, -0.03750, 1.00000, 0.26250,
. 0.50000, 0.46250, 0.18750, 0.26250,
. 0.58750, 0.12500, 1.05000, -0.06125,
. 0.10000, 0.11250/
C
C Node set 4: Random 100-node set.
C
DATA (X4(I),Y4(I), I = 1,20)/
. 0.0096326, 0.3083158, 0.0216348, 0.2450434,
. 0.0298360, 0.8613847, 0.0417447, 0.0977864,
. 0.0470462, 0.3648355, 0.0562965, 0.7156339,
. 0.0646857, 0.5311312, 0.0740377, 0.9755672,
. 0.0873907, 0.1781117, 0.0934832, 0.5452797,
. 0.1032216, 0.1603881, 0.1110176, 0.7837139,
. 0.1181193, 0.9982015, 0.1251704, 0.6910589,
. 0.1327330, 0.1049580, 0.1439536, 0.8184662,
. 0.1564861, 0.7086405, 0.1651043, 0.4456593,
. 0.1786039, 0.1178342, 0.1886405, 0.3189021/
DATA (X4(I),Y4(I), I = 21,40)/
. 0.2016706, 0.9668446, 0.2099886, 0.7571834,
. 0.2147003, 0.2016598, 0.2204141, 0.3232444,
. 0.2343715, 0.4368583, 0.2409660, 0.8907869,
. 0.2527740, 0.0647260, 0.2570839, 0.5692618,
. 0.2733365, 0.2947027, 0.2853833, 0.4332426,
. 0.2901755, 0.3347464, 0.2964854, 0.7436284,
. 0.3019725, 0.1066265, 0.3125695, 0.8845357,
. 0.3307163, 0.5158730, 0.3378504, 0.9425637,
. 0.3439061, 0.4799701, 0.3529922, 0.1783069,
. 0.3635507, 0.1146760, 0.3766172, 0.8225797/
DATA (X4(I),Y4(I), I = 41,60)/
. 0.3822429, 0.2270688, 0.3869838, 0.4073598,
. 0.3973137, 0.8875080, 0.4170708, 0.7631616,
. 0.4255588, 0.9972804, 0.4299218, 0.4959884,
. 0.4372839, 0.3410421, 0.4705033, 0.2498120,
. 0.4736655, 0.6409007, 0.4879299, 0.1058690,
. 0.4940260, 0.5411969, 0.5055324, 0.0089792,
. 0.5162593, 0.8784268, 0.5219219, 0.5515874,
. 0.5348529, 0.4038952, 0.5483213, 0.1654023,
. 0.5569571, 0.2965158, 0.5638611, 0.3660356,
. 0.5784908, 0.0366554, 0.5863950, 0.9502420/
DATA (X4(I),Y4(I), I = 61,80)/
. 0.5929148, 0.2638101, 0.5987839, 0.9277386,
. 0.6117561, 0.5377694, 0.6252296, 0.7374676,
. 0.6331381, 0.4674627, 0.6399048, 0.9186109,
. 0.6488972, 0.0416884, 0.6558537, 0.1291029,
. 0.6677405, 0.6763676, 0.6814074, 0.8444238,
. 0.6887812, 0.3273328, 0.6940896, 0.1893879,
. 0.7061687, 0.0645923, 0.7160957, 0.0180147,
. 0.7317445, 0.8904992, 0.7370798, 0.4160648,
. 0.7462030, 0.4688995, 0.7566957, 0.2174508,
. 0.7699998, 0.5734231, 0.7879347, 0.8853319/
DATA (X4(I),Y4(I), I = 81,100)/
. 0.7944014, 0.8018436, 0.8164468, 0.6388941,
. 0.8192794, 0.8931002, 0.8368405, 0.1000558,
. 0.8500993, 0.2789506, 0.8588255, 0.9082948,
. 0.8646496, 0.3259159, 0.8792329, 0.8318747,
. 0.8837536, 0.0508513, 0.8900077, 0.9708450,
. 0.8969894, 0.5120548, 0.9044917, 0.2859716,
. 0.9083947, 0.9581641, 0.9203972, 0.6183429,
. 0.9347906, 0.3779934, 0.9434519, 0.4010423,
. 0.9490328, 0.9478657, 0.9569571, 0.7425486,
. 0.9772067, 0.8883287, 0.9983493, 0.5496750/
C
C Node set 5: 9 by 9 uniform grid.
C
DATA (X5(I),Y5(I), I = 1,20)/
. 0.125, 0.000, 0.000, 0.125,
. 0.000, 0.250, 0.000, 0.375,
. 0.000, 0.500, 0.000, 0.625,
. 0.000, 0.750, 0.000, 0.875,
. 0.000, 1.000, 0.000, 0.000,
. 0.125, 0.125, 0.125, 0.250,
. 0.125, 0.375, 0.125, 0.500,
. 0.125, 0.625, 0.125, 0.750,
. 0.125, 0.875, 0.125, 1.000,
. 0.250, 0.000, 0.250, 0.125/
DATA (X5(I),Y5(I), I = 21,40)/
. 0.250, 0.250, 0.250, 0.375,
. 0.250, 0.500, 0.250, 0.625,
. 0.250, 0.750, 0.250, 0.875,
. 0.250, 1.000, 0.375, 0.000,
. 0.375, 0.125, 0.375, 0.250,
. 0.375, 0.375, 0.375, 0.500,
. 0.375, 0.625, 0.375, 0.750,
. 0.375, 0.875, 0.375, 1.000,
. 0.500, 0.000, 0.500, 0.125,
. 0.500, 0.250, 0.500, 0.375/
DATA (X5(I),Y5(I), I = 41,60)/
. 0.500, 0.500, 0.500, 0.625,
. 0.500, 0.750, 0.500, 0.875,
. 0.500, 1.000, 0.625, 0.000,
. 0.625, 0.125, 0.625, 0.250,
. 0.625, 0.375, 0.625, 0.500,
. 0.625, 0.625, 0.625, 0.750,
. 0.625, 0.875, 0.625, 1.000,
. 0.750, 0.000, 0.750, 0.125,
. 0.750, 0.250, 0.750, 0.375,
. 0.750, 0.500, 0.750, 0.625/
DATA (X5(I),Y5(I), I = 61,81)/
. 0.750, 0.750, 0.750, 0.875,
. 0.750, 1.000, 0.875, 0.000,
. 0.875, 0.125, 0.875, 0.250,
. 0.875, 0.375, 0.875, 0.500,
. 0.875, 0.625, 0.875, 0.750,
. 0.875, 0.875, 0.875, 1.000,
. 1.000, 0.000, 1.000, 0.125,
. 1.000, 0.250, 1.000, 0.375,
. 1.000, 0.500, 1.000, 0.625,
. 1.000, 0.750, 1.000, 0.875,
. 1.000, 1.000/
C
C Store node set K in (X,Y).
C
IF (K .EQ. 1) THEN
DO 1 I = 1,100
X(I) = X1(I)
Y(I) = Y1(I)
1 CONTINUE
N = 100
ELSEIF (K .EQ. 2) THEN
DO 2 I = 1,33
X(I) = X2(I)
Y(I) = Y2(I)
2 CONTINUE
N = 33
ELSEIF (K .EQ. 3) THEN
DO 3 I = 1,25
X(I) = X3(I)
Y(I) = Y3(I)
3 CONTINUE
N = 25
ELSEIF (K .EQ. 4) THEN
DO 4 I = 1,100
X(I) = X4(I)
Y(I) = Y4(I)
4 CONTINUE
N = 100
ELSEIF (K .EQ. 5) THEN
DO 5 I = 1,81
X(I) = X5(I)
Y(I) = Y5(I)
5 CONTINUE
N = 81
ELSE
N = 0
ENDIF
RETURN
END
SUBROUTINE TSTFN1 (K,X,Y,IFLAG, F,FX,FY)
INTEGER K, IFLAG
DOUBLE PRECISION X, Y, F, FX, FY
C
C***********************************************************
C
C Robert J. Renka
C Dept. of Computer Science
C Univ. of North Texas
C renka@cs.unt.edu
C 10/14/98
C
C This subroutine computes the value and, optionally, the
C first partial derivatives of one of ten bivariate test
C functions. The first six functions were chosen by Richard
C Franke to test interpolation software (See the reference
C below). The last four functions represent more chal-
C lenging surface fitting problems.
C
C On input:
C
C K = Integer in the range 1 to 10 which determines
C the choice of function as follows:
C
C K = 1 - Exponential
C K = 2 - Cliff
C K = 3 - Saddle
C K = 4 - Gentle
C K = 5 - Steep
C K = 6 - Sphere
C K = 7 - Trig
C K = 8 - Synergistic Gaussian
C K = 9 - Cloverleaf Asymmetric Peak/Valley
C K = 10 - Cosine Peak
C
C Note that function 6 is only defined inside a circle of
C radius 8/9 centered at (.5,.5). Thus, if (X-.5)**2 +
C (Y-.5)**2 .GE. 64/81, the value (and partials if IFLAG=1)
C are set to 0 for this function. Also, the first partial
C derivatives of function 10 are not defined at (.5,.5) --
C again, zeros are returned.
C
C X,Y = Coordinates of the point at which the selected
C function is to be evaluated.
C
C IFLAG = Derivative option indicator:
C IFLAG = 0 if only a function value is
C required.
C IFLAG = 1 if both the function and its first
C partial derivatives are to be
C evaluated.
C
C Input parameters are not altered by this routine.
C
C On output:
C
C F = Value of function K at (X,Y).
C
C FX,FY = First partial derivatives of function K at
C (X,Y) if IFLAG = 1, unaltered otherwise.
C
C Intrinsic functions called by TSTFN1: COS, EXP, SIN,
C SQRT, TANH
C
C Reference: R. Franke, A Critical Comparison of Some
C Methods for Interpolation of Scattered Data,
C Naval Postgraduate School Technical Report,
C NPS-53-79-003, 1979.
C
C***********************************************************
C
DOUBLE PRECISION T1, T2, T3, T4
IF (K .LT. 1 .OR. K .GT. 10) RETURN
GO TO (1,2,3,4,5,6,7,8,9,10), K
C
C Exponential:
C
1 F = .75*EXP(-((9.*X-2.)**2 + (9.*Y-2.)**2)/4.) +
. .75*EXP(-((9.*X+1.)**2)/49. - (9.*Y+1.)/10.) +
. .5*EXP(-((9.*X-7.)**2 + (9.*Y-3.)**2)/4.) -
. .2*EXP(-(9.*X-4.)**2 - (9.*Y-7.)**2)
IF (IFLAG .NE. 1) RETURN
T1 = EXP(-((9.*X-2.)**2 + (9.*Y-2.)**2)/4.)
T2 = EXP(-((9.*X+1.)**2)/49. - (9.*Y+1.)/10.)
T3 = EXP(-((9.*X-7.)**2 + (9.*Y-3.)**2)/4.)
T4 = EXP(-(9.*X-4.)**2 - (9.*Y-7.)**2)
FX = -3.375*(9.*X-2.)*T1 - (27./98.)*(9.*X+1.)*T2
. -2.25*(9.*X-7.)*T3 + 3.6*(9.*X-4.)*T4
FY = -3.375*(9.*Y-2.)*T1 - .675*T2
. -2.25*(9.*Y-3.)*T3 + 3.6*(9.*Y-7.)*T4
RETURN
C
C Cliff:
C
2 F = (TANH(9.0*(Y-X)) + 1.0)/9.0
IF (IFLAG .NE. 1) RETURN
T1 = 18.0*(Y-X)
FX = -4.0/(EXP(T1) + 2.0 + EXP(-T1))
FY = -FX
RETURN
C
C Saddle:
C
3 F = (1.25 + COS(5.4*Y))/(6.0 + 6.0*(3.0*X-1.0)**2)
IF (IFLAG .NE. 1) RETURN
T1 = 5.4*Y
T2 = 1.0 + (3.0*X-1.)**2
FX = -(3.0*X-1.0)*(1.25 + COS(T1))/(T2**2)
FY = -.9*SIN(T1)/T2
RETURN
C
C Gentle:
C
4 F = EXP(-5.0625*((X-.5)**2 + (Y-.5)**2))/3.0
IF (IFLAG .NE. 1) RETURN
T1 = X - .5
T2 = Y - .5
T3 = -3.375*EXP(-5.0625*(T1**2 + T2**2))
FX = T1*T3
FY = T2*T3
RETURN
C
C Steep:
C
5 F = EXP(-20.25*((X-.5)**2 + (Y-.5)**2))/3.0
IF (IFLAG .NE. 1) RETURN
T1 = X - .5
T2 = Y - .5
T3 = -13.5*EXP(-20.25*(T1**2 + T2**2))
FX = T1*T3
FY = T2*T3
RETURN
C
C Sphere:
C
6 T4 = 64.0 - 81.0*((X-.5)**2 + (Y-.5)**2)
F = 0.
IF (T4 .GE. 0.) F = SQRT(T4)/9.0 - .5
IF (IFLAG .NE. 1) RETURN
T1 = X - .5
T2 = Y - .5
T3 = 0.
IF (T4 .GT. 0.) T3 = -9.0/SQRT(T4)
FX = T1*T3
FY = T2*T3
RETURN
C
C Trig:
C
7 F = 2.0*COS(10.0*X)*SIN(10.0*Y) + SIN(10.0*X*Y)
IF (IFLAG .NE. 1) RETURN
T1 = 10.0*X
T2 = 10.0*Y
T3 = 10.0*COS(10.0*X*Y)
FX = -20.0*SIN(T1)*SIN(T2) + T3*Y
FY = 20.0*COS(T1)*COS(T2) + T3*X
RETURN
C
C Gaussx(1,.5,.1) + Gaussy(.75,.5,.1) + Gaussx(1,.5,.1)*
C Gaussy(.75,.5,.1), where Gaussx(a,b,c) is the Gaussian
C function of x with amplitude a, center (mean) b, and
C width (standard deviation) c.
C
8 T1 = 5.0 - 10.0*X
T2 = 5.0 - 10.0*Y
T3 = EXP(-.5*T1*T1)
T4 = EXP(-.5*T2*T2)
F = T3 + .75*T4*(1.0+T3)
IF (IFLAG .NE. 1) RETURN
FX = T1*T3*(10.0 + 7.5*T4)
FY = T2*T4*(7.5 + 7.5*T3)
RETURN
C
C Cloverleaf Asymmetric Hill/Valley:
C
9 T1 = EXP((10.0 - 20.0*X)/3.0)
T2 = EXP((10.0 - 20.0*Y)/3.0)
T3 = 1.0/(1.0 + T1)
T4 = 1.0/(1.0 + T2)
F = ((20.0/3.0)**3 * T1*T2)**2 * (T3*T4)**5 *
. (T1-2.0*T3)*(T2-2.0*T4)
IF (IFLAG .NE. 1) RETURN
FX = ((20.0/3.0)*T1)**2 * ((20.0/3.0)*T3)**5 *
. (2.0*T1-3.0*T3-5.0+12.0*T3*T3)*T2*T2*T4**5 *
. (T2-2.0*T4)
FY = ((20.0/3.0)*T1)**2 * ((20.0/3.0)*T3)**5 *
. (2.0*T2-3.0*T4-5.0+12.0*T4*T4)*T2*T2*T4**5 *
. (T1-2.0*T3)
RETURN
C
C Cosine Peak:
C
10 T1 = SQRT( (80.0*X - 40.0)**2 + (90.0*Y - 45.0)**2 )
T2 = EXP(-.04*T1)
T3 = COS(.15*T1)
F = T2*T3
IF (IFLAG .NE. 1) RETURN
T4 = SIN(.15*T1)
FX = 0.
FY = 0.
IF (T1 .EQ. 0.) RETURN
T4 = SIN(.15*T1)
FX = -T2*(12.0*T4 + 3.2*T3)*(80.0*X - 40.0)/T1
FY = -T2*(13.5*T4 + 3.6*T3)*(90.0*Y - 45.0)/T1
RETURN
END
SUBROUTINE TSTFN2 (K,X,Y,IFLAG, F,FX,FY,FXX,FXY,FYY)
INTEGER K, IFLAG
DOUBLE PRECISION X, Y, F, FX, FY, FXX, FXY, FYY
C
C***********************************************************
C
C Robert J. Renka
C Dept. of Computer Science
C Univ. of North Texas
C renka@cs.unt.edu
C 10/14/98
C
C This subroutine computes the value and, optionally, the
C first and/or second partial derivatives of one of ten
C bivariate test functions. The first six functions were
C chosen by Richard Franke to test interpolation software.
C (See the reference below.) The last four functions repre-
C sent more challenging surface fitting problems.
C
C On input:
C
C K = Integer in the range 1 to 10 which determines
C the choice of function as follows:
C
C K = 1 - Exponential
C K = 2 - Cliff
C K = 3 - Saddle
C K = 4 - Gentle
C K = 5 - Steep
C K = 6 - Sphere
C K = 7 - Trig
C K = 8 - Synergistic Gaussian
C K = 9 - Cloverleaf Asymmetric Peak/Valley
C K = 10 - Cosine Peak
C
C Note that function 6 is only defined inside a circle of
C radius 8/9 centered at (.5,.5). Thus, if (X-.5)**2 +
C (Y-.5)**2 .GE. 64/81, the value (and partials if IFLAG=1)
C are set to 0 for this function. Also, the first partial
C derivatives of function 10 are not defined at (.5,.5) --
C again, zeros are returned.
C
C X,Y = Coordinates of the point at which the selected
C function is to be evaluated.
C
C IFLAG = Derivative option indicator:
C IFLAG = 0 if only a function value is
C required.
C IFLAG = 1 if both the function and its first
C partial derivatives are to be
C evaluated.
C IFLAG = 2 if the function, its first partial
C derivatives, and its second partial
C derivatives are to be evaluated.
C
C Input parameters are not altered by this routine.
C
C On output:
C
C F = Value of function K at (X,Y).
C
C FX,FY = First partial derivatives of function K at
C (X,Y) if IFLAG >= 1, unaltered otherwise.
C
C FXX,FXY,FYY = Second partial derivatives of function
C K at (X,Y) if IFLAG >= 2, unaltered
C otherwise.
C
C Intrinsic functions called by TSTFN2: COS, EXP, SIN,
C SQRT, TANH
C
C Reference: R. Franke, A Critical Comparison of Some
C Methods for Interpolation of Scattered Data,
C Naval Postgraduate School Technical Report,
C NPS-53-79-003, 1979.
C
C***********************************************************
C
DOUBLE PRECISION T1, T2, T3, T4, T5, T6
IF (K .LT. 1 .OR. K .GT. 10) RETURN
GO TO (1,2,3,4,5,6,7,8,9,10), K
C
C Exponential:
C
1 F = .75*EXP(-((9.*X-2.)**2 + (9.*Y-2.)**2)/4.) +
. .75*EXP(-((9.*X+1.)**2)/49. - (9.*Y+1.)/10.) +
. .5*EXP(-((9.*X-7.)**2 + (9.*Y-3.)**2)/4.) -
. .2*EXP(-(9.*X-4.)**2 - (9.*Y-7.)**2)
IF (IFLAG .LT. 1) RETURN
T1 = EXP(-((9.*X-2.)**2 + (9.*Y-2.)**2)/4.)
T2 = EXP(-((9.*X+1.)**2)/49. - (9.*Y+1.)/10.)
T3 = EXP(-((9.*X-7.)**2 + (9.*Y-3.)**2)/4.)
T4 = EXP(-(9.*X-4.)**2 - (9.*Y-7.)**2)
FX = -3.375*(9.*X-2.)*T1 - (27./98.)*(9.*X+1.)*T2
. -2.25*(9.*X-7.)*T3 + 3.6*(9.*X-4.)*T4
FY = -3.375*(9.*Y-2.)*T1 - .675*T2
. -2.25*(9.*Y-3.)*T3 + 3.6*(9.*Y-7.)*T4
IF (IFLAG .LT. 2) RETURN
FXX = 15.1875*((9.*X-2.)**2 - 2.)*T1 +
. 60.75*((9.*X+1.)**2 - 24.5)*T2 +
. 10.125*((9.*X-7.)**2 - 2.)*T3 -
. 64.8*((9.*X-4.)**2 - .5)*T4
FXY = 15.1875*(9.*X-2.)*(9.*Y-2.)*T1 +
. (243./980.)*(9.*X+1.)*T2 +
. 10.125*(9.*X-7.)*(9.*Y-3.)*T3 -
. 64.8*(9.*X-4.)*(9.*Y-7.)*T4
FYY = 15.1875*((9.*Y-2.)**2 - 2.)*T1 +
. .6075*T2 +
. 10.125*((9.*Y-3.)**2 - 2.)*T3 -
. 64.8*((9.*Y-7.)**2 - .5)*T4
RETURN
C
C Cliff:
C
2 F = (TANH(9.0*(Y-X)) + 1.0)/9.0
IF (IFLAG .LT. 1) RETURN
T1 = 18.0*(Y-X)
FX = -4.0/(EXP(T1) + 2.0 + EXP(-T1))
FY = -FX
IF (IFLAG .LT. 2) RETURN
FXX = 18.0*TANH(0.5*T1)*FX
FXY = -FXX
FYY = FXX
RETURN
C
C Saddle:
C
3 F = (1.25 + COS(5.4*Y))/(6.0 + 6.0*(3.0*X-1.0)**2)
IF (IFLAG .LT. 1) RETURN
T1 = 5.4*Y
T2 = 1.0 + (3.0*X-1.)**2
FX = -(3.0*X-1.0)*(1.25 + COS(T1))/(T2**2)
FY = -.9*SIN(T1)/T2
IF (IFLAG .LT. 2) RETURN
FXX = 3.0*(1.25 + COS(T1))*(3.0*T2-4.0)/(T2**3)
FXY = 5.4*(3.0*X-1.0)*SIN(T1)/(T2**2)
FYY = -4.86*COS(T1)/T2
RETURN
C
C Gentle:
C
4 F = EXP(-5.0625*((X-.5)**2 + (Y-.5)**2))/3.0
IF (IFLAG .LT. 1) RETURN
T1 = X - .5
T2 = Y - .5
T3 = -3.375*EXP(-5.0625*(T1**2 + T2**2))
FX = T1*T3
FY = T2*T3
IF (IFLAG .LT. 2) RETURN
FXX = (1.0 - 10.125*T1*T1)*T3
FXY = -10.125*T1*T2*T3
FYY = (1.0 - 10.125*T2*T2)*T3
RETURN
C
C Steep:
C
5 F = EXP(-20.25*((X-.5)**2 + (Y-.5)**2))/3.0
IF (IFLAG .LT. 1) RETURN
T1 = X - .5
T2 = Y - .5
T3 = -13.5*EXP(-20.25*(T1**2 + T2**2))
FX = T1*T3
FY = T2*T3
IF (IFLAG .LT. 2) RETURN
FXX = (1.0 - 40.5*T1*T1)*T3
FXY = -40.5*T1*T2*T3
FYY = (1.0 - 40.5*T2*T2)*T3
RETURN
C
C Sphere:
C
6 T4 = 64.0 - 81.0*((X-.5)**2 + (Y-.5)**2)
F = 0.
IF (T4 .GE. 0.) F = SQRT(T4)/9.0 - .5
IF (IFLAG .LT. 1) RETURN
T1 = X - .5
T2 = Y - .5
T3 = 0.
IF (T4 .GT. 0.) T3 = -9.0/SQRT(T4)
FX = T1*T3
FY = T2*T3
IF (IFLAG .LT. 2) RETURN
FXX = (1.0 + FX*FX)*T3
FXY = FX*FY
FYY = (1.0 + FY*FY)*T3
RETURN
C
C Trig:
C
7 F = 2.0*COS(10.0*X)*SIN(10.0*Y) + SIN(10.0*X*Y)
IF (IFLAG .LT. 1) RETURN
T1 = 10.0*X
T2 = 10.0*Y
T3 = 10.0*COS(10.0*X*Y)
FX = -20.0*SIN(T1)*SIN(T2) + T3*Y
FY = 20.0*COS(T1)*COS(T2) + T3*X
IF (IFLAG .LT. 2) RETURN
T4 = 100.0*SIN(10.0*X*Y)
FXX = -200.0*COS(T1)*SIN(T2) - T4*Y*Y
FXY = -200.0*SIN(T1)*COS(T2) + T3 - T4*X*Y
FYY = -200.0*COS(T1)*SIN(T2) - T4*X*X
RETURN
C
C Gaussx(1,.5,.1) + Gaussy(.75,.5,.1) + Gaussx(1,.5,.1)*
C Gaussy(.75,.5,.1), where Gaussx(a,b,c) is the Gaussian
C function of x with amplitude a, center (mean) b, and
C width (standard deviation) c.
C
8 T1 = 5.0 - 10.0*X
T2 = 5.0 - 10.0*Y
T3 = EXP(-.5*T1*T1)
T4 = EXP(-.5*T2*T2)
F = T3 + .75*T4*(1.0+T3)
IF (IFLAG .LT. 1) RETURN
FX = T1*T3*(10.0 + 7.5*T4)
FY = T2*T4*(7.5 + 7.5*T3)
IF (IFLAG .LT. 2) RETURN
FXX = T3*(T1*T1-1.0)*(100.0 + 75.0*T4)
FXY = 75.0*T1*T2*T3*T4
FYY = T4*(T2*T2-1.0)*(75.0 + 75.0*T3)
RETURN
C
C Cloverleaf Asymmetric Hill/Valley:
C
9 T1 = EXP((10.0 - 20.0*X)/3.0)
T2 = EXP((10.0 - 20.0*Y)/3.0)
T3 = 1.0/(1.0 + T1)
T4 = 1.0/(1.0 + T2)
T5 = 20.0/3.0
F = (T5**3 * T1*T2)**2 * (T3*T4)**5 *
. (T1-2.0*T3)*(T2-2.0*T4)
IF (IFLAG .LT. 1) RETURN
T6 = (T5*T1*T2)**2 * (T5*T3*T4)**5
FX = T6 * (T2-2.0*T4) *
. ((12.0*T3 - 3.0)*T3 + 2.0*T1 - 5.0)
FY = T6 * (T1-2.0*T3) *
. ((12.0*T4 - 3.0)*T4 + 2.0*T2 - 5.0)
IF (IFLAG .LT. 2) RETURN
FXX = T5*T6 * (T2-2.0*T4) *
. (((-84.0*T3 + 78.0)*T3 + 23.0)*T3 + 4.0*T1-25.0)
FXY = T5*T6 *
. ((12.0*T4 - 3.0)*T4 + 2.0*T2 - 5.0) *
. ((12.0*T3 - 3.0)*T3 + 2.0*T1 - 5.0)
FYY = T5*T6 * (T1-2.0*T3) *
. (((-84.0*T4 + 78.0)*T4 + 23.0)*T4 + 4.0*T2-25.0)
RETURN
C
C Cosine Peak:
C
10 T1 = SQRT( (80.0*X - 40.0)**2 + (90.0*Y - 45.0)**2 )
T2 = EXP(-.04*T1)
T3 = COS(.15*T1)
F = T2*T3
IF (IFLAG .LT. 1) RETURN
T4 = SIN(.15*T1)
FX = 0.
FY = 0.
IF (T1 .EQ. 0.) RETURN
T4 = SIN(.15*T1)
FX = -T2*(12.0*T4 + 3.2*T3)*(80.0*X - 40.0)/T1
FY = -T2*(13.5*T4 + 3.6*T3)*(90.0*Y - 45.0)/T1
IF (IFLAG .LT. 2) RETURN
FXX = 0.
FXY = 0.
FYY = 0.
IF (T1 .EQ. 0.) RETURN
T5 = T2/(T1**3)
FXX = T5*(T1*(76.8*T4 - 133.76*T3)*(80.0*X-40.0)**2 -
. (960.0*T4 + 256.0*T3)*(90.0*Y-45.0)**2 )
FXY = T5*(T1*(86.4*T4 - 150.48*T3) + 1080.0*T4 +
. 288.0*T3)*(80.0*X-40.0)*(90.0*Y-45.0)
FYY = T5*(T1*(97.2*T4 - 169.29*T3)*(90.0*Y-45.0)**2 -
. (1215.0*T4 + 324.0*T3)*(80.0*X-40.0)**2)
RETURN
END
SHAR_EOF
fi # end of overwriting check
cd ..
cd ..
cd ..
# End of shell archive
exit 0