C ALGORITHM 833, COLLECTED ALGORITHMS FROM ACM. C THIS WORK PUBLISHED IN TRANSACTIONS ON MATHEMATICAL SOFTWARE, C VOL. 30, NO. 2, June, 2004, P. 200--211. #! /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/csrfpack.doc # Fortran77/ # Fortran77/Drivers/ # Fortran77/Drivers/Makefile # Fortran77/Drivers/csrftest.eps # Fortran77/Drivers/csrftest.prt # Fortran77/Drivers/data # Fortran77/Drivers/driver.f # Fortran77/Drivers/gplot.eps # Fortran77/Drivers/res # Fortran77/Drivers/rplot.eps # Fortran77/Drivers/tgplot.eps # Fortran77/Drivers/tplot.eps # Fortran77/Src/ # Fortran77/Src/src.f # This archive created: Wed Aug 25 13:20:05 2004 export PATH; PATH=/bin:$PATH if test ! -d 'Doc' then mkdir 'Doc' fi cd 'Doc' if test -f 'csrfpack.doc' then echo shar: will not over-write existing file "'csrfpack.doc'" else cat << "SHAR_EOF" > 'csrfpack.doc' CSRFPACK.DOC 01/31/2004 Overview: CSRFPACK is a Fortran-77 software package which constructs a once-continuously differentiable convex bivariate function F that interpolates a set of N strictly convex data points (data points for which the piecewise linear interpolant on some triangulation is convex, and no four points are coplanar). The interpolant F is obtained by applying convo- lution smoothing to a piecewise linear Hermite interpolant H of the data values and a set of convexity-preserving nodal gradients. Usage: The program is distributed in the form of two source code files and a sample input data set: csrftest.f, csrfpack.f, and csrftest.dat. The driver, csrftest.f, prompts for the name of an input data set which consists of the number of data points N (read with format I5) followed by a sequence of N ordered triples of data point coordinates (x,y,z) read with format 3E23.15: the x, y, and z components are read from columns 1 to 23, 24 to 46, and 47 to 69, respecively, and each must include a decimal point (and optional expo- nent). The output consists of the following files. csrftest.prt: Print file with parameter values and error messages. csrftest.out: Data set (suitable for a surface plotting package) consisting of interpolated values on a triangulation of a uniform grid in the bounding box (the smallest rectangle with sides parallel to the axes that contains the nodes). csrftest.eps: Encapsulated Postscript file containing a contour plot of the interpolatory surface. tplot.eps: Encapsulated Postscript file containing a plot of the convexity-preserving triangulation of the nodes. gplot.eps: Encapsulated Postscript file containing a plot of the gradient feasibility diagram and nodal gradients. tgplot.eps: Encapsulated Postscript file containing a plot of the gradient triangulation. rplot.eps: Encapsulated Postscript file containing a plot of the cell diagram and nodes. Various options governing the behavior of the code are easily specified by altering PARAMETER and DATA statements in the driver. Computational procedure and code structure: The procedure consists of the following ten steps: 1) Construct a convexity-preserving triangulation T, if it exists, of the data points (nodes and data values): Subroutine TRMSHC. Remove extraneous boundary edges (nearly null triangles): Subroutine DELBE. 2) Construct a gradient feasibility diagram (the straight line dual of T): a set of N convex polygons that parti- tion the plane and such that, for any choice of nodal gradients, one from each polygon, the piecewise linear Hermite interpolant H of the nodal values and gradients is convex: Subroutine VLIST. 3) Select a set of nodal gradients by taking the centroids of the gradient feasibility regions or (in the case of boundary nodes) truncated regions: Subroutine GLIST. 4) Optionally, create level-2 Encapsulated PostScript files containing plots of T and/or the dual of T (along with the nodal gradients): Subroutines PLTTR and PLTGR. 5) Optionally, compute a scale factor EPS defining a quad- ratic function q(p) = EPS*, and adjust the data values and gradients by subtracting nodal values and gradients of q, where EPS is defined so that a strictly convex Hermite interpolant of the original data can be constructed from a convex Hermite interpolant of the adjusted data (Subroutine ADDQT). 6) Construct a convexity-preserving triangulation Tg of the (possibly adjusted) nodal gradients and negative z- intercepts of the affine nodal functions defined by the (adjusted) data values and nodal gradients: Subroutine TRMSHC. 7) Construct a cell diagram (the straight line dual of Tg): a set of N convex polygons R_i that partition the plane and such that the restriction of H to R_i is the i-th affine nodal function f_i, where f_i(p) = + z_i for nodes p_i, (adjusted) data values z_i and (adjusted) gradients g_i for i = 1 to N and H(p) = Max(i=1,N){f_i(p)}: Subroutine VLIST. 8) Optionally, create level-2 Encapsulated PostScript files containing plots of Tg and/or the dual of Tg (along with the nodes): Subroutines PLTTR and PLTGR. 9) Compute the distance D_i from each node p_i to the nearest boundary point of cell R_i: Subroutine DELTAI. 10) Compute weights W_i and quadrature points q_ij defining the quadrature rule Q(p) = Sum(i=1,NR) [ W_i*Sum(j=1,NA) H(p+q_ij) ] for approximating F(p) = Integral[ H(p+q)* Phi(q) ]dq, where the integral is over the disk D0 of radius Dmin = Min(i=1,N){D_i} centered at the origin, and Phi(q) = phi(Norm(q)/Dmin), normalized to have inte- gral 1, for phi(t) = 1 - 3*t**2 + 2*t**3 (t in [0,1]): Subroutine GETQW. Subprograms: The subprograms in csrfpack.f are alphabetically ordered except for Subroutine CSURF which appears at the beginning of the file. The header comments in CSURF duplicate much of the documentation provided here. The subprograms that are likely to be called directly by a driver program are as follows. CSURF: Subroutine which computes parameters defining the smooth interpolatory surface F, and optionally creates plots of the triangulations and their duals, by executing the 10-step procedure outlined above. FGRID: Subroutine which uses FVAL to evaluate F at the vertices of a rectangular grid. FVAL: Subroutine which, given a point p and the output parameters from Subroutine CSURF, returns the value and gradient of F at p. PLTCNT: Subroutine which, given a set of function values z = F(x,y) at the vertices of a rectangular grid, creates an Encapsulated PostScript file contain- ing a contour plot of the piecewise bilinear interpolant of the function values. PLTTR: Subroutine which creates an Encapsulated PostScript file containing a plot of a triangulation created by Subroutine TRMSHC. TRMSHC: Subroutine which constructs a convexity-preserving triangulation, if it exists, of a set of N data points. Additional high-level subprograms that are called by CSURF are the following. ADDQT: Subroutine which, given a set of strictly convex Hermite data (nodes, data values, and nodal gradi- ents such that there exists a convex Hermite interpolant of the data values and gradients, and no four data points are coplanar), adjusts the data values and gradients (using a quadratic correction term) such that a strictly convex Her- mite interpolant of the data can be constructed from a convex Hermite interpolant of the adjusted data. DELBE: Subroutine which removes extraneous boundary edges (nearly null triangles) to avoid large errors caused by steep gradients on the boundary. The calling program specifies a lower bound on the acceptable aspect ratio (ratio of the radius of the inscribed circle to that of the circumcircle) of boundary triangles. DELTAI: Subroutine which, given a set of nodes p_i, data values z_i, and a data structure representing a convexity-preserving gradient triangulation Tg (Subroutine TRMSHC), returns the perpendicular distances D_i from each node p_i to the boundary of R_i. GETQW: Subroutine which computes the weights and quadra- ture points defining the quadrature rule for approximating values of F. GLIST: Subroutine which, given a convexity-preserving triangulation and its straight-line dual computed by Subroutine VLIST, returns a set of nodal gradi- ents for which there exists a convex Hermite interpolant of the data values and gradients. The nodal gradients are taken to be the centroids of the gradient feasibility regions (or truncated regions). PLTGR: Subroutine which creates an Encapsulated PostScript file containing a plot of the straight-line dual of a triangulation along with the set of nodal gradients. VLIST: Subroutine which, given a convexity-preserving triangulation, computes its straight-line dual, returning the vertices and adjacency information defining the gradient feasibility regions or cell diagram. The header comments in Subroutine TRMSHC include a list of additional user-callable subprograms related to the triangu- lations. Many of these are, with the exception of minor changes in comments, identical to subprograms in TRIPACK (ACM TOMS Algorithm 751). The triangulation data structure is the same as that of TRIPACK, and the structure defining T is overwritten by the structure for Tg. 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-0.907983 lineto stroke 1.000000 0.907983 moveto 0.964410 0.948718 lineto 0.948718 0.960571 lineto 0.897436 0.996239 lineto 0.891669 1.000000 lineto stroke -0.891669 1.000000 moveto -0.897436 0.996239 lineto -0.948718 0.960571 lineto -0.964410 0.948718 lineto -1.000000 0.907983 lineto stroke showpage %%EOF  SHAR_EOF fi # end of overwriting check if test -f 'csrftest.prt' then echo shar: will not over-write existing file "'csrftest.prt'" else cat << "SHAR_EOF" > 'csrftest.prt' CSRFTEST.PRT N = 8 XMIN = -0.100D+01 XMAX = 0.100D+01 YMIN = -0.100D+01 YMAX = 0.100D+01 TOLBE = 0.100D-01 STRICT = T NR = 8 NA = 24 Number of boundary edge deletions: ND = 0 Scale factor for quadratic term: EPS = 0.865D+00 Min. distance node to cell boundary: DMIN = 0.537D-01 SHAR_EOF fi # end of overwriting check if test -f 'data' then echo shar: will not over-write existing file "'data'" else cat << "SHAR_EOF" > 'data' 8 -1.00 -1.00 2.0000 1.00 -1.00 2.0000 0.00 -0.60 0.1296 -0.40 0.00 0.0256 0.40 0.00 0.0256 0.00 0.60 0.1296 -1.00 1.00 2.0000 1.00 1.00 2.0000 SHAR_EOF fi # end of overwriting check if test -f 'driver.f' then echo shar: will not over-write existing file "'driver.f'" else cat << "SHAR_EOF" > 'driver.f' C C C CSRFTEST C 06/25/02 C C This program is a test driver for the software package C CSRFPACK which constructs a convex once-continuously C differentiable bivariate function that interpolates a set C of data values at arbitrarily distributed nodes in the C plane. C C The user must provide an input data set consisting of C the number of data points N (Format I5) and a sequence of C N ordered triples (X,Y,Z) with nodal coordinates (X,Y) and C data values Z: Format 3E23.15. C C Provided the data defines a convex function, the output C consists of the following files: C C csrftest.prt: Print file with parameter values and error C messages. C csrftest.eps: Encapsulated Postscript file containing a C contour plot of the interpolatory surface. C csrftest.out: Data set suitable for a surface plotting C package consisting of interpolated values C on a triangulation of a uniform grid in the C bounding box (the smallest rectangle with C sides parallel to the axes that contains C the nodes). C tplot.eps: Encapsulated Postscript file containing the C convexity-preserving triangulation of the C nodes. C gplot.eps: Encapsulated Postscript file containing the C gradient feasibility diagram and nodal C gradients C tgplot.eps: Encapsulated Postscript file containing the C gradient triangulation. C rplot.eps: Encapsulated Postscript file containing the C cell diagram and nodes. C C Previously existing files with the above names are over- C written. C INTEGER LC, LWK, N2, N6, NA, NMAX, NR, NX, NY C C Maximum number of nodes NMAX: C PARAMETER (NMAX=300, N2=2*NMAX, N6=6*NMAX) C C Number of grid points NX (horizontally) and NY C (vertically) defining the uniform grid. C PARAMETER (NX=40, NY=40, LWK=NX*NY+(NX*NY)/2, . LC=NX*NY+LWK) C C Number of quadrature points in the radial and angular C directions: C PARAMETER (NR=8, NA=24) C CHARACTER*60 FNAME DOUBLE PRECISION C(NMAX), D(NMAX), DMIN, DX, DXL(N2), . DY, DYL(N2), EPS, F(NX,NY), GX(NMAX), . GY(NMAX), PLTSIZ, PX(NX), PY(NY), . QX(NR,NA), QY(NR,NA), TOLBE, W(NR), . X(NMAX), XC(LC), XMAX, XMIN, Y(NMAX), . YC(LC), YMAX, YMIN, Z(NMAX) INTEGER I, IER, IPLOT, IWK(LWK), J, K, LCON, . LEND(NMAX), LIN, LIST(N6), LISTV(N6), LOUT, . LPRT, LPTR(N6), LNEW, N, NCON, ND, NEAR(NMAX), . NEXT(NMAX), NV LOGICAL STRICT C C Data: C C Plot option in the range 0 to 15: IPLOT = (b3,b2,b1,b0), C with bits specifying which Postscript files are created: C C b0: tplot.eps C b1: gplot.eps C b2: tgplot.eps C b3: rplot.eps C DATA IPLOT/15/ C C Number of contour values in the contour plot. C DATA NCON/10/ C C PostScript plot size: 1.0 <= PLTSIZ <= 7.5. C DATA PLTSIZ/3.0/ C C Strict convexity option (quadratic correction term C requiring an O(N**2) algorithm). C DATA STRICT/.TRUE./ C C Positive tolerance for removing boundary triangles. C DATA TOLBE/1.D-2/ C C LIN,LOUT = Logical unit numbers for input and output. C DATA LCON/1/, LIN/5/, LOUT/6/, LPRT/4/ C C Input/output formats: C 300 FORMAT (I5) 310 FORMAT (3E23.15) 320 FORMAT (A60) 330 FORMAT (3I6) C C Get an input file name and open it. C C 1 WRITE (*,100) C 100 FORMAT (///13X,'CSRFTEST: CSRFPACK Test Program'// C . 5X,'Specify a data set file name (at most 60', C . ' characters):'/) C READ (*,320,ERR=1) FNAME C OPEN (LIN,FILE=FNAME,STATUS='OLD',ERR=1) C C Read N and the nodal coordinates and data values from C unit LIN. C READ (LIN,300) N IF (N .LT. 3 .OR. N .GT. NMAX) THEN WRITE (*,110) NMAX 110 FORMAT (//5X,'Invalid data. N must be in the ', . 'range 3 to ',I5,'.') STOP ENDIF READ (LIN,310) (X(I),Y(I),Z(I), I = 1,N) C C Compute the bounding box corner coordinates (XMIN,YMIN) C and (XMAX,YMAX) and the uniform grid points C (((PX(i),PY(j)), i = 1,NX), j = 1,NY). C XMIN = X(1) XMAX = X(1) YMIN = Y(1) YMAX = Y(1) DO 2 I = 2,N IF (X(I) .LT. XMIN) XMIN = X(I) IF (X(I) .GT. XMAX) XMAX = X(I) IF (Y(I) .LT. YMIN) YMIN = Y(I) IF (Y(I) .GT. YMAX) YMAX = Y(I) 2 CONTINUE DX = (XMAX-XMIN)/DBLE(NX-1) DY = (YMAX-YMIN)/DBLE(NY-1) DO 3 I = 1,NX PX(I) = XMIN + DBLE(I-1)*DX 3 CONTINUE DO 4 J = 1,NY PY(J) = YMIN + DBLE(J-1)*DY 4 CONTINUE C C Create a print file with parameter values. C OPEN (LPRT,FILE='csrftest.prt') WRITE (LPRT,120) N, XMIN, XMAX, YMIN, YMAX, TOLBE, . STRICT, NR, NA 120 FORMAT (///27X,'CSRFTEST.PRT'// . 10X,'N = ',I5/ . 10X,'XMIN = ',D10.3,3X,'XMAX = ',D10.3/ . 10X,'YMIN = ',D10.3,3X,'YMAX = ',D10.3/ . 10X,'TOLBE = ',D10.3/ . 10X,'STRICT = ',L1/ . 10X,'NR = ',I2/ . 10X,'NA = ',I2) C C Construct the interpolatory surface F. C CALL CSURF (N,X,Y,TOLBE,IPLOT,PLTSIZ,STRICT,NR,NA, ND, . Z,C,LIST,LPTR,LEND,LNEW,NEAR,NEXT,NV, . LISTV,DXL,DYL,GX,GY,EPS,D,DMIN,W,QX,QY, . IER) IF (IER .NE. 0) THEN IF (IER .NE. -11) GO TO 10 C C DMIN = 0. C WRITE (*,130) WRITE (LPRT,130) ENDIF 130 FORMAT (//10X,'*** The surface is not strictly ', . 'convex: DMIN = 0. ***'//) C C Output computed parameter values to the print file. C WRITE (LPRT,140) ND, EPS, DMIN WRITE (*,140) ND, EPS, DMIN 140 FORMAT (//10X,'Number of boundary edge deletions: ', . 'ND = ',I3/ . 10X,'Scale factor for quadratic term: ', . 'EPS = ',D10.3/ . 10X,'Min. distance node to cell boundary: ', . ' DMIN = ',D10.3) C C Compute interpolated values on the uniform grid. C CALL FGRID (NX,NY,PX,PY,EPS,GX,GY,C,LIST,LPTR, . LEND,DMIN,NR,NA,W,QX,QY, F,IER) C C Create a contour plot of the interpolatory surface on the C NX by NY uniform grid. C OPEN (LCON,FILE='csrftest.eps') CALL PLTCNT (LCON,PLTSIZ,NX,NY,PX,PY,F,NCON,IWK,XC, . YC, IER) C C Create the output data set using the natural ordering of C the rectangular grid points: left-to-right within C bottom-to-top. C C Write the number of vertices followed by the sequence of C vertex coordinate triples. C C OPEN (LOUT,FILE='csrftest.out') WRITE (LOUT,300) NX*NY DO 6 J = 1,NY DO 5 I = 1,NX WRITE (LOUT,310) PX(I), PY(J), F(I,J) 5 CONTINUE 6 CONTINUE C C Write the number of triangles followed by the sequence of C triangles (CCW-ordered vertex indices). For each of the C (NX-1)*(NY-1) cells, k is the vertex index of the upper C right corner, and the cell is partitioned by the C diagonal with slope 1. C WRITE (LOUT,300) 2*(NX-1)*(NY-1) DO 8 J = 2,NY DO 7 I = 2,NX K = NX*(J-1) + I WRITE (LOUT,330) K-NX-1, K-NX, K WRITE (LOUT,330) K-NX-1, K, K-1 7 CONTINUE 8 CONTINUE STOP C C Error in CSURF. C 10 WRITE (LPRT,400) IER WRITE (*,400) IER 400 FORMAT (///10X,'Error flag ',I4,' returned by CSURF.') IF (IER .EQ. -2) THEN WRITE (LPRT,410) WRITE (*,410) ENDIF 410 FORMAT (/10X,'The first three nodes are collinear.') IF (IER .EQ. -3) THEN WRITE (LPRT,420) WRITE (*,420) ENDIF 420 FORMAT (/10X,'A convex triangulation does not exist ', . '(ADDNDC error -3).') IF (IER .EQ. -4) THEN WRITE (LPRT,430) WRITE (*,430) ENDIF 430 FORMAT (/10X,'A convex triangulation does not exist ', . '(ADDNDC error -4).') IF (IER .GT. 0) THEN WRITE (LPRT,440) WRITE (*,440) ENDIF 440 FORMAT (/10X,'Duplicate nodes.') 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