*DECK PCHIM SUBROUTINE PCHIM (N, X, F, D, INCFD, IERR) C***BEGIN PROLOGUE PCHIM C***PURPOSE Set derivatives needed to determine a monotone piecewise C cubic Hermite interpolant to given data. Boundary values C are provided which are compatible with monotonicity. The C interpolant will have an extremum at each point where mono- C tonicity switches direction. (See PCHIC if user control is C desired over boundary or switch conditions.) C***LIBRARY SLATEC (PCHIP) C***CATEGORY E1A C***TYPE SINGLE PRECISION (PCHIM-S, DPCHIM-D) C***KEYWORDS CUBIC HERMITE INTERPOLATION, MONOTONE INTERPOLATION, C PCHIP, PIECEWISE CUBIC INTERPOLATION C***AUTHOR Fritsch, F. N., (LLNL) C Lawrence Livermore National Laboratory C P.O. Box 808 (L-316) C Livermore, CA 94550 C FTS 532-4275, (510) 422-4275 C***DESCRIPTION C C PCHIM: Piecewise Cubic Hermite Interpolation to C Monotone data. C C Sets derivatives needed to determine a monotone piecewise cubic C Hermite interpolant to the data given in X and F. C C Default boundary conditions are provided which are compatible C with monotonicity. (See PCHIC if user control of boundary con- C ditions is desired.) C C If the data are only piecewise monotonic, the interpolant will C have an extremum at each point where monotonicity switches direc- C tion. (See PCHIC if user control is desired in such cases.) C C To facilitate two-dimensional applications, includes an increment C between successive values of the F- and D-arrays. C C The resulting piecewise cubic Hermite function may be evaluated C by PCHFE or PCHFD. C C ---------------------------------------------------------------------- C C Calling sequence: C C PARAMETER (INCFD = ...) C INTEGER N, IERR C REAL X(N), F(INCFD,N), D(INCFD,N) C C CALL PCHIM (N, X, F, D, INCFD, IERR) C C Parameters: C C N -- (input) number of data points. (Error return if N.LT.2 .) C If N=2, simply does linear interpolation. C C X -- (input) real array of independent variable values. The C elements of X must be strictly increasing: C X(I-1) .LT. X(I), I = 2(1)N. C (Error return if not.) C C F -- (input) real array of dependent variable values to be inter- C polated. F(1+(I-1)*INCFD) is value corresponding to X(I). C PCHIM is designed for monotonic data, but it will work for C any F-array. It will force extrema at points where mono- C tonicity switches direction. If some other treatment of C switch points is desired, PCHIC should be used instead. C ----- C D -- (output) real array of derivative values at the data points. C If the data are monotonic, these values will determine a C a monotone cubic Hermite function. C The value corresponding to X(I) is stored in C D(1+(I-1)*INCFD), I=1(1)N. C No other entries in D are changed. C C INCFD -- (input) increment between successive values in F and D. C This argument is provided primarily for 2-D applications. C (Error return if INCFD.LT.1 .) C C IERR -- (output) error flag. C Normal return: C IERR = 0 (no errors). C Warning error: C IERR.GT.0 means that IERR switches in the direction C of monotonicity were detected. C "Recoverable" errors: C IERR = -1 if N.LT.2 . C IERR = -2 if INCFD.LT.1 . C IERR = -3 if the X-array is not strictly increasing. C (The D-array has not been changed in any of these cases.) C NOTE: The above errors are checked in the order listed, C and following arguments have **NOT** been validated. C C***REFERENCES 1. F. N. Fritsch and J. Butland, A method for construc- C ting local monotone piecewise cubic interpolants, SIAM C Journal on Scientific and Statistical Computing 5, 2 C (June 1984), pp. 300-304. C 2. F. N. Fritsch and R. E. Carlson, Monotone piecewise C cubic interpolation, SIAM Journal on Numerical Ana- C lysis 17, 2 (April 1980), pp. 238-246. C***ROUTINES CALLED PCHST, XERMSG C***REVISION HISTORY (YYMMDD) C 811103 DATE WRITTEN C 820201 1. Introduced PCHST to reduce possible over/under- C flow problems. C 2. Rearranged derivative formula for same reason. C 820602 1. Modified end conditions to be continuous functions C of data when monotonicity switches in next interval. C 2. Modified formulas so end conditions are less prone C of over/underflow problems. C 820803 Minor cosmetic changes for release 1. C 870813 Updated Reference 1. C 890411 Added SAVE statements (Vers. 3.2). C 890531 Changed all specific intrinsics to generic. (WRB) C 890703 Corrected category record. (WRB) C 890831 Modified array declarations. (WRB) C 890831 REVISION DATE from Version 3.2 C 891214 Prologue converted to Version 4.0 format. (BAB) C 900315 CALLs to XERROR changed to CALLs to XERMSG. (THJ) C 920429 Revised format and order of references. (WRB,FNF) C***END PROLOGUE PCHIM C Programming notes: C C 1. The function PCHST(ARG1,ARG2) is assumed to return zero if C either argument is zero, +1 if they are of the same sign, and C -1 if they are of opposite sign. C 2. To produce a double precision version, simply: C a. Change PCHIM to DPCHIM wherever it occurs, C b. Change PCHST to DPCHST wherever it occurs, C c. Change all references to the Fortran intrinsics to their C double precision equivalents, C d. Change the real declarations to double precision, and C e. Change the constants ZERO and THREE to double precision. C C DECLARE ARGUMENTS. C INTEGER N, INCFD, IERR REAL X(*), F(INCFD,*), D(INCFD,*) C C DECLARE LOCAL VARIABLES. C INTEGER I, NLESS1 REAL DEL1, DEL2, DMAX, DMIN, DRAT1, DRAT2, DSAVE, * H1, H2, HSUM, HSUMT3, THREE, W1, W2, ZERO SAVE ZERO, THREE REAL PCHST DATA ZERO /0./, THREE /3./ C C VALIDITY-CHECK ARGUMENTS. C C***FIRST EXECUTABLE STATEMENT PCHIM IF ( N.LT.2 ) GO TO 5001 IF ( INCFD.LT.1 ) GO TO 5002 DO 1 I = 2, N IF ( X(I).LE.X(I-1) ) GO TO 5003 1 CONTINUE C C FUNCTION DEFINITION IS OK, GO ON. C IERR = 0 NLESS1 = N - 1 H1 = X(2) - X(1) DEL1 = (F(1,2) - F(1,1))/H1 DSAVE = DEL1 C C SPECIAL CASE N=2 -- USE LINEAR INTERPOLATION. C IF (NLESS1 .GT. 1) GO TO 10 D(1,1) = DEL1 D(1,N) = DEL1 GO TO 5000 C C NORMAL CASE (N .GE. 3). C 10 CONTINUE H2 = X(3) - X(2) DEL2 = (F(1,3) - F(1,2))/H2 C C SET D(1) VIA NON-CENTERED THREE-POINT FORMULA, ADJUSTED TO BE C SHAPE-PRESERVING. C HSUM = H1 + H2 W1 = (H1 + HSUM)/HSUM W2 = -H1/HSUM D(1,1) = W1*DEL1 + W2*DEL2 IF ( PCHST(D(1,1),DEL1) .LE. ZERO) THEN D(1,1) = ZERO ELSE IF ( PCHST(DEL1,DEL2) .LT. ZERO) THEN C NEED DO THIS CHECK ONLY IF MONOTONICITY SWITCHES. DMAX = THREE*DEL1 IF (ABS(D(1,1)) .GT. ABS(DMAX)) D(1,1) = DMAX ENDIF C C LOOP THROUGH INTERIOR POINTS. C DO 50 I = 2, NLESS1 IF (I .EQ. 2) GO TO 40 C H1 = H2 H2 = X(I+1) - X(I) HSUM = H1 + H2 DEL1 = DEL2 DEL2 = (F(1,I+1) - F(1,I))/H2 40 CONTINUE C C SET D(I)=0 UNLESS DATA ARE STRICTLY MONOTONIC. C D(1,I) = ZERO IF ( PCHST(DEL1,DEL2) ) 42, 41, 45 C C COUNT NUMBER OF CHANGES IN DIRECTION OF MONOTONICITY. C 41 CONTINUE IF (DEL2 .EQ. ZERO) GO TO 50 IF ( PCHST(DSAVE,DEL2) .LT. ZERO) IERR = IERR + 1 DSAVE = DEL2 GO TO 50 C 42 CONTINUE IERR = IERR + 1 DSAVE = DEL2 GO TO 50 C C USE BRODLIE MODIFICATION OF BUTLAND FORMULA. C 45 CONTINUE HSUMT3 = HSUM+HSUM+HSUM W1 = (HSUM + H1)/HSUMT3 W2 = (HSUM + H2)/HSUMT3 DMAX = MAX( ABS(DEL1), ABS(DEL2) ) DMIN = MIN( ABS(DEL1), ABS(DEL2) ) DRAT1 = DEL1/DMAX DRAT2 = DEL2/DMAX D(1,I) = DMIN/(W1*DRAT1 + W2*DRAT2) C 50 CONTINUE C C SET D(N) VIA NON-CENTERED THREE-POINT FORMULA, ADJUSTED TO BE C SHAPE-PRESERVING. C W1 = -H2/HSUM W2 = (H2 + HSUM)/HSUM D(1,N) = W1*DEL1 + W2*DEL2 IF ( PCHST(D(1,N),DEL2) .LE. ZERO) THEN D(1,N) = ZERO ELSE IF ( PCHST(DEL1,DEL2) .LT. ZERO) THEN C NEED DO THIS CHECK ONLY IF MONOTONICITY SWITCHES. DMAX = THREE*DEL2 IF (ABS(D(1,N)) .GT. ABS(DMAX)) D(1,N) = DMAX ENDIF C C NORMAL RETURN. C 5000 CONTINUE RETURN C C ERROR RETURNS. C 5001 CONTINUE C N.LT.2 RETURN. IERR = -1 CALL XERMSG ('SLATEC', 'PCHIM', + 'NUMBER OF DATA POINTS LESS THAN TWO', IERR, 1) RETURN C 5002 CONTINUE C INCFD.LT.1 RETURN. IERR = -2 CALL XERMSG ('SLATEC', 'PCHIM', 'INCREMENT LESS THAN ONE', IERR, + 1) RETURN C 5003 CONTINUE C X-ARRAY NOT STRICTLY INCREASING. IERR = -3 CALL XERMSG ('SLATEC', 'PCHIM', 'X-ARRAY NOT STRICTLY INCREASING' + , IERR, 1) RETURN C------------- LAST LINE OF PCHIM FOLLOWS ------------------------------ END