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*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
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