*DECK PCHIC SUBROUTINE PCHIC (IC, VC, SWITCH, N, X, F, D, INCFD, WK, NWK, + IERR) C***BEGIN PROLOGUE PCHIC C***PURPOSE Set derivatives needed to determine a piecewise monotone C piecewise cubic Hermite interpolant to given data. C User control is available over boundary conditions and/or C treatment of points where monotonicity switches direction. C***LIBRARY SLATEC (PCHIP) C***CATEGORY E1A C***TYPE SINGLE PRECISION (PCHIC-S, DPCHIC-D) C***KEYWORDS CUBIC HERMITE INTERPOLATION, MONOTONE INTERPOLATION, C PCHIP, PIECEWISE CUBIC INTERPOLATION, C SHAPE-PRESERVING 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 PCHIC: Piecewise Cubic Hermite Interpolation Coefficients. C C Sets derivatives needed to determine a piecewise monotone piece- C wise cubic interpolant to the data given in X and F satisfying the C boundary conditions specified by IC and VC. C C The treatment of points where monotonicity switches direction is C controlled by argument SWITCH. 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 IC(2), N, NWK, IERR C REAL VC(2), SWITCH, X(N), F(INCFD,N), D(INCFD,N), WK(NWK) C C CALL PCHIC (IC, VC, SWITCH, N, X, F, D, INCFD, WK, NWK, IERR) C C Parameters: C C IC -- (input) integer array of length 2 specifying desired C boundary conditions: C IC(1) = IBEG, desired condition at beginning of data. C IC(2) = IEND, desired condition at end of data. C C IBEG = 0 for the default boundary condition (the same as C used by PCHIM). C If IBEG.NE.0, then its sign indicates whether the boundary C derivative is to be adjusted, if necessary, to be C compatible with monotonicity: C IBEG.GT.0 if no adjustment is to be performed. C IBEG.LT.0 if the derivative is to be adjusted for C monotonicity. C C Allowable values for the magnitude of IBEG are: C IBEG = 1 if first derivative at X(1) is given in VC(1). C IBEG = 2 if second derivative at X(1) is given in VC(1). C IBEG = 3 to use the 3-point difference formula for D(1). C (Reverts to the default b.c. if N.LT.3 .) C IBEG = 4 to use the 4-point difference formula for D(1). C (Reverts to the default b.c. if N.LT.4 .) C IBEG = 5 to set D(1) so that the second derivative is con- C tinuous at X(2). (Reverts to the default b.c. if N.LT.4.) C This option is somewhat analogous to the "not a knot" C boundary condition provided by PCHSP. C C NOTES (IBEG): C 1. An error return is taken if ABS(IBEG).GT.5 . C 2. Only in case IBEG.LE.0 is it guaranteed that the C interpolant will be monotonic in the first interval. C If the returned value of D(1) lies between zero and C 3*SLOPE(1), the interpolant will be monotonic. This C is **NOT** checked if IBEG.GT.0 . C 3. If IBEG.LT.0 and D(1) had to be changed to achieve mono- C tonicity, a warning error is returned. C C IEND may take on the same values as IBEG, but applied to C derivative at X(N). In case IEND = 1 or 2, the value is C given in VC(2). C C NOTES (IEND): C 1. An error return is taken if ABS(IEND).GT.5 . C 2. Only in case IEND.LE.0 is it guaranteed that the C interpolant will be monotonic in the last interval. C If the returned value of D(1+(N-1)*INCFD) lies between C zero and 3*SLOPE(N-1), the interpolant will be monotonic. C This is **NOT** checked if IEND.GT.0 . C 3. If IEND.LT.0 and D(1+(N-1)*INCFD) had to be changed to C achieve monotonicity, a warning error is returned. C C VC -- (input) real array of length 2 specifying desired boundary C values, as indicated above. C VC(1) need be set only if IC(1) = 1 or 2 . C VC(2) need be set only if IC(2) = 1 or 2 . C C SWITCH -- (input) indicates desired treatment of points where C direction of monotonicity switches: C Set SWITCH to zero if interpolant is required to be mono- C tonic in each interval, regardless of monotonicity of data. C NOTES: C 1. This will cause D to be set to zero at all switch C points, thus forcing extrema there. C 2. The result of using this option with the default boun- C dary conditions will be identical to using PCHIM, but C will generally cost more compute time. C This option is provided only to facilitate comparison C of different switch and/or boundary conditions. C Set SWITCH nonzero to use a formula based on the 3-point C difference formula in the vicinity of switch points. C If SWITCH is positive, the interpolant on each interval C containing an extremum is controlled to not deviate from C the data by more than SWITCH*DFLOC, where DFLOC is the C maximum of the change of F on this interval and its two C immediate neighbors. C If SWITCH is negative, no such control is to be imposed. C C N -- (input) number of data points. (Error return if N.LT.2 .) 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 C D -- (output) real array of derivative values at the data points. C These values will determine a monotone cubic Hermite func- C tion on each subinterval on which the data are monotonic, C except possibly adjacent to switches in monotonicity. 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 WK -- (scratch) real array of working storage. The user may wish C to know that the returned values are: C WK(I) = H(I) = X(I+1) - X(I) ; C WK(N-1+I) = SLOPE(I) = (F(1,I+1) - F(1,I)) / H(I) C for I = 1(1)N-1. C C NWK -- (input) length of work array. C (Error return if NWK.LT.2*(N-1) .) C C IERR -- (output) error flag. C Normal return: C IERR = 0 (no errors). C Warning errors: C IERR = 1 if IBEG.LT.0 and D(1) had to be adjusted for C monotonicity. C IERR = 2 if IEND.LT.0 and D(1+(N-1)*INCFD) had to be C adjusted for monotonicity. C IERR = 3 if both of the above are true. 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 IERR = -4 if ABS(IBEG).GT.5 . C IERR = -5 if ABS(IEND).GT.5 . C IERR = -6 if both of the above are true. C IERR = -7 if NWK.LT.2*(N-1) . 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, Piecewise Cubic Hermite Interpolation C Package, Report UCRL-87285, Lawrence Livermore Nation- C al Laboratory, July 1982. [Poster presented at the C SIAM 30th Anniversary Meeting, 19-23 July 1982.] C 2. 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 3. 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 PCHCE, PCHCI, PCHCS, XERMSG C***REVISION HISTORY (YYMMDD) C 820218 DATE WRITTEN C 820804 Converted to SLATEC library version. C 870813 Updated Reference 2. 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 PCHIC