SLATEC Routines --- PCHDOC ---
*DECK PCHDOC
SUBROUTINE PCHDOC
C***BEGIN PROLOGUE PCHDOC
C***PURPOSE Documentation for PCHIP, a Fortran package for piecewise
C cubic Hermite interpolation of data.
C***LIBRARY SLATEC (PCHIP)
C***CATEGORY E1A, Z
C***TYPE ALL (PCHDOC-A)
C***KEYWORDS CUBIC HERMITE INTERPOLATION, DOCUMENTATION,
C MONOTONE INTERPOLATION, PCHIP,
C 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 PCHIP: Piecewise Cubic Hermite Interpolation Package
C
C This document describes the contents of PCHIP, which is a
C Fortran package for piecewise cubic Hermite interpolation of data.
C It features software to produce a monotone and "visually pleasing"
C interpolant to monotone data. As is demonstrated in Reference 4,
C such an interpolant may be more reasonable than a cubic spline if
C the data contains both "steep" and "flat" sections. Interpola-
C tion of cumulative probability distribution functions is another
C application. (See References 2-4 for examples.)
C
C
C All piecewise cubic functions in PCHIP are represented in
C cubic Hermite form; that is, f(x) is determined by its values
C F(I) and derivatives D(I) at the breakpoints X(I), I=1(1)N.
C Throughout the package a PCH function is represented by the
C five variables N, X, F, D, INCFD:
C N - number of data points;
C X - abscissa values for the data points;
C F - ordinates (function values) for the data points;
C D - slopes (derivative values) at the data points;
C INCFD - increment between successive elements in the F- and
C D-arrays (more on this later).
C These appear together and in the same order in all calls.
C
C The double precision equivalents of the PCHIP routines are
C obtained from the single precision names by prefixing the
C single precision names with a D. For example, the double
C precision equivalent of PCHIM is DPCHIM.
C
C The contents of the package are as follows:
C
C 1. Determine Derivative Values.
C
C NOTE: These routines provide alternate ways of determining D
C if these values are not already known.
C
C PCHIM -- Piecewise Cubic Hermite Interpolation to Monotone
C data.
C Used if the data are monotonic or if the user wants
C to guarantee that the interpolant stays within the
C limits of the data. (See Reference 3.)
C
C PCHIC -- Piecewise Cubic Hermite Interpolation Coefficients.
C Used if neither of the above conditions holds, or if
C the user wishes control over boundary derivatives.
C Will generally reproduce monotonicity on subintervals
C over which the data are monotonic.
C
C PCHSP -- Piecewise Cubic Hermite Spline.
C Produces a cubic spline interpolator in cubic Hermite
C form. Provided primarily for easy comparison of the
C spline with other piecewise cubic interpolants. (A
C modified version of de Boor's CUBSPL, Reference 1.)
C
C 2. Evaluate, Differentiate, or Integrate Resulting PCH Function.
C
C NOTE: If derivative values are available from some other
C source, these routines can be used without calling
C any of the previous routines.
C
C CHFEV -- Cubic Hermite Function EValuator.
C Evaluates a single cubic Hermite function at an array
C of points. Used when the interval is known, as in
C graphing applications. Called by PCHFE.
C
C PCHFE -- Piecewise Cubic Hermite Function Evaluator.
C Used when the interval is unknown or the evaluation
C array spans more than one data interval.
C
C CHFDV -- Cubic Hermite Function and Derivative Evaluator.
C Evaluates a single cubic Hermite function and its
C first derivative at an array of points. Used when
C the interval is known, as in graphing applications.
C Called by PCHFD.
C
C PCHFD -- Piecewise Cubic Hermite Function and Derivative
C Evaluator.
C Used when the interval is unknown or the evaluation
C array spans more than one data interval.
C
C PCHID -- Piecewise Cubic Hermite Integrator, Data Limits.
C Computes the definite integral of a piecewise cubic
C Hermite function when the integration limits are data
C points.
C
C PCHIA -- Piecewise Cubic Hermite Integrator, Arbitrary Limits.
C Computes the definite integral of a piecewise cubic
C Hermite function over an arbitrary finite interval.
C
C 3. Utility routines.
C
C PCHBS -- Piecewise Cubic Hermite to B-Spline converter.
C Converts a PCH function to B-representation, so that
C it can be used with other elements of the B-spline
C package (see BSPDOC).
C
C PCHCM -- Piecewise Cubic Hermite, Check Monotonicity of.
C Checks the monotonicity of an arbitrary PCH function.
C Might be used with PCHSP to build a polyalgorithm for
C piecewise C-2 interpolation.
C
C 4. Internal routines.
C
C CHFIE -- Cubic Hermite Function Integral Evaluator.
C (Real function called by PCHIA.)
C
C CHFCM -- Cubic Hermite Function, Check Monotonicity of.
C (Integer function called by PCHCM.)
C
C PCHCE -- PCHIC End Derivative Setter.
C (Called by PCHIC.)
C
C PCHCI -- PCHIC Initial Derivative Setter.
C (Called by PCHIC.)
C
C PCHCS -- PCHIC Monotonicity Switch Derivative Setter.
C (Called by PCHIC.)
C
C PCHDF -- PCHIP Finite Difference Formula.
C (Real function called by PCHCE and PCHSP.)
C
C PCHST -- PCHIP Sign Testing Routine.
C (Real function called by various PCHIP routines.)
C
C PCHSW -- PCHCS Switch Excursion Adjuster.
C (Called by PCHCS.)
C
C The calling sequences for these routines are described in the
C prologues of the respective routines.
C
C
C INCFD, the increment between successive elements in the F-
C and D-arrays is included in the representation of a PCH function
C in this package to facilitate two-dimensional applications. For
C "normal" usage INCFD=1, and F and D are one-dimensional arrays.
C one would call PCHxx (where "xx" is "IM", "IC", or "SP") with
C
C N, X, F, D, 1 .
C
C Suppose, however, that one has data on a rectangular mesh,
C
C F2D(I,J) = value at (X(I), Y(J)), I=1(1)NX,
C J=1(1)NY.
C Assume the following dimensions:
C
C REAL X(NXMAX), Y(NYMAX)
C REAL F2D(NXMAX,NYMAX), FX(NXMAX,NYMAX), FY(NXMAX,NYMAX)
C
C where 2.LE.NX.LE.NXMAX AND 2.LE.NY.LE.NYMAX . To interpolate
C in X along the line Y = Y(J), call PCHxx with
C
C NX, X, F2D(1,J), FX(1,J), 1 .
C
C To interpolate along the line X = X(I), call PCHxx with
C
C NY, Y, F2D(I,1), FY(I,1), NXMAX .
C
C (This example assumes the usual columnwise storage of 2-D arrays
C in Fortran.)
C
C***REFERENCES 1. Carl de Boor, A Practical Guide to Splines, Springer-
C Verlag, New York, 1978 (esp. Chapter IV, pp.49-62).
C 2. F. N. Fritsch, Piecewise Cubic Hermite Interpolation
C Package, Report UCRL-87285, Lawrence Livermore Natio-
C nal Laboratory, July 1982. [Poster presented at the
C SIAM 30th Anniversary Meeting, 19-23 July 1982.]
C 3. 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 4. 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 (NONE)
C***REVISION HISTORY (YYMMDD)
C 811106 DATE WRITTEN
C 870930 Updated Reference 3.
C 890414 Changed PCHMC and CHFMC to PCHCM and CHFCM, respectively,
C and augmented description of PCHCM.
C 891214 Prologue converted to Version 4.0 format. (BAB)
C 910826 1. Revised purpose, clarified role of argument INCFD,
C corrected error in example, and removed redundant
C reference list.
C 2. Added description of PCHBS. (FNF)
C 920429 Revised format and order of references. (WRB,FNF)
C 930505 Changed CHFIV to CHFIE. (FNF)
C***END PROLOGUE PCHDOC