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213 lines
9.1 KiB
Fortran
213 lines
9.1 KiB
Fortran
*DECK PCHDOC
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SUBROUTINE PCHDOC
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C***BEGIN PROLOGUE PCHDOC
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C***PURPOSE Documentation for PCHIP, a Fortran package for piecewise
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C cubic Hermite interpolation of data.
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C***LIBRARY SLATEC (PCHIP)
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C***CATEGORY E1A, Z
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C***TYPE ALL (PCHDOC-A)
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C***KEYWORDS CUBIC HERMITE INTERPOLATION, DOCUMENTATION,
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C MONOTONE INTERPOLATION, PCHIP,
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C PIECEWISE CUBIC INTERPOLATION
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C***AUTHOR Fritsch, F. N., (LLNL)
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C Lawrence Livermore National Laboratory
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C P.O. Box 808 (L-316)
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C Livermore, CA 94550
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C FTS 532-4275, (510) 422-4275
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C***DESCRIPTION
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C
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C PCHIP: Piecewise Cubic Hermite Interpolation Package
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C
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C This document describes the contents of PCHIP, which is a
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C Fortran package for piecewise cubic Hermite interpolation of data.
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C It features software to produce a monotone and "visually pleasing"
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C interpolant to monotone data. As is demonstrated in Reference 4,
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C such an interpolant may be more reasonable than a cubic spline if
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C the data contains both "steep" and "flat" sections. Interpola-
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C tion of cumulative probability distribution functions is another
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C application. (See References 2-4 for examples.)
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C
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C
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C All piecewise cubic functions in PCHIP are represented in
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C cubic Hermite form; that is, f(x) is determined by its values
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C F(I) and derivatives D(I) at the breakpoints X(I), I=1(1)N.
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C Throughout the package a PCH function is represented by the
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C five variables N, X, F, D, INCFD:
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C N - number of data points;
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C X - abscissa values for the data points;
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C F - ordinates (function values) for the data points;
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C D - slopes (derivative values) at the data points;
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C INCFD - increment between successive elements in the F- and
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C D-arrays (more on this later).
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C These appear together and in the same order in all calls.
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C
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C The double precision equivalents of the PCHIP routines are
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C obtained from the single precision names by prefixing the
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C single precision names with a D. For example, the double
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C precision equivalent of PCHIM is DPCHIM.
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C
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C The contents of the package are as follows:
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C
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C 1. Determine Derivative Values.
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C
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C NOTE: These routines provide alternate ways of determining D
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C if these values are not already known.
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C
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C PCHIM -- Piecewise Cubic Hermite Interpolation to Monotone
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C data.
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C Used if the data are monotonic or if the user wants
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C to guarantee that the interpolant stays within the
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C limits of the data. (See Reference 3.)
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C
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C PCHIC -- Piecewise Cubic Hermite Interpolation Coefficients.
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C Used if neither of the above conditions holds, or if
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C the user wishes control over boundary derivatives.
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C Will generally reproduce monotonicity on subintervals
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C over which the data are monotonic.
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C
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C PCHSP -- Piecewise Cubic Hermite Spline.
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C Produces a cubic spline interpolator in cubic Hermite
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C form. Provided primarily for easy comparison of the
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C spline with other piecewise cubic interpolants. (A
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C modified version of de Boor's CUBSPL, Reference 1.)
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C
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C 2. Evaluate, Differentiate, or Integrate Resulting PCH Function.
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C
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C NOTE: If derivative values are available from some other
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C source, these routines can be used without calling
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C any of the previous routines.
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C
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C CHFEV -- Cubic Hermite Function EValuator.
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C Evaluates a single cubic Hermite function at an array
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C of points. Used when the interval is known, as in
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C graphing applications. Called by PCHFE.
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C
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C PCHFE -- Piecewise Cubic Hermite Function Evaluator.
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C Used when the interval is unknown or the evaluation
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C array spans more than one data interval.
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C
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C CHFDV -- Cubic Hermite Function and Derivative Evaluator.
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C Evaluates a single cubic Hermite function and its
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C first derivative at an array of points. Used when
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C the interval is known, as in graphing applications.
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C Called by PCHFD.
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C
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C PCHFD -- Piecewise Cubic Hermite Function and Derivative
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C Evaluator.
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C Used when the interval is unknown or the evaluation
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C array spans more than one data interval.
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C
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C PCHID -- Piecewise Cubic Hermite Integrator, Data Limits.
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C Computes the definite integral of a piecewise cubic
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C Hermite function when the integration limits are data
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C points.
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C
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C PCHIA -- Piecewise Cubic Hermite Integrator, Arbitrary Limits.
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C Computes the definite integral of a piecewise cubic
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C Hermite function over an arbitrary finite interval.
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C
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C 3. Utility routines.
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C
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C PCHBS -- Piecewise Cubic Hermite to B-Spline converter.
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C Converts a PCH function to B-representation, so that
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C it can be used with other elements of the B-spline
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C package (see BSPDOC).
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C
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C PCHCM -- Piecewise Cubic Hermite, Check Monotonicity of.
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C Checks the monotonicity of an arbitrary PCH function.
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C Might be used with PCHSP to build a polyalgorithm for
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C piecewise C-2 interpolation.
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C
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C 4. Internal routines.
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C
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C CHFIE -- Cubic Hermite Function Integral Evaluator.
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C (Real function called by PCHIA.)
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C
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C CHFCM -- Cubic Hermite Function, Check Monotonicity of.
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C (Integer function called by PCHCM.)
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C
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C PCHCE -- PCHIC End Derivative Setter.
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C (Called by PCHIC.)
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C
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C PCHCI -- PCHIC Initial Derivative Setter.
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C (Called by PCHIC.)
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C
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C PCHCS -- PCHIC Monotonicity Switch Derivative Setter.
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C (Called by PCHIC.)
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C
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C PCHDF -- PCHIP Finite Difference Formula.
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C (Real function called by PCHCE and PCHSP.)
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C
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C PCHST -- PCHIP Sign Testing Routine.
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C (Real function called by various PCHIP routines.)
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C
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C PCHSW -- PCHCS Switch Excursion Adjuster.
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C (Called by PCHCS.)
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C
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C The calling sequences for these routines are described in the
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C prologues of the respective routines.
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C
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C
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C INCFD, the increment between successive elements in the F-
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C and D-arrays is included in the representation of a PCH function
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C in this package to facilitate two-dimensional applications. For
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C "normal" usage INCFD=1, and F and D are one-dimensional arrays.
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C one would call PCHxx (where "xx" is "IM", "IC", or "SP") with
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C
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C N, X, F, D, 1 .
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C
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C Suppose, however, that one has data on a rectangular mesh,
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C
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C F2D(I,J) = value at (X(I), Y(J)), I=1(1)NX,
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C J=1(1)NY.
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C Assume the following dimensions:
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C
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C REAL X(NXMAX), Y(NYMAX)
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C REAL F2D(NXMAX,NYMAX), FX(NXMAX,NYMAX), FY(NXMAX,NYMAX)
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C
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C where 2.LE.NX.LE.NXMAX AND 2.LE.NY.LE.NYMAX . To interpolate
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C in X along the line Y = Y(J), call PCHxx with
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C
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C NX, X, F2D(1,J), FX(1,J), 1 .
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C
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C To interpolate along the line X = X(I), call PCHxx with
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C
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C NY, Y, F2D(I,1), FY(I,1), NXMAX .
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C
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C (This example assumes the usual columnwise storage of 2-D arrays
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C in Fortran.)
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C
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C***REFERENCES 1. Carl de Boor, A Practical Guide to Splines, Springer-
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C Verlag, New York, 1978 (esp. Chapter IV, pp.49-62).
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C 2. F. N. Fritsch, Piecewise Cubic Hermite Interpolation
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C Package, Report UCRL-87285, Lawrence Livermore Natio-
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C nal Laboratory, July 1982. [Poster presented at the
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C SIAM 30th Anniversary Meeting, 19-23 July 1982.]
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C 3. F. N. Fritsch and J. Butland, A method for construc-
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C ting local monotone piecewise cubic interpolants, SIAM
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C Journal on Scientific and Statistical Computing 5, 2
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C (June 1984), pp. 300-304.
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C 4. F. N. Fritsch and R. E. Carlson, Monotone piecewise
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C cubic interpolation, SIAM Journal on Numerical Ana-
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C lysis 17, 2 (April 1980), pp. 238-246.
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C***ROUTINES CALLED (NONE)
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C***REVISION HISTORY (YYMMDD)
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C 811106 DATE WRITTEN
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C 870930 Updated Reference 3.
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C 890414 Changed PCHMC and CHFMC to PCHCM and CHFCM, respectively,
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C and augmented description of PCHCM.
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C 891214 Prologue converted to Version 4.0 format. (BAB)
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C 910826 1. Revised purpose, clarified role of argument INCFD,
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C corrected error in example, and removed redundant
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C reference list.
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C 2. Added description of PCHBS. (FNF)
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C 920429 Revised format and order of references. (WRB,FNF)
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C 930505 Changed CHFIV to CHFIE. (FNF)
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C***END PROLOGUE PCHDOC
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C-----------------------------------------------------------------------
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C THIS IS A DUMMY SUBROUTINE, AND SHOULD NEVER BE CALLED.
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C
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C***FIRST EXECUTABLE STATEMENT PCHDOC
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RETURN
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C------------- LAST LINE OF PCHDOC FOLLOWS -----------------------------
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END
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