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  Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
  UPDATED VERSIONs can be found on my WEB PAGE:


  Estimate Chebyshev polynomial coefficients of a function on an interval

Major Topics

  Curve and Surface Fitting

Calling Sequence

                PRECISION=prec, ERROR=err, NMAX=nmax, INTERVAL=interval, $
                REDUCE_ALGORITHM=, STATUS=)


  CHEBCOEF estimates the coefficients for a finite sum of Chebyshev
  polynomials approximating the function FUNC(x) over an interval.
  The user can choose the desired precision and maximum number of
  chebyshev coefficients.
  This routine is intended for functions which can be evaluated to
  full machine precision at arbitrary abcissae, and which are smooth
  enough to ensure that the coefficients are a decreasing sequence.
  For already-tabulated or potentially noisy data, the routines
  CHEBGRID or CHEBFIT should be used instead.
  The function to be approximated may either be the name of an IDL
  function (the default behavior), or an IDL expression (using the
  /EXPRESSION keyword).
  The procedure uses a modified form of the classic algorithm for
  determining the coefficients, which relies the orthogonality
  relation for Chebyshev polynomials. The interval [a,b] is
  subdivided successively into sets of subintervals of length
  2^(-k)*(b-a),(k = 0,1,2...). After each subdivision the
  orthogonality properties of the Chebyshev polynomials with respect
  to summation over equally-spaced points are used to compute two
  sets of approximate values of the coefficients cj, one set
  computed using the end-points of the subintervals, and one set
  using the mid-points. Certain convergence requirements must be
  met before terminating. If the routine fails to converge with 64
  coefficents, then the current best-fitting coefficients are
  returned, along with an error estimate in the ERROR keyword.
  CHEBCOEF never returns more than 64 coefficients.
  The coefficients may be further refined. If the keyword
  REDUCE_ALGORITHM is set to a value of 1, then any high order
  coefficients below a certain threshold are discarded. If
  REDUCE_ALGORITHM is set to 2 (the default), then all coefficients
  below the threshold are discarded rather than just the high order
  ones. The threshold is determined by the PRECISION keyword.


  FUNC - a scalar string, the name of the function to be
          approximated, or an IDL string containing an expression to
          be approximated (if /EXPRESSION is set).
  PRIVATE - any optional variable to be passed on to the function to
            be integrated. For functions, PRIVATE is passed as the
            second positional parameter; for expressions, PRIVATE can
            be referenced by the variable 'P'. CHEBCOEF does not
            examine or alter PRIVATE.


  An array of Chebyshev coefficients which can be passed to
  CHEBEVAL. NOTE: the convention employed here is such that the
  constant term in the expansion is P(0)*T0(x) (i.e., the convention
  of Luke), and not P(0)/2 * T0(x).

Keyword Parameters

  DOUBLE - if set, then computations are done in double precision
            rather than single precision.
  ERROR - upon return, this keyword contains an estimate of the
          maximum absolute error in the approximation.
  EXPRESSION - if set, then FUNC is an IDL expression to be
                approximated, rather than the name of a function.
  FUNCTARGS - A structure which contains the parameters to be passed
              to the user-supplied function specified by FUNCT via
              the _EXTRA mechanism. This is the way you can pass
              additional data to your user-supplied function without
              using common blocks. By default, no extra parameters
              are passed to the user-supplied function.
  INTERVAL - a 2-element vector describing the interval over which
              the polynomial is to be evaluated.
              Default: [-1, 1]
  NMAX - a scalar, the maximum number of coefficients to be
          estimated. This number may not exceed 64.
          Default: 64
  PRECISION - a scalar, the requested precision in the
              approximation. Any terms which do not contribute
              significantly, as defined by this threshold, are
              discarded. If the function to be estimated is not
              well-behaved, then the precision is not guaranteed to
              reach the desired level. Default: 1E-7
  REDUCE_ALGORITHM - a scalar integer, describes how insignificant
              terms are removed from the fit. If 0, then all terms
              are kept, and none are dicarded. If 1, then only
              trailing terms less than PRECISION are discarded. If
              2, then both trailing and intermediate terms less than
              PRECISION are discarded.
              Default: 2
  STATUS - upon return, this keyword contains information about the
            status of the approximation. A value of -1 indicates bad
            input values; a value of 0 indicates the required
            accuracy was not obtained; a value of 1 indicates


  x = dindgen(1000)/100 ; Range of 0 to 10
  p = chebcoef('COS(x)', /expr, interval=[0d, 10d]) ;; Compute coefs
  y = chebeval(x, p, interval=[0d,10d]) ;; Eval Cheby poly
  plot, x, y - cos(x) ; Plot residuals


  Abramowitz, M. & Stegun, I., 1965, *Handbook of Mathematical
    Functions*, 1965, U.S. Government Printing Office, Washington,
    D.C. (Applied Mathematical Series 55)
  CERN, 1995, CERN Program Library, Function E406
  Luke, Y. L., *The Special Functions and Their Approximations*,
    1969, Academic Press, New York

Modification History

  Written and documented, CM, June 2001
  Copyright license terms changed, CM, 30 Dec 2001
  Added usage message, CM, 20 Mar 2002
  Changed docs slightly, CM, 25 Mar 2002
  $Id: chebcoef.pro,v 1.6 2002/05/03 18:40:27 craigm Exp $

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