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CHEBFIT

CHEBFIT

Name


  CHEBFIT

Author


  Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
  craigm@lheamail.gsfc.nasa.gov
  UPDATED VERSIONs can be found on my WEB PAGE:
      http://cow.physics.wisc.edu/~craigm/idl/idl.html

Purpose


  Fit Chebyshev polynomial coefficients to a tabulated function

Major Topics


  Curve and Surface Fitting

Calling Sequence


  p = CHEBFIT(X, Y, ERR, INTERVAL=interval, NMAX=nmax,
              PRECISION=prec, /EVEN, /ODD, REDUCE_ALGORITHM=)

Description



  CHEBFIT fits a series of Chebyshev polynomials to a set of
  tabulated and possibly noisy data points. The functions MPFIT and
  CHEBEVAL, available from the above web page, must also be in your
  IDL path for this function to work properly. The user can choose
  the desired precision and maximum number of chebyshev
  coefficients.
  This function is intended for use on already-tabulated data which
  are potentially noisy. The user should never expect more than
  NPOINTS terms, where NPOINTS is the number of (x,y) pairs. For
  functions which can be evaluated to full machine precision at
  arbitrary abcissae, the routine CHEBCOEF should be used instead.
  For exact data tabulated on a regular grid, the routine CHEBGRID
  should be tried.
  The user can also specify that the function is even or odd, using
  the keywords EVEN or ODD. This saves computation time because
  certain terms in the expansion can be ignored. For the purposes
  of this function even and odd refer to the symmetry about the
  center of the interval.
  The algorithm is employed in three steps. In the first step, the
  coefficients are estimated at a crude level. In the second step,
  it is determined whether certain coefficients are deemed
  "ignoreable", i.e., they do not contribute significantly to the
  function and are discarded. The operation of this step is
  determined by the REDUCE_ALGORITHM keyword. Finally, the
  remaining "good" coefficients are re-fitted to achieve the best
  fit.

Inputs



  X, Y - the x- and y- tabulated values to be fitted.
  ERR - (optional) the y-error bar associated with each (x,y) pair.
        Default: 1

Returns



  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



  EVEN, ODD - if set, then the fitting routine assumes the function
              is even or odd, about the center of the interval.
  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 polynomial terms to be
          fitted at one time.
          Default: 16
  PRECISION - a scalar, the requested precision in the fit. Any
              terms which do not contribute significantly, as
              defined by this threshold, are discarded. If the
              function to be fitted 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

Example



  x = dindgen(1000)/100 ; Range of 0 to 10
  y = cos(x) + randomn(seed,1000)*0.01 ; Function with some noise
  p = chebfit(x, y, interval=[0d,10d])
  plot, x, y - chebeval(x,p, interval=[0d,10d])

References



  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 E407
  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
  Slight docs change, CM, 25 Mar 2002
  $Id: chebfit.pro,v 1.7 2003/07/20 05:53:44 craigm Exp $



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