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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 a grid

Major Topics

  Curve and Surface Fitting

Calling Sequence

                RMS=, DRMS=, RESIDUALS=, DRESIDUALS= , $
                XMATRIX=, DXMATRIX=, RESET=,
                DERIV_WEIGHT= ] )


  CHEBGRID estimates the coefficients for a finite sum of Chebyshev
  polynomials approximating a continuous tabulated function over an
  interval. The function (and optionally its derivative) must be
  tabulated on a regularly sampled grid. The implementation of this
  function is taken from a method described by X. X. Newhall, used
  in estimating coefficients for ephemerides in the solar system.
  The tabulated function is assumed to be continuous over the entire
  interval. A Chebyshev series is fitted to the function over small
  segments, called granules. The size of each granule, the number
  of points in each granule, and the number of Chebyshev polynomials
  are all configurable.
  Users may specify either the function alone, or the function and
  its first derivative. By also giving the tabulated derivative, a
  more accurate Chebyshev polynomial can be developed. Aside from
  the constraints mentioned in the next paragraph, the polynomial
  that is returned is the best-fit polynomial in a least-squares
  Here is a definition of terms:
      GRANULE - a single continuous fitted segment. The length of the
        granule, NGRANULE, is specified in units of the tabulated
        grid size. Because of the continuity requirements developed
        below, granules will always overlap at their endpoints.
        Thus, then length of a granule should be a factor of
        N_ELEMENTS(X)-1. For simple functions over short intervals,
        the granule size can be equal to N_ELEMENTS(X)-1
      NUMBER OF POINTS the number of points, NPOINTS, within a
        granule to be fitted to the polynomial, not necessarily
        equal to the granule size. The greater the number of
        points, the more computation time and storage is required.
        This number *must* be a factor of NGRANULE. Typically
        NPOINTS is a number between 8 and 12. Because of the
        single-point overlap between granules (see below), the
        actual number of points per fit is NPOINTS+1.
      NUMBER OF POLYNOMIALS the number of Chebyshev polynomial terms,
        NPOLYNOMIAL, to be fitted per granule. The greater the
        number of polynomial terms, the more computation time and
        storage is required, but also the greater the approximating
        precision of the fit.
  The particular set of Chebyshev polynomial coefficients developed
  by this function have some special properties. If both the
  function and its derivative are specified, then the value and
  derivative of the interpolating polynomial at the granule
  endpoints will be exactly equal to the tabulated endpoint values.
  This feature allows many approximations to be strung together
  piecewise, and the function value and first derivative will be
  continuous across granule boundaries.
  If only the function value is specified, then only the function
  value will be continuous at the granule endpoints, and not the
  An extensive set of statistics are computed to assess the quality
  of the Chebyshev polynomial fit. The keywords RESIDUALS and
  DRESIDUALS return the residuals of the fit after subtracting the
  interpolation. The RMS and DRMS keywords return the root mean
  squared deviations between data and model.
  If the user does not know how many granules, points, or polynomial
  coefficients to use, then he or she should try several
  combinations and see which minimizes the r.m.s. value with the
  fewest number of coefficients.
  If the XMATRIX and DXMATRIX keywords are passed, then CHEBGRID
  attempts to avoid recomputing several of the matrices it uses in
  estimating the coefficients. If multiple calls to CHEBGRID are to
  be made, some compution time savings can be made. In the first
  call CHEBGRID the required matrices are computed and returned. In
  subsequent calls, CHEBGRID detects the XMATRIX and DXMATRIX
  keyword values and uses those values if it can.
  The user can also estimate their own coefficients. The matrices
  returned are (NPOINTS+1)x(NPOLYNOMIAL). The coefficients from a
  NPOINTS+1 tabulation, X, are found by:
  if derivative information is known, or
      PCHEB = XMATRIX ## X
  if no derivative information is known. [ Note: the matrices are
  different, depending on whether derivative information is known or
  not. ]


  T - array of regularly sampled *independent* variables. The number
      of elements in T should be a multiple of NGRANULE, plus one.
  X - array of regularly sampled *dependent* variables. The number
      of elements in X should be equal to the number of elements in
  DXDT - optionally, a tabulated array of first derivatives of X
          with respect to T, at the same grid points.

Keyword Parameters

  NGRANULE - size of a "granule", in grid intervals. NGRANULE must
              be at least 2, and a factor of N_ELEMENTS(T)-1.
              Default: 8
  NPOINTS - number of points per granule that are fitted. NPOINTS
            must be at least 2, and a factor of NGRANULE.
            Default: NGRANULE
  NPOLYNOMIAL - number of Chebyshev polynomial terms per fit.
                NPOLYNOMIAL must be at least 2 and less than
                2*(NPOINTS+1), when derivative information is
                specified; or less than NPOINTS+1, when no
                derivative information is specified.
                Default: 7
  RESIDUALS - upon return, an array of size N_ELEMENTS(T), with
              residuals of the tabulated function minus the
              interpolated function.
  DRESIDUALS - same as RESIDUALS, but for the function's first
  RMS - upon return, the root mean square of the function value
  DRMS - same as RMS, but for the function's first derivative.
  XMATRIX - upon return, the matrix used to compute Chebyshev
            polynomial coefficients from the function value.
            Upon input, CHEBGRID determines if XMATRIX will apply to
            the data, and if so, XMATRIX is reused rather than
            computed. If XMATRIX cannot be reused, then it is
            computed afresh, and the new value is returned in the
            XMATRIX keyword.
            The user should not modify the contents of this array.
  DXMATRIX - same as XMATRIX, but for the function's first
  RESET - if set, force a recomputation of XMATRIX and/or DXMATRIX.
  DERIV_WEIGHT - amount of weight to give to function derivative,
                relative to the function value.
                Default: 0.16d


  An array of coefficient values. The dimensions of the array are
  NPOLYNOMIALxNSEGS, where NSEGS is the number of granules in the
  entire interval.


  ;; Estimate Chebyshev coefficients for the function SIN(X), on the
  ;; interval [-1,+1].
  xx = dindgen(9)/4d - 1d ;; Regular grid from -1 to 1 (9 points)
  yy = sin(xx) ;; Function values, sin(x), ...
  dy = cos(xx) ;; ... and derivatives
  ;; Estimate coefficients using CHEBGRID (single granule of 8 intervals)
  p = chebgrid(xx, yy, dy, npoints=8, ngranule=8, npoly=10)
  xxx = dindgen(1001)/500 - 1d ;; New grid for testing
  res = sin(xxx) - chebeval(xxx, p)
  plot, xxx, res
  ;; Same as example above, except extended range to [-1, +15],
  using eight granules.
  xx2 = dindgen(65)/4d - 1
  yy2 = sin(xx2)
  dy2 = cos(xx2)
  p = chebgrid(xx2, yy2, dy2, ngranule=8, npoint=8, npoly=10)
  help, p
      P DOUBLE = Array[10, 8]
  ;; (i.e., 10 polynomial coefficients over 8 granules)


  Abramowitz, M. & Stegun, I., 1965, *Handbook of Mathematical
    Functions*, 1965, U.S. Government Printing Office, Washington,
    D.C. (Applied Mathematical Series 55)
  Newhall, X. X. 1989, Celestial Mechanics, 45, p. 305-310

Modification History

  Written, CM, Feb 2002
  Documented, CM, 24 Mar 2002
  Corrected documentation, CM, 28 Apr 2002
  Typo correction, CM, 10 Oct 2002

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