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IMSL Name:  RATCH/DRATCH (Single/Double precision version)
 
Revised:    May 31, 1991
 
Purpose:    Compute a rational weighted Chebyshev approximation to
            a continuous function on an interval.
 
Usage:      CALL RATCH (F, PHI, WEIGHT, A, B, N, M, P, Q, ERROR)
 
Arguments:
   F      - User-supplied FUNCTION to be approximated.  The form is
            F(X), where
            X      - Independent variable.  (Input)
            F      - The function value.  (Output)
            F must be declared EXTERNAL in the calling program.
   PHI    - User-supplied FUNCTION to supply the variable
            transformation which must be continuous and monotonic.
            The form is
            PHI(X), where
            X      - Independent variable.  (Input)
            PHI    - The function value.  (Output)
            PHI must be declared EXTERNAL in the calling program.
   WEIGHT - User-supplied FUNCTION to scale the maximum error.  It
            must be continuous and nonvanishing on the closed
            interval (A,B).  The form is
            WEIGHT(X), where
            X      - Independent variable.  (Input)
            WEIGHT - The function value.  (Output)
            WEIGHT must be declared EXTERNAL in the calling
            program.
   A      - Lower end of the interval on which the approximation is
            desired.  (Input)
   B      - Upper end of the interval on which the approximation is
            desired.  (Input)
   N      - The degree of the numerator.  (Input)
   M      - The degree of the denominator.  (Input)
   P      - Vector of length N+1 containing the coefficients of
            the numerator polynomial.  (Output)
   Q      - Vector of length M+1 containing the coefficients of
            the denominator polynomial.  (Output)
   ERROR  - Min-max error of approximation.  (Output)
 
Remarks:
 
1. Automatic workspace usage is
            RATCH    (N+M+8)*(N+M+2)+N+M+2 units, or
            DRATCH   2*(N+M+8)*(N+M+2)+N+M+2 units.
   Workspace may be explicitly provided, if desired, by use of
   R2TCH/DR2TCH.  The reference is
            CALL R2TCH (F, PHI, WEIGHT, A, B, N, M, P, Q, ERROR,
                        ITMAX, IWK, WK)
   The additional arguments are as follows:
   ITMAX  - Maximum number of iterations.  (Input)
            The default value is 20.
   IWK    - Workspace vector of length (N+M+2).  (Workspace)
   WK     - Workspace vector of length (N+M+8)*(N+M+2).
            (Workspace)
 
2. Informational errors
   Type Code
     3   1  The maximum number of iterations has been reached.  The
            routine R2TCH may be called directly to set a larger
            value for ITMAX.
     3   2  The error was reduced as far as numerically possible.
            A good approximation is returned in P and Q, but this
            does not necessarily give the Chebyshev approximation.
     4   3  The linear system that defines P and Q was found to be
            algorithmically singular.  This indicates the
            possibility of a degenerate approximation.
     4   4  A sequence of critical points that was not monotonic
            generated.  This indicates the possibility of a
            degenerate approximation.
     4   5  The value of the error curve at some critical point is
            too large.  This indicates the possibility of poles in
            the rational function.
     4   6  The weight function cannot be zero on the closed
            interval (A,B).
 
Keywords:   Rational L approximation; L-infinity; Minimax
 
GAMS:       K2
 
Chapter:    MATH/LIBRARY Interpolation and Approximation
 
Page No.:   MATH/LIBRARY User's Manual page 676