Chebyshev Curve Fitting

The operator family implements approximation and evaluation of functions by the Chebyshev method. Let be the Chebyshev polynomial of order transformed to the interval . Then a function can be approximated in by a series

The operator computes this approximation and returns a list, which has as first element the sum expressed as a polynomial and as second element the sequence of Chebyshev coefficients . and transform a Chebyshev coefficient list into the coefficients of the corresponding derivative or integral respectively. For evaluating a Chebyshev approximation at a given point in the basic interval the operator can be used. is based on a recurrence relation which is in general more stable than a direct evaluation of the complete polynomial.

CHEBYSHEV_FIT

CHEBYSHEV_EVAL

CHEBYSHEV_DF

CHEBYSHEV_INT

where is an algebraic expression (the target function), is the variable of , and are numerical real values which describe an interval (), the integer is the approximation order (set to 20 if missing), is a number in the interval and is a series of Chebyshev coefficients, computed by one of , or .

Example:

on rounded;

w:=chebyshev_fit(sin x/x,x=(1 .. 3),5);

               3           2                                              
w := {0.03824*x  - 0.2398*x  + 0.06514*x + 0.9778,                        
                                                                     
      {0.8991,-0.4066,-0.005198,0.009464,-0.00009511}}                    

chebyshev_eval(second w, x=(1 .. 3), x=2.1);   

0.4111