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The operator family \(Chebyshev\_\ldots \) implements approximation and evaluation of functions by the Chebyshev method. Let \(T_n^{(a,b)}(x)\) be the Chebyshev polynomial of order \(n\) transformed to the interval \((a,b)\). Then a function \(f(x)\) can be approximated in \((a,b)\) by a series
chebyshev_fit 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 \(c_{i}\). chebyshev_df and chebyshev_int
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 chebyshev_eval can be used. Note that
Chebyshev_eval is based on a recurrence relation which is in general more stable
than a direct evaluation of the complete polynomial.
\((fcn,var=(lo .. hi),n)\)
\((coeffs,var=(lo .. hi),var=pt)\)
\((coeffs,var=(lo .. hi))\)
\((coeffs,var=(lo .. hi))\)
where 〈fcn⟩ is an algebraic expression (the function to be fitted), 〈var⟩
is the variable of 〈fcn⟩, 〈lo⟩ and 〈hi⟩ are numerical real values which
describe an interval (\(lo < hi\)), 〈n⟩ is the approximation order, a positive integer,
set to 20 if missing, 〈pt⟩ is a numerical value in the interval and 〈coeffs⟩
is a series of Chebyshev coefficients, computed by one of the operators
chebyshev_coeff, chebyshev_df, or chebyshev_int.
Example:
on rounded;
w:=chebyshev_fit(sin x/x,x=(1 .. 3),5);
w := {0.0382345446975*x - 0.239802588672*x
+ 0.0651206939005*x + 0.977836217464,
{0.899091895826,-0.406599215895,
-0.00519766024352,0.00946374143079,
-0.0000948947435876}}
chebyshev_eval(second w, x=(1 .. 3), x=2.1);
0.411091086819
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