chebyshev - Maple Help

numapprox

 chebyshev
 Chebyshev series expansion

 Calling Sequence chebyshev(f, x=a..b, eps) chebyshev(f, x, eps) chebyshev(f, a..b, eps)

Parameters

 f - procedure or expression representing the function x - variable name appearing in f, if f is an expression a, b - numerical values specifying the interval of approximation eps - (optional) numeric value

Description

 • This function computes the Chebyshev series expansion of f, with respect to the variable x on the interval $a..b$, valid to accuracy eps.
 • If the second argument is simply a name x then the equation $x=-1..1$ is implied.
 • If the second argument is a range then the first argument is assumed to be a Maple operator and the result will be returned as an operator. Otherwise, the first argument is assumed to be an expression and the result will be returned as an expression.
 • If the third argument eps is present then it specifies the desired accuracy;  otherwise, the value used is $\mathrm{eps}={10}^{-\mathrm{Digits}}$. It is an error to specify eps less than 10^(-Digits).
 • The expression or operator f must evaluate to a numerical value when x takes on a numerical value.  Moreover, it must represent a function which is analytic in a region surrounding the interval $a..b$.
 • The resulting series is expressed in terms of the Chebyshev polynomials $T\left(k,x\right),...$ with floating-point series coefficients. If 'ser' is the Chebyshev series then conversion to ordinary polynomial form can be accomplished via eval(ser, T=orthopoly[T]).
 • The series computed is the infinite'' Chebyshev series, truncated by dropping all terms with coefficients smaller than eps multiplied by the largest coefficient.
 • Note:  The name T used in representing the Chebyshev polynomials is a global name, so the user must ensure that this name has no previous value.
 • The command with(numapprox,chebyshev) allows the use of the abbreviated form of this command.

Examples

 > $\mathrm{with}\left(\mathrm{numapprox}\right):$
 > $\mathrm{Digits}≔5:$
 > $\mathrm{chebyshev}\left(\mathrm{cos}\left(x\right),x\right)$
 ${0.76520}{}{T}{}\left({0}{,}{x}\right){-}{0.22981}{}{T}{}\left({2}{,}{x}\right){+}{0.0049533}{}{T}{}\left({4}{,}{x}\right){-}{0.000041877}{}{T}{}\left({6}{,}{x}\right)$ (1)
 > $\mathrm{chebyshev}\left(\mathrm{exp}\left(x\right),x=0..1,0.001\right)$
 ${1.7534}{}{T}{}\left({0}{,}{2}{}{x}{-}{1}\right){+}{0.85039}{}{T}{}\left({1}{,}{2}{}{x}{-}{1}\right){+}{0.10521}{}{T}{}\left({2}{,}{2}{}{x}{-}{1}\right){+}{0.0087221}{}{T}{}\left({3}{,}{2}{}{x}{-}{1}\right)$ (2)
 > $\mathrm{chebyshev}\left(\mathrm{exp},0..1,0.001\right)$
 ${x}{↦}{1.7534}{\cdot }{T}{}\left({0}{,}{2}{\cdot }{x}{-}{1}\right){+}{0.85039}{\cdot }{T}{}\left({1}{,}{2}{\cdot }{x}{-}{1}\right){+}{0.10521}{\cdot }{T}{}\left({2}{,}{2}{\cdot }{x}{-}{1}\right){+}{0.0087222}{\cdot }{T}{}\left({3}{,}{2}{\cdot }{x}{-}{1}\right)$ (3)
 > $\mathrm{chebyshev}\left(\mathrm{sin}+\mathrm{cos},-1..1\right)$
 ${x}{↦}{0.76520}{\cdot }{T}{}\left({0}{,}{x}\right){+}{0.88010}{\cdot }{T}{}\left({1}{,}{x}\right){-}{0.22981}{\cdot }{T}{}\left({2}{,}{x}\right){-}{0.039127}{\cdot }{T}{}\left({3}{,}{x}\right){+}{0.0049533}{\cdot }{T}{}\left({4}{,}{x}\right){+}{0.00049952}{\cdot }{T}{}\left({5}{,}{x}\right){-}{0.000041877}{\cdot }{T}{}\left({6}{,}{x}\right)$ (4)