Chebyshev展开是将有限区间上的光滑函数以Chebyshev多项式为基做展开。和Taylor展开不同的是，它对展开函数的光滑性要求较低，只需连续即可。著名的Chebfun系统基础之一就是Chebyshev展开。下面是Mathematica上的一个简单的Chebyshev展开，展开系数使用Gauss-Chebyshev积分计算，积分的代数精度是2*M+1，这里M是展开的阶数。
(************************************************************************)
(* Chebyshev approximation *)
(************************************************************************)
ChebyshevApproximation[f_, x_, a_, b_, M_] := Module[{tmp, i, j, z, F, A},
(*M:{1,x,x^2,...,x^M}*)
(* Comment: A[j] is accurate for polynomials of degree less than 2M+1 *)
If[FreeQ[f,x],Return[f]];
tmp[0] = f /. x -> a + (b - a)*(z + 1)/2;
If[Length[nodes] != M + 1,
(* global nodes and T to avoid reduplicate computation*)
nodes = Table[Cos[(2*i + 1)*Pi/(2*(M + 1))] // GetDigit, {i, 0, M}];
T = Table[ChebyshevT[i, nodes[[j]]], {i, 0, M}, {j, 1, M + 1}]
];
F = tmp[0] /. z -> nodes // Expand;
For[j = 0, j <= M, j++,
A[j] = 2/(M + 1)*F.T[[j + 1]] // GetDigit;
];
tmp[1] = A[0]/2 + Sum[A[j]*ChebyshevT[j, z], {j, 1, M}];
tmp[2] = tmp[1] /. z -> -1 + 2*(x - a)/(b - a) (*// Expand//GetDigit*)
];这里GetDigit是自定义的一个函数，其作用是保持数字的有效位，默认是100位。
(************************************************************************)
(* Define GetDigit[] *)
(************************************************************************)
(* 2012.01.05, 12:21, happy!! *)
Naccu=100;
GetDigit[p_Plus] := Map[GetDigit, p];
GetDigit[p_List] := Map[GetDigit, p];
GetDigit[p_Complex]:=GetDigit[Re[p]]+I*GetDigit[Im[p]];
GetDigit[c_Real] := 0 /;Abs[c] < 10^(-Naccu+1);
GetDigit[c_*f_] := GetDigit[c]*f /; NumericQ[c];
GetDigit[f_] := f /; !NumericQ[f];
GetDigit[c_] := SetPrecision[Apply[Rationalize[#1]*10^#2&,MantissaExponent[c]],Naccu]/;NumericQ[c]
Maple中的Chebyshev 级数展开，展开长度M由eps自适应决定，帮助文件说“The series computed is the ‘infinite’ Chebyshev series, truncated by dropping all terms with coefficients smaller than eps muliplied the largest coefficient”，函数chebyshev的源代码如下：
> with(numapprox);
>interface(verboseproc = 2);
>print(chebyshev);
proc(f::{algebraic,procedure},eqn::{name,name=algebraic..algebraic,algebraic..algebraic},eps::numeric,size::integer)
option `Copyright (c) 1992 by the University of Waterloo. All rights reserved.`;
local f_is_operator,x,r,epsilon,oldDigits,a,b,evalhfOK,fproc,err,nofun,c,intf,tol,maxcoef,k,K,s,y,flags,dim,g;
if nargs<2 or 40 or max(epsilon*abs(intf),0.001*epsilon)+Float(1,-Digits)evalf(v/d),[seq(c[k],k=1..nofun)],nofun - 1);
maxcoef:=max(1,op(c));
tol:=1/5*epsilon*maxcoef;
for K from nofun by -1 to 1 while abs(c[K])0. or b - 1.0<>0. then
userinfo(2,'`numapprox:-chebyshev`',nprintf("even coefficents are zero -- transform to interval -1..1"));
a:=op(1,r);
b:=op(2,r);
g:=eval(f,x=1/2*(b - a)*y+1/2*a+1/2*b);
s:=procname(g,y=-1..1,epsilon,dim);
s:=subs(y=2*x/(b - a) - (a+b)/(b - a),s)
else
userinfo(2,'`numapprox:-chebyshev`',nprintf("function is odd -- try f(x)/x"));
Digits:=Digits+2;
s:=procname(f/x,x=r,1/100*epsilon);
s:=numapprox:-chebmult('T'(1,x),s);
Digits:=Digits - 2;
if type(s,`+`) then s:=map(proc(t,tol) if abs(lcoeff(t))