Rosetta Code

Chebyshev coefficients

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Chebyshev coefficients are the basis of polynomial approximations of functions.

Chebyshev coefficients is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.


Task

Write a program to generate Chebyshev coefficients.

Calculate coefficients:   cosine function,   10   coefficients,   interval   0   1

11l

Translation of: Python

<lang 11l>F test_func(Float x) 

  R cos(x)

F mapper(x, min_x, max_x, min_to, max_to) 

  R (x - min_x) / (max_x - min_x) * (max_to - min_to) + min_to

F cheb_coef(func, n, min, max) 

  V coef = [0.0] * n
  L(i) 0 .< n
     V f = func(mapper(cos(math:pi * (i + 0.5) / n), -1, 1, min, max)) * 2 / n
     L(j) 0 .< n
        coef[j] += f * cos(math:pi * j * (i + 0.5) / n)
  R coef

F cheb_approx(=x, n, min, max, coef) 

  V a = 1.0
  V b = mapper(x, min, max, -1, 1)
  V res = coef[0] / 2 + coef[1] * b

  x = 2 * b
  V i = 2
  L i < n
     V c = x * b - a
     res = res + coef[i] * c
     (a, b) = (b, c)
     i++

  R res

V n = 10 V minv = 0 V maxv = 1 V c = cheb_coef(test_func, n, minv, maxv)

print(‘Coefficients:’) L(i) 0 .< n 

  print(c[i])

print("\n\nApproximation:\n x func(x) approx diff") L(i) 20 

  V x = mapper(i, 0.0, 20.0, minv, maxv)
  V f = test_func(x)
  V approx = cheb_approx(x, n, minv, maxv, c)
  print(‘#.3 #.10 #.10 #.’.format(x, f, approx, format_float_exp(approx - f, 2, 9)))</lang>
Output:
Coefficients:
1.64717
-0.232299
-0.0537151
0.00245824
0.000282119
-7.72223e-06
-5.89856e-07
1.15214e-08
6.5963e-10
-1.00219e-11


Approximation:
    x      func(x)       approx      diff
0.000 1.0000000000 1.0000000000  4.68e-13
0.050 0.9987502604 0.9987502604 -9.36e-14
0.100 0.9950041653 0.9950041653  4.62e-13
0.150 0.9887710779 0.9887710779 -4.73e-14
0.200 0.9800665778 0.9800665778 -4.60e-13
0.250 0.9689124217 0.9689124217 -2.32e-13
0.300 0.9553364891 0.9553364891  2.62e-13
0.350 0.9393727128 0.9393727128  4.61e-13
0.400 0.9210609940 0.9210609940  1.98e-13
0.450 0.9004471024 0.9004471024 -2.47e-13
0.500 0.8775825619 0.8775825619 -4.58e-13
0.550 0.8525245221 0.8525245221 -2.46e-13
0.600 0.8253356149 0.8253356149  1.96e-13
0.650 0.7960837985 0.7960837985  4.53e-13
0.700 0.7648421873 0.7648421873  2.54e-13
0.750 0.7316888689 0.7316888689 -2.28e-13
0.800 0.6967067093 0.6967067093 -4.47e-13
0.850 0.6599831459 0.6599831459 -4.37e-14
0.900 0.6216099683 0.6216099683  4.46e-13
0.950 0.5816830895 0.5816830895 -8.98e-14


BASIC

BASIC256

Translation of: FreeBASIC

Given the limitations of the language, only 8 coefficients are calculated <lang BASIC256>a = 0: b = 1: n = 8 dim cheby(n) dim coef(n)

for i = 0 to n-1 coef[i] = cos(cos(pi/n*(i+1/2))*(b-a)/2+(b+a)/2) next i

for i = 0 to n-1 w = 0 for j = 0 to n-1 w += coef[j] * cos(pi/n*i*(j+1/2)) next j cheby[i] = w*2/n print i; " : "; cheby[i] next i end</lang>

Output:
0 : 1.64716947539
1 : -0.23229937162
2 : -0.05371511462
3 : 0.00245823527
4 : 0.00028211906
5 : -0.00000772223
6 : -5.89855645106e-07
7 : 1.15214275009e-08

QBasic

Works with: QBasic
Works with: QuickBasic version 4.5
Translation of: FreeBASIC

<lang qbasic>pi = 4 * ATN(1) a = 0: b = 1: n = 10 DIM cheby!(n) DIM coef!(n)

FOR i = 0 TO n - 1 

   coef(i) = COS(COS(pi / n * (i + 1 / 2)) * (b - a) / 2 + (b + a) / 2)

NEXT i

FOR i = 0 TO n - 1 

   w = 0
   FOR j = 0 TO n - 1
       w = w + coef(j) * COS(pi / n * i * (j + 1 / 2))
   NEXT j
   cheby(i) = w * 2 / n
   PRINT USING " # : ##.#####################"; i; cheby(i)

NEXT i END</lang>

Output:
 0 :  1.647169470787048000000
 1 : -0.232299402356147800000
 2 : -0.053715050220489500000
 3 :  0.002458173315972090000
 4 :  0.000282166845863685000
 5 : -0.000007787576578266453
 6 : -0.000000536595905487047
 7 :  0.000000053614126471757
 8 :  0.000000079823998078155
 9 : -0.000000070922546058227

FreeBASIC

<lang freebasic>Const pi As Double = 4 * Atn(1) Dim As Double i, w, j Dim As Double a = 0, b = 1, n = 10 Dim As Double cheby(10), coef(10)

For i = 0 To n-1 

   coef(i) = Cos(Cos(pi/n*(i+1/2))*(b-a)/2+(b+a)/2)

Next i

For i = 0 To n-1 

   w = 0
   For j = 0 To n-1
       w += coef(j) * Cos(pi/n*i*(j+1/2))
   Next j
   cheby(i) = w*2/n
   Print i; " : "; cheby(i)

Next i Sleep</lang>

Output:
 0 :  1.647169475390314
 1 : -0.2322993716151719
 2 : -0.05371511462204768
 3 :  0.002458235266981634
 4 :  0.0002821190574339161
 5 : -7.7222291556156e-006
 6 : -5.898556451056081e-007
 7 :  1.152142750093788e-008
 8 :  6.596299062522348e-010
 9 : -1.002201654998203e-011

Yabasic

Translation of: FreeBASIC

<lang yabasic>a = 0: b = 1: n = 10 dim cheby(n) dim coef(n)

for i = 0 to n-1 

   coef(i) = cos(cos(pi/n*(i+1/2))*(b-a)/2+(b+a)/2)

next i

for i = 0 to n-1 

   w = 0
   for j = 0 to n-1
       w = w + coef(j) * cos(pi/n*i*(j+1/2))
   next j
   cheby(i) = w*2/n
   print i, " : ", cheby(i)

next i end</lang>

Output:
0 : 1.64717
1 : -0.232299
2 : -0.0537151
3 : 0.00245824
4 : 0.000282119
5 : -7.72223e-06
6 : -5.89856e-07
7 : 1.15214e-08
8 : 6.5963e-10
9 : -1.0022e-11


C

C99. <lang C>#include <stdio.h>

  1. include <string.h>
  2. include <math.h>
  1. ifndef M_PI
  2. define M_PI 3.14159265358979323846
  3. endif

double test_func(double x) { //return sin(cos(x)) * exp(-(x - 5)*(x - 5)/10); return cos(x); }

// map x from range [min, max] to [min_to, max_to] double map(double x, double min_x, double max_x, double min_to, double max_to) { return (x - min_x)/(max_x - min_x)*(max_to - min_to) + min_to; }

void cheb_coef(double (*func)(double), int n, double min, double max, double *coef) { memset(coef, 0, sizeof(double) * n); for (int i = 0; i < n; i++) { double f = func(map(cos(M_PI*(i + .5f)/n), -1, 1, min, max))*2/n; for (int j = 0; j < n; j++) coef[j] += f*cos(M_PI*j*(i + .5f)/n); } }

// f(x) = sum_{k=0}^{n - 1} c_k T_k(x) - c_0/2 // Note that n >= 2 is assumed; probably should check for that, however silly it is. double cheb_approx(double x, int n, double min, double max, double *coef) { double a = 1, b = map(x, min, max, -1, 1), c; double res = coef[0]/2 + coef[1]*b;

x = 2*b; for (int i = 2; i < n; a = b, b = c, i++) // T_{n+1} = 2x T_n - T_{n-1} res += coef[i]*(c = x*b - a);

return res; }

int main(void) {

  1. define N 10

double c[N], min = 0, max = 1; cheb_coef(test_func, N, min, max, c);

printf("Coefficients:"); for (int i = 0; i < N; i++) printf(" %lg", c[i]);

puts("\n\nApproximation:\n x func(x) approx diff"); for (int i = 0; i <= 20; i++) { double x = map(i, 0, 20, min, max); double f = test_func(x); double approx = cheb_approx(x, N, min, max, c);

printf("% 10.8lf % 10.8lf % 10.8lf % 4.1le\n", x, f, approx, approx - f); }

return 0; }</lang>

C#

Translation of: C++

<lang csharp>using System; using System.Collections.Generic; using System.Linq; using System.Text; using System.Threading.Tasks;

namespace Chebyshev { 

   class Program {
       struct ChebyshevApprox {
           public readonly List<double> coeffs;
           public readonly Tuple<double, double> domain;

           public ChebyshevApprox(Func<double, double> func, int n, Tuple<double, double> domain) {
               coeffs = ChebCoef(func, n, domain);
               this.domain = domain;
           }

           public double Call(double x) {
               return ChebEval(coeffs, domain, x);
           }
       }

       static double AffineRemap(Tuple<double, double> from, double x, Tuple<double, double> to) {
           return to.Item1 + (x - from.Item1) * (to.Item2 - to.Item1) / (from.Item2 - from.Item1);
       }

       static List<double> ChebCoef(List<double> fVals) {
           int n = fVals.Count;
           double theta = Math.PI / n;
           List<double> retval = new List<double>();
           for (int i = 0; i < n; i++) {
               retval.Add(0.0);
           }
           for (int ii = 0; ii < n; ii++) {
               double f = fVals[ii] * 2.0 / n;
               double phi = (ii + 0.5) * theta;
               double c1 = Math.Cos(phi);
               double s1 = Math.Sin(phi);
               double c = 1.0;
               double s = 0.0;
               for (int j = 0; j < n; j++) {
                   retval[j] += f * c;
                   // update c -> cos(j*phi) for next value of j
                   double cNext = c * c1 - s * s1;
                   s = c * s1 + s * c1;
                   c = cNext;
               }
           }
           return retval;
       }

       static List<double> ChebCoef(Func<double, double> func, int n, Tuple<double, double> domain) {
           double remap(double x) {
               return AffineRemap(new Tuple<double, double>(-1.0, 1.0), x, domain);
           }
           double theta = Math.PI / n;
           List<double> fVals = new List<double>();
           for (int i = 0; i < n; i++) {
               fVals.Add(0.0);
           }
           for (int ii = 0; ii < n; ii++) {
               fVals[ii] = func(remap(Math.Cos((ii + 0.5) * theta)));
           }
           return ChebCoef(fVals);
       }

       static double ChebEval(List<double> coef, double x) {
           double a = 1.0;
           double b = x;
           double c;
           double retval = 0.5 * coef[0] + b * coef[1];
           var it = coef.GetEnumerator();
           it.MoveNext();
           it.MoveNext();
           while (it.MoveNext()) {
               double pc = it.Current;
               c = 2.0 * b * x - a;
               retval += pc * c;
               a = b;
               b = c;
           }
           return retval;
       }

       static double ChebEval(List<double> coef, Tuple<double, double> domain, double x) {
           return ChebEval(coef, AffineRemap(domain, x, new Tuple<double, double>(-1.0, 1.0)));
       }

       static void Main() {
           const int N = 10;
           ChebyshevApprox fApprox = new ChebyshevApprox(Math.Cos, N, new Tuple<double, double>(0.0, 1.0));
           Console.WriteLine("Coefficients: ");
           foreach (var c in fApprox.coeffs) {
               Console.WriteLine("\t{0: 0.00000000000000;-0.00000000000000;zero}", c);
           }

           Console.WriteLine("\nApproximation:\n    x       func(x)        approx      diff");
           const int nX = 20;
           const int min = 0;
           const int max = 1;
           for (int i = 0; i < nX; i++) {
               double x = AffineRemap(new Tuple<double, double>(0, nX), i, new Tuple<double, double>(min, max));
               double f = Math.Cos(x);
               double approx = fApprox.Call(x);
               Console.WriteLine("{0:0.000} {1:0.00000000000000} {2:0.00000000000000} {3:E}", x, f, approx, approx - f);
           }
       }
   }

}</lang>

Output:
Coefficients:
         1.64716947539031
        -0.23229937161517
        -0.05371511462205
         0.00245823526698
         0.00028211905743
        -0.00000772222916
        -0.00000058985565
         0.00000001152143
         0.00000000065963
        -0.00000000001002

Approximation:
    x       func(x)        approx      diff
0.000 1.00000000000000 1.00000000000047 4.689582E-013
0.050 0.99875026039497 0.99875026039487 -9.370282E-014
0.100 0.99500416527803 0.99500416527849 4.622969E-013
0.150 0.98877107793604 0.98877107793600 -4.662937E-014
0.200 0.98006657784124 0.98006657784078 -4.604095E-013
0.250 0.96891242171065 0.96891242171041 -2.322587E-013
0.300 0.95533648912561 0.95533648912587 2.609024E-013
0.350 0.93937271284738 0.93937271284784 4.606315E-013
0.400 0.92106099400289 0.92106099400308 1.980638E-013
0.450 0.90044710235268 0.90044710235243 -2.473577E-013
0.500 0.87758256189037 0.87758256188991 -4.586331E-013
0.550 0.85252452205951 0.85252452205926 -2.461364E-013
0.600 0.82533561490968 0.82533561490988 1.961764E-013
0.650 0.79608379854906 0.79608379854951 4.536371E-013
0.700 0.76484218728449 0.76484218728474 2.553513E-013
0.750 0.73168886887382 0.73168886887359 -2.267075E-013
0.800 0.69670670934717 0.69670670934672 -4.467537E-013
0.850 0.65998314588498 0.65998314588494 -4.485301E-014
0.900 0.62160996827066 0.62160996827111 4.444223E-013
0.950 0.58168308946388 0.58168308946379 -8.992806E-014

C++

Based on the C99 implementation above. The main improvement is that, because C++ containers handle memory for us, we can use a more functional style.

The two overloads of cheb_coef show a useful idiom for working with C++ templates; the non-template code, which does all the mathematical work, can be placed in a source file so that it is compiled only once (reducing code bloat from repeating substantial blocks of code). The template function is a minimal wrapper to call the non-template implementation.

The wrapper class ChebyshevApprox_ supports very terse user code.

<lang CPP>

  1. include <iostream>
  2. include <iomanip>
  3. include <string>
  4. include <cmath>
  5. include <utility>
  6. include <vector>

using namespace std;

static const double PI = acos(-1.0);

double affine_remap(const pair<double, double>& from, double x, const pair<double, double>& to) { return to.first + (x - from.first) * (to.second - to.first) / (from.second - from.first); }

vector<double> cheb_coef(const vector<double>& f_vals) { const int n = f_vals.size(); const double theta = PI / n; vector<double> retval(n, 0.0); for (int ii = 0; ii < n; ++ii) { double f = f_vals[ii] * 2.0 / n; const double phi = (ii + 0.5) * theta; double c1 = cos(phi), s1 = sin(phi); double c = 1.0, s = 0.0; for (int j = 0; j < n; j++) { retval[j] += f * c; // update c -> cos(j*phi) for next value of j const double cNext = c * c1 - s * s1; s = c * s1 + s * c1; c = cNext; } } return retval; }

template<class F_> vector<double> cheb_coef(const F_& func, int n, const pair<double, double>& domain) { auto remap = [&](double x){return affine_remap({ -1.0, 1.0 }, x, domain); }; const double theta = PI / n; vector<double> fVals(n); for (int ii = 0; ii < n; ++ii) fVals[ii] = func(remap(cos((ii + 0.5) * theta))); return cheb_coef(fVals); }

double cheb_eval(const vector<double>& coef, double x) { double a = 1.0, b = x, c; double retval = 0.5 * coef[0] + b * coef[1]; for (auto pc = coef.begin() + 2; pc != coef.end(); a = b, b = c, ++pc) { c = 2.0 * b * x - a; retval += (*pc) * c; } return retval; } double cheb_eval(const vector<double>& coef, const pair<double, double>& domain, double x) { return cheb_eval(coef, affine_remap(domain, x, { -1.0, 1.0 })); }

struct ChebyshevApprox_ { vector<double> coeffs_; pair<double, double> domain_;

double operator()(double x) const { return cheb_eval(coeffs_, domain_, x); }

template<class F_> ChebyshevApprox_ (const F_& func, int n, const pair<double, double>& domain) : coeffs_(cheb_coef(func, n, domain)), domain_(domain) { } };


int main(void) { static const int N = 10; ChebyshevApprox_ fApprox(cos, N, { 0.0, 1.0 }); cout << "Coefficients: " << setprecision(14); for (const auto& c : fApprox.coeffs_) cout << "\t" << c << "\n";

for (;;) { cout << "Enter x, or non-numeric value to quit:\n"; double x; if (!(cin >> x)) return 0; cout << "True value: \t" << cos(x) << "\n"; cout << "Approximate: \t" << fApprox(x) << "\n"; } } </lang>

D

This imperative code retains some of the style of the original C version. <lang d>import std.math: PI, cos;

/// Map x from range [min, max] to [min_to, max_to]. real map(in real x, in real min_x, in real max_x, in real min_to, in real max_to) pure nothrow @safe @nogc { return (x - min_x) / (max_x - min_x) * (max_to - min_to) + min_to; }


void chebyshevCoef(size_t N)(in real function(in real) pure nothrow @safe @nogc func, 

                            in real min, in real max, ref real[N] coef)

pure nothrow @safe @nogc { 

   coef[] = 0.0;
   
   foreach (immutable i; 0 .. N) {
       immutable f = func(map(cos(PI * (i + 0.5f) / N), -1, 1, min, max)) * 2 / N;
       foreach (immutable j, ref cj; coef)
           cj += f * cos(PI * j * (i + 0.5f) / N);

} }


/// f(x) = sum_{k=0}^{n - 1} c_k T_k(x) - c_0/2 real chebyshevApprox(size_t N)(in real x, in real min, in real max, in ref real[N] coef) pure nothrow @safe @nogc if (N >= 2) { 

   real a = 1.0L,
        b = map(x, min, max, -1, 1),
        result = coef[0] / 2 + coef[1] * b;

immutable x2 = 2 * b; 

   foreach (immutable ci; coef[2 .. $]) {

// T_{n+1} = 2x T_n - T_{n-1} 

       immutable c = x2 * b - a;
       result += ci * c;
       a = b;
       b = c;
   }

   return result;

}


void main() @safe { 

   import std.stdio: writeln, writefln;    
   enum uint N = 10;

real[N] c; 

   real min = 0, max = 1;
   static real test(in real x) pure nothrow @safe @nogc { return x.cos; }

chebyshevCoef(&test, min, max, c); 

   writefln("Coefficients:\n%(  %+2.25g\n%)", c);

   enum nX = 20;

writeln("\nApproximation:\n x func(x) approx diff"); 

   foreach (immutable i; 0 .. nX) {
       immutable x = map(i, 0, nX, min, max);

immutable f = test(x); immutable approx = chebyshevApprox(x, min, max, c);

writefln("%1.3f % 10.10f % 10.10f % 4.2e", x, f, approx, approx - f); } }</lang>

Output:
Coefficients:
  +1.6471694753903136868
  -0.23229937161517194216
  -0.053715114622047555044
  +0.0024582352669814797779
  +0.00028211905743400579387
  -7.7222291558103533853e-06
  -5.898556452178771968e-07
  +1.1521427332860788728e-08
  +6.5963000382704222411e-10
  -1.0022591914390921452e-11

Approximation:
    x                 func(x)                  approx      diff
0.000  1.00000000000000000000  1.00000000000046961190  4.70e-13
0.050  0.99875026039496624654  0.99875026039487216781 -9.41e-14
0.100  0.99500416527802576609  0.99500416527848803832  4.62e-13
0.150  0.98877107793604228670  0.98877107793599569749 -4.66e-14
0.200  0.98006657784124163110  0.98006657784078136889 -4.60e-13
0.250  0.96891242171064478408  0.96891242171041249593 -2.32e-13
0.300  0.95533648912560601967  0.95533648912586667367  2.61e-13
0.350  0.93937271284737892005  0.93937271284783928305  4.60e-13
0.400  0.92106099400288508277  0.92106099400308274515  1.98e-13
0.450  0.90044710235267692169  0.90044710235242891114 -2.48e-13
0.500  0.87758256189037271615  0.87758256188991362600 -4.59e-13
0.550  0.85252452205950574283  0.85252452205925896211 -2.47e-13
0.600  0.82533561490967829723  0.82533561490987400509  1.96e-13
0.650  0.79608379854905582896  0.79608379854950937939  4.54e-13
0.700  0.76484218728448842626  0.76484218728474395029  2.56e-13
0.750  0.73168886887382088633  0.73168886887359430061 -2.27e-13
0.800  0.69670670934716542091  0.69670670934671868322 -4.47e-13
0.850  0.65998314588498217039  0.65998314588493717370 -4.50e-14
0.900  0.62160996827066445648  0.62160996827110870299  4.44e-13
0.950  0.58168308946388349416  0.58168308946379353278 -9.00e-14

The same code, with N = 16: 

Coefficients:
  +1.6471694753903136868
  -0.23229937161517194214
  -0.053715114622047555035
  +0.0024582352669814797982
  +0.00028211905743400571932
  -7.722229155810705751e-06
  -5.898556452177348953e-07
  +1.1521427330794028337e-08
  +6.5963022091481034181e-10
  -1.0016894235462866363e-11
  -4.5865582517937500406e-13
  +5.6974586994888026802e-15
  +2.1752822525027137867e-16
  -2.3140940118987485263e-18
  -1.0333801956502464137e-19
  +2.5410988417629010172e-20

Approximation:
    x                 func(x)                  approx      diff
0.000  1.00000000000000000000  1.00000000000000000030  3.25e-19
0.050  0.99875026039496624654  0.99875026039496624646 -1.08e-19
0.100  0.99500416527802576609  0.99500416527802576557 -5.42e-19
0.150  0.98877107793604228670  0.98877107793604228636 -3.79e-19
0.200  0.98006657784124163110  0.98006657784124163127  1.08e-19
0.250  0.96891242171064478408  0.96891242171064478451  3.79e-19
0.300  0.95533648912560601967  0.95533648912560601967  0.00e+00
0.350  0.93937271284737892005  0.93937271284737891962 -3.79e-19
0.400  0.92106099400288508277  0.92106099400288508260 -2.17e-19
0.450  0.90044710235267692169  0.90044710235267692169  5.42e-20
0.500  0.87758256189037271615  0.87758256189037271632  2.17e-19
0.550  0.85252452205950574283  0.85252452205950574274 -5.42e-20
0.600  0.82533561490967829723  0.82533561490967829697 -2.17e-19
0.650  0.79608379854905582896  0.79608379854905582861 -3.25e-19
0.700  0.76484218728448842626  0.76484218728448842630  5.42e-20
0.750  0.73168886887382088633  0.73168886887382088637  5.42e-20
0.800  0.69670670934716542091  0.69670670934716542087 -5.42e-20
0.850  0.65998314588498217039  0.65998314588498217022 -1.63e-19
0.900  0.62160996827066445648  0.62160996827066445674  2.71e-19
0.950  0.58168308946388349416  0.58168308946388349403 -1.63e-19

EasyLang

<lang>numfmt 0 5 a = 0 b = 1 n = 10 len coef[] n len cheby[] n for i range n 

 coef[i] = cos (180 / pi * (cos (180 / n * (i + 1 / 2)) * (b - a) / 2 + (b + a) / 2))

. for i range n 

 w = 0
 for j range n
   w += coef[j] * cos (180 / n * i * (j + 1 / 2))
 .
 cheby[i] = w * 2 / n
 print cheby[i]

.</lang>

Go

Wikipedia gives a formula for coefficients in a section "Example 1". Read past the bit about the inner product to where it gives the technique based on the discrete orthogonality condition. The N of the WP formulas is the parameter nNodes in the code here. It is not necessarily the same as n, the number of polynomial coefficients, the parameter nCoeff here.

The evaluation method is the Clenshaw algorithm.

Two variances here from the WP presentation and most mathematical presentations follow other examples on this page and so keep output directly comparable. One variance is that the Kronecker delta factor is dropped, which has the effect of doubling the first coefficient. This simplifies both coefficient generation and polynomial evaluation. A further variance is that there is no scaling for the range of function values. The result is that coefficients are not necessarily bounded by 1 (2 for the first coefficient) but by the maximum function value over the argument range from min to max (or twice that for the first coefficient.) <lang go>package main

import ( 

   "fmt"
   "math"

)

type cheb struct { 

   c        []float64
   min, max float64

}

func main() { 

   fn := math.Cos
   c := newCheb(0, 1, 10, 10, fn)
   fmt.Println("coefficients:")
   for _, c := range c.c {
       fmt.Printf("% .15f\n", c)
   }
   fmt.Println("\nx     computed    approximated    computed-approx")
   const n = 10
   for i := 0.; i <= n; i++ {
       x := (c.min*(n-i) + c.max*i) / n
       computed := fn(x)
       approx := c.eval(x)
       fmt.Printf("%.1f %12.8f  %12.8f   % .3e\n",
           x, computed, approx, computed-approx)
   }

}

func newCheb(min, max float64, nCoeff, nNodes int, fn func(float64) float64) *cheb { 

   c := &cheb{
       c:   make([]float64, nCoeff),
       min: min,
       max: max,
   }
   f := make([]float64, nNodes)
   p := make([]float64, nNodes)
   z := .5 * (max + min)
   r := .5 * (max - min)
   for k := 0; k < nNodes; k++ {
       p[k] = math.Pi * (float64(k) + .5) / float64(nNodes)
       f[k] = fn(z + math.Cos(p[k])*r)
   }
   n2 := 2 / float64(nNodes)
   for j := 0; j < nCoeff; j++ {
       sum := 0.
       for k := 0; k < nNodes; k++ {
           sum += f[k] * math.Cos(float64(j)*p[k])
       }
       c.c[j] = sum * n2
   }
   return c

}

func (c *cheb) eval(x float64) float64 { 

   x1 := (2*x - c.min - c.max) / (c.max - c.min)
   x2 := 2 * x1
   var s, t float64
   for j := len(c.c) - 1; j >= 1; j-- {
       t, s = x2*t-s+c.c[j], t
   }
   return x1*t - s + .5*c.c[0]

}</lang>

Output:
coefficients:
 1.647169475390314
-0.232299371615172
-0.053715114622048
 0.002458235266982
 0.000282119057434
-0.000007722229156
-0.000000589855645
 0.000000011521427
 0.000000000659630
-0.000000000010022

x     computed    approximated    computed-approx
0.0   1.00000000    1.00000000   -4.685e-13
0.1   0.99500417    0.99500417   -4.620e-13
0.2   0.98006658    0.98006658    4.601e-13
0.3   0.95533649    0.95533649   -2.607e-13
0.4   0.92106099    0.92106099   -1.972e-13
0.5   0.87758256    0.87758256    4.587e-13
0.6   0.82533561    0.82533561   -1.965e-13
0.7   0.76484219    0.76484219   -2.552e-13
0.8   0.69670671    0.69670671    4.470e-13
0.9   0.62160997    0.62160997   -4.449e-13
1.0   0.54030231    0.54030231   -4.476e-13

Groovy

Translation of: Java

<lang groovy>class ChebyshevCoefficients { 

   static double map(double x, double min_x, double max_x, double min_to, double max_to) {
       return (x - min_x) / (max_x - min_x) * (max_to - min_to) + min_to
   }

   static void chebyshevCoef(Closure<Double> func, double min, double max, double[] coef) {
       final int N = coef.length
       for (int i = 0; i < N; i++) {
           double m = map(Math.cos(Math.PI * (i + 0.5f) / N), -1, 1, min, max)
           double f = func(m) * 2 / N

           for (int j = 0; j < N; j++) {
               coef[j] += f * Math.cos(Math.PI * j * (i + 0.5f) / N)
           }
       }
   }

   static void main(String[] args) {
       final int N = 10
       double[] c = new double[N]
       double min = 0, max = 1
       chebyshevCoef(Math.&cos, min, max, c)

       println("Coefficients:")
       for (double d : c) {
           println(d)
       }
   }

}</lang>

Output:
Coefficients:
1.6471694753903139
-0.23229937161517178
-0.0537151146220477
0.002458235266981773
2.8211905743405485E-4
-7.722229156320592E-6
-5.898556456745974E-7
1.1521427770166959E-8
6.59630183807991E-10
-1.0021913854352249E-11

J

From 'J for C Programmers: Calculating Chebyshev Coefficients [[1]] <lang J> chebft =: adverb define

f =. u 0.5 * (+/y) - (-/y) * 2 o. o. (0.5 + i. x) % x 

  (2 % x) * +/ f * 2 o. o. (0.5 + i. x) *"0 1 (i. x) % x

) </lang> Calculate coefficients: <lang J> 

     10 (2&o.) chebft 0 1

1.64717 _0.232299 _0.0537151 0.00245824 0.000282119 _7.72223e_6 _5.89856e_7 1.15214e_8 6.59629e_10 _1.00227e_11 </lang>

Java

Partial translation of C via D

Works with: Java version 8

<lang java>import static java.lang.Math.*; import java.util.function.Function;

public class ChebyshevCoefficients { 

   static double map(double x, double min_x, double max_x, double min_to,
           double max_to) {
       return (x - min_x) / (max_x - min_x) * (max_to - min_to) + min_to;
   }

   static void chebyshevCoef(Function<Double, Double> func, double min,
           double max, double[] coef) {

       int N = coef.length;

       for (int i = 0; i < N; i++) {

           double m = map(cos(PI * (i + 0.5f) / N), -1, 1, min, max);
           double f = func.apply(m) * 2 / N;

           for (int j = 0; j < N; j++) {
               coef[j] += f * cos(PI * j * (i + 0.5f) / N);
           }
       }
   }

   public static void main(String[] args) {
       final int N = 10;
       double[] c = new double[N];
       double min = 0, max = 1;
       chebyshevCoef(x -> cos(x), min, max, c);

       System.out.println("Coefficients:");
       for (double d : c)
           System.out.println(d);
   }

}</lang> 

Coefficients:
1.6471694753903139
-0.23229937161517178
-0.0537151146220477
0.002458235266981773
2.8211905743405485E-4
-7.722229156320592E-6
-5.898556456745974E-7
1.1521427770166959E-8
6.59630183807991E-10
-1.0021913854352249E-11

jq

Adapted from Wren

Works with: jq

Works with gojq, the Go implementation of jq

Preliminaries <lang jq>def lpad($len): tostring | ($len - length) as $l | (" " * $l) + .; def rpad($len; $fill): tostring | ($len - length) as $l | . + ($fill * $l)[:$l];

  1. Format a decimal number so that there are at least `left` characters
  2. to the left of the decimal point, and at most `right` characters to its right.
  3. No left-truncation occurs, so `left` can be specified as 0 to prevent left-padding.
  4. If tostring has an "e" then eparse as defined below is used.

def pp(left; right): 

 def lpad: if (left > length) then ((left - length) * " ") + . else . end;
 def eparse: index("e") as $ix | (.[:$ix]|pp(left;right)) + .[$ix:];
 tostring as $s
 | $s
 | if test("e") then eparse
   else index(".") as $ix
   | ((if $ix then $s[0:$ix] else $s end) | lpad) + "." +
     (if $ix then $s[$ix+1:] | .[0:right] else "" end)
   end;</lang>

Chebyshev Coefficients <lang jq>def mapRange($x; $min; $max; $minTo; $maxTo): 

 (($x - $min)/($max - $min))*($maxTo - $minTo) + $minTo;

def chebCoeffs(func; n; min; max): 

 (1 | atan * 4) as $pi
 | reduce range(0;n) as $i ([]; # coeffs
     ((mapRange( ($pi * ($i + 0.5) / n)|cos; -1; 1; min; max) | func) * 2 / n) as $f
     | reduce range(0;n) as $j (.;
         .[$j] +=  $f * ($pi * $j * (($i + 0.5) / n)|cos)) );

def chebApprox(x; n; min; max; coeffs): 

 if n < 2 or (coeffs|length) < 2 then "'n' can't be less than 2." | error
 else { a: 1,
        b: mapRange(x; min; max; -1; 1) }
 | .res = coeffs[0]/2 + coeffs[1]*.b
 | .xx = 2 * .b
 | reduce range(2;n) as $i (.;
      (.xx * .b - .a) as $c
      | .res += coeffs[$i]*$c)
      | .a = .b
      | .b = $c)
 | .res
 end ;

def task: 

   [10, 0, 1] as [$n, $min, $max]
 |  chebCoeffs(cos; $n; $min; $max) as $coeffs
 | "Coefficients:",
    ($coeffs[]|pp(2;14)),
    "\nApproximations:\n  x      func(x)    approx       diff",
    (range(0;21) as $i
     | mapRange($i; 0; 20; $min; $max) as $x
     | ($x|cos) as $f
     | chebApprox($x; $n; $min; $max; $coeffs) as $approx
     | ($approx - $f) as $diff
     | [ ($x|pp(0;3)|rpad( 4;"0")),
         ($f|pp(0;8)|rpad(10;"0")),

($approx|pp(0;8)), 

         ($diff  |pp(2;2)) ]
     | join("  ") );

task</lang>

Output:
Coefficients:
 1.64716947539031
-0.23229937161517
-0.05371511462204
 0.00245823526698
 0.00028211905743
-7.72222915562670e-06
-5.89855645688475e-07
 1.15214280338449e-08
 6.59629580124221e-10
-1.00220526322303e-11

Approximations:
  x      func(x)    approx       diff
0.00  1.00000000  1.00000000   4.66e-13
0.05  0.99875026  0.99875026  -9.21e-14
0.10  0.99500416  0.99500416   4.62e-13
0.15  0.98877107  0.98877107  -4.74e-14
0.20  0.98006657  0.98006657  -4.60e-13
0.25  0.96891242  0.96891242  -2.32e-13
0.30  0.95533648  0.95533648   2.61e-13
0.35  0.93937271  0.93937271   4.60e-13
0.40  0.92106099  0.92106099   1.98e-13
0.45  0.90044710  0.90044710  -2.47e-13
0.50  0.87758256  0.87758256  -4.59e-13
0.55  0.85252452  0.85252452  -2.46e-13
0.60  0.82533561  0.82533561   1.95e-13
0.65  0.79608379  0.79608379   4.53e-13
0.70  0.76484218  0.76484218   2.55e-13
0.75  0.73168886  0.73168886  -2.26e-13
0.80  0.69670670  0.69670670  -4.46e-13
0.85  0.65998314  0.65998314  -4.45e-14
0.90  0.62160996  0.62160996   4.44e-13
0.95  0.58168308  0.58168308  -9.01e-14
1.00  0.54030230  0.54030230   4.47e-13

Julia

Works with: Julia version 0.6
Translation of: Go

<lang julia>mutable struct Cheb 

   c::Vector{Float64}
   min::Float64
   max::Float64

end

function Cheb(min::Float64, max::Float64, ncoeff::Int, nnodes::Int, fn::Function)::Cheb 

   c = Cheb(Vector{Float64}(ncoeff), min, max)
   f = Vector{Float64}(nnodes)
   p = Vector{Float64}(nnodes)
   z = (max + min) / 2
   r = (max - min) / 2
   for k in 0:nnodes-1
       p[k+1] = π * (k + 0.5) / nnodes
       f[k+1] = fn(z + cos(p[k+1]) * r)
   end
   n2 = 2 / nnodes
   for j in 0:nnodes-1
       s = sum(fk * cos(j * pk) for (fk, pk) in zip(f, p))
       c.c[j+1] = s * n2
   end
   return c

end

function evaluate(c::Cheb, x::Float64)::Float64 

   x1 = (2x - c.max - c.min) / (c.max - c.min)
   x2 = 2x1
   t = s = 0
   for j in length(c.c):-1:2
       t, s = x2 * t - s + c.c[j], t
   end
   return x1 * t - s + c.c[1] / 2

end

fn = cos c = Cheb(0.0, 1.0, 10, 10, fn)

  1. coefs

println("Coefficients:") for x in c.c 

   @printf("% .15f\n", x)

end

  1. values

println("\nx computed approximated computed-approx") const n = 10 for i in 0.0:n 

   x = (c.min * (n - i) + c.max * i) / n
   computed = fn(x)
   approx   = evaluate(c, x)
   @printf("%.1f %12.8f  %12.8f   % .3e\n", x, computed, approx, computed - approx)

end</lang>

Output:
Coefficients:
 1.647169475390314
-0.232299371615172
-0.053715114622048
 0.002458235266981
 0.000282119057434
-0.000007722229156
-0.000000589855645
 0.000000011521427
 0.000000000659630
-0.000000000010022

x     computed    approximated    computed-approx
0.0   1.00000000    1.00000000   -4.685e-13
0.1   0.99500417    0.99500417   -4.620e-13
0.2   0.98006658    0.98006658    4.601e-13
0.3   0.95533649    0.95533649   -2.605e-13
0.4   0.92106099    0.92106099   -1.970e-13
0.5   0.87758256    0.87758256    4.586e-13
0.6   0.82533561    0.82533561   -1.967e-13
0.7   0.76484219    0.76484219   -2.551e-13
0.8   0.69670671    0.69670671    4.470e-13
0.9   0.62160997    0.62160997   -4.449e-13
1.0   0.54030231    0.54030231   -4.476e-13

Kotlin

Translation of: C

<lang scala>// version 1.1.2

typealias DFunc = (Double) -> Double

fun mapRange(x: Double, min: Double, max: Double, minTo: Double, maxTo:Double): Double { 

   return (x - min) / (max - min) * (maxTo - minTo) + minTo

}

fun chebCoeffs(func: DFunc, n: Int, min: Double, max: Double): DoubleArray { 

   val coeffs = DoubleArray(n)
   for (i in 0 until n) {
       val f = func(mapRange(Math.cos(Math.PI * (i + 0.5) / n), -1.0, 1.0, min, max)) * 2.0 / n
       for (j in 0 until n) coeffs[j] += f * Math.cos(Math.PI * j * (i + 0.5) / n)
   }
   return coeffs

}

fun chebApprox(x: Double, n: Int, min: Double, max: Double, coeffs: DoubleArray): Double { 

   require(n >= 2 && coeffs.size >= 2)
   var a = 1.0
   var b = mapRange(x, min, max, -1.0, 1.0)
   var res = coeffs[0] / 2.0 + coeffs[1] * b
   val xx = 2 * b
   var i = 2
   while (i < n) {
       val c = xx * b - a
       res += coeffs[i] * c
       a = b
       b = c
       i++
   }
   return res

}

fun main(args: Array<String>) { 

   val n = 10
   val min = 0.0
   val max = 1.0
   val coeffs = chebCoeffs(Math::cos, n, min, max)
   println("Coefficients:")
   for (coeff in coeffs) println("%+1.15g".format(coeff))
   println("\nApproximations:\n  x      func(x)     approx       diff")
   for (i in 0..20) {
       val x = mapRange(i.toDouble(), 0.0, 20.0, min, max)
       val f = Math.cos(x)
       val approx = chebApprox(x, n, min, max, coeffs)
       System.out.printf("%1.3f  %1.8f  %1.8f  % 4.1e\n", x, f, approx, approx - f)
   }

}</lang>

Output:
Coefficients:
+1.64716947539031
-0.232299371615172
-0.0537151146220477
+0.00245823526698177
+0.000282119057434055
-7.72222915632059e-06
-5.89855645674597e-07
+1.15214277701670e-08
+6.59630183807991e-10
-1.00219138543522e-11

Approximations:
  x      func(x)     approx       diff
0.000  1.00000000  1.00000000   4.7e-13
0.050  0.99875026  0.99875026  -9.4e-14
0.100  0.99500417  0.99500417   4.6e-13
0.150  0.98877108  0.98877108  -4.7e-14
0.200  0.98006658  0.98006658  -4.6e-13
0.250  0.96891242  0.96891242  -2.3e-13
0.300  0.95533649  0.95533649   2.6e-13
0.350  0.93937271  0.93937271   4.6e-13
0.400  0.92106099  0.92106099   2.0e-13
0.450  0.90044710  0.90044710  -2.5e-13
0.500  0.87758256  0.87758256  -4.6e-13
0.550  0.85252452  0.85252452  -2.5e-13
0.600  0.82533561  0.82533561   2.0e-13
0.650  0.79608380  0.79608380   4.5e-13
0.700  0.76484219  0.76484219   2.5e-13
0.750  0.73168887  0.73168887  -2.3e-13
0.800  0.69670671  0.69670671  -4.5e-13
0.850  0.65998315  0.65998315  -4.4e-14
0.900  0.62160997  0.62160997   4.5e-13
0.950  0.58168309  0.58168309  -9.0e-14
1.000  0.54030231  0.54030231   4.5e-13

Lua

Translation of: Java

<lang lua>function map(x, min_x, max_x, min_to, max_to) 

   return (x - min_x) / (max_x - min_x) * (max_to - min_to) + min_to

end

function chebyshevCoef(func, minn, maxx, coef) 

   local N = table.getn(coef)
   for j=1,N do
       local i = j - 1
       local m = map(math.cos(math.pi * (i + 0.5) / N), -1, 1, minn, maxx)
       local f = func(m) * 2 / N

       for k=1,N do
           local p = k  -1
           coef[k] = coef[k] + f * math.cos(math.pi * p * (i + 0.5) / N)
       end
   end

end

function main() 

   local N = 10
   local c = {}
   local minn = 0.0
   local maxx = 1.0

   for i=1,N do
       table.insert(c, 0)
   end

   chebyshevCoef(function (x) return math.cos(x) end, minn, maxx, c)

   print("Coefficients:")
   for i,d in pairs(c) do
       print(d)
   end

end

main() </lang>

Output:
Coefficients:
1.6471694753903
-0.23229937161517
-0.053715114622048
0.0024582352669818
0.00028211905743405
-7.7222291563483e-006
-5.898556456746e-007
1.1521427756289e-008
6.5963018380799e-010
-1.0021913854352e-011

Microsoft Small Basic

Translation of: Perl

<lang smallbasic>' N Chebyshev coefficients for the range 0 to 1 - 18/07/2018 

 pi=Math.pi
 a=0
 b=1
 n=10
 For i=0 To n-1
   coef[i]=Math.cos(Math.cos(pi/n*(i+1/2))*(b-a)/2+(b+a)/2)
 EndFor
 For i=0 To n-1
   w=0
   For j=0 To n-1
     w=w+coef[j]*Math.cos(pi/n*i*(j+1/2))
   EndFor
   cheby[i]=w*2/n
   t=" "
   If cheby[i]<=0 Then
     t=""
   EndIf
   TextWindow.WriteLine(i+" : "+t+cheby[i])
 EndFor</lang>
Output:
0 :  1,6471694753903136
1 : -0,2322993716151700684187787635
2 : -0,0537151146220494010749946688
3 :  0,0024582352669837594966069584
4 :  0,0002821190574317389206759282
5 : -0,0000077222291539069653168878
6 : -0,0000005898556481086082412444
7 :  0,0000000115214300591398939205
8 :  0,0000000006596278553822696656
9 : -0,0000000000100189955816952521

Nim

Translation of: Go

<lang Nim>import lenientops, math, strformat, sugar

type Cheb = object 

 c: seq[float]
 min, max: float


func initCheb(min, max: float; nCoeff, nNodes: int; fn: float -> float): Cheb = 

 result = Cheb(c: newSeq[float](nCoeff), min: min, max: max)
 var f, p = newSeq[float](nNodes)
 let z = 0.5 * (max + min)
 let r = 0.5 * (max - min)
 for k in 0..<nNodes:
   p[k] = PI * (k + 0.5) / nNodes
   f[k] = fn(z + cos(p[k]) * r)

 let n2 = 2 / nNodes
 for j in 0..<nCoeff:
   var sum = 0.0
   for k in 0..<nNodes:
     sum += f[k] * cos(j * p[k])
   result.c[j] = sum * n2


func eval(cheb: Cheb; x: float): float = 

 let x1 = (2 * x - cheb.min - cheb.max) / (cheb.max - cheb.min)
 let x2 = 2 * x1
 var s, t: float
 for j in countdown(cheb.c.high, 1):
   s = x2 * t - s + cheb.c[j]
   swap s, t
 result = x1 * t - s + 0.5 * cheb.c[0]


when isMainModule: 

 let fn: float -> float = cos
 let cheb = initCheb(0, 1, 10, 10, fn)
 echo "Coefficients:"
 for c in cheb.c:
   echo &"{c: .15f}"

 echo "\n x     computed    approximated   computed-approx"
 const N = 10
 for i in 0..N:
   let x = (cheb.min * (N - i) + cheb.max * i) / N
   let computed = fn(x)
   let approx = cheb.eval(x)
   echo &"{x:.1f} {computed:12.8f}  {approx:12.8f}      {computed-approx: .3e}"</lang>
Output:
Coefficients:
 1.647169475390314
-0.232299371615172
-0.053715114622048
 0.002458235266981
 0.000282119057434
-0.000007722229156
-0.000000589855645
 0.000000011521427
 0.000000000659630
-0.000000000010022

 x     computed    approximated   computed-approx
0.0   1.00000000    1.00000000      -4.685e-13
0.1   0.99500417    0.99500417      -4.620e-13
0.2   0.98006658    0.98006658       4.601e-13
0.3   0.95533649    0.95533649      -2.605e-13
0.4   0.92106099    0.92106099      -1.970e-13
0.5   0.87758256    0.87758256       4.586e-13
0.6   0.82533561    0.82533561      -1.967e-13
0.7   0.76484219    0.76484219      -2.551e-13
0.8   0.69670671    0.69670671       4.470e-13
0.9   0.62160997    0.62160997      -4.450e-13
1.0   0.54030231    0.54030231      -4.476e-13

Perl

Translation of: C

<lang perl>use constant PI => 3.141592653589793;

sub chebft { 

 my($func, $a, $b, $n) = @_;
 my($bma, $bpa) = ( 0.5*($b-$a), 0.5*($b+$a) );

 my @pin = map { ($_ + 0.5) * (PI/$n) } 0..$n-1;
 my @f = map { $func->( cos($_) * $bma + $bpa ) } @pin;
 my @c = (0) x $n;
 for my $j (0 .. $n-1) {
   $c[$j] += $f[$_] * cos($j * $pin[$_])   for 0..$n-1;
   $c[$j] *= (2.0/$n);
 }
 @c;

}

print "$_\n" for chebft(sub{cos($_[0])}, 0, 1, 10);</lang>

Output:
1.64716947539031
-0.232299371615172
-0.0537151146220477
0.00245823526698163
0.000282119057433938
-7.72222915566001e-06
-5.89855645105608e-07
1.15214274787334e-08
6.59629917354465e-10
-1.00219943455215e-11

Phix

Translation of: Go
function Cheb(atom cmin, cmax, integer ncoeff, nnodes)
    sequence c = repeat(0,ncoeff),
             f = repeat(0,nnodes),
             p = repeat(0,nnodes)
    atom z = (cmax + cmin) / 2,
         r = (cmax - cmin) / 2
    for k=1 to nnodes do
        p[k] = PI * ((k-1) + 0.5) / nnodes
        f[k] = cos(z + cos(p[k]) * r)
    end for
    atom n2 = 2 / nnodes
    for j=1 to nnodes do
        atom s := 0
        for k=1 to nnodes do
            s += f[k] * cos((j-1)*p[k])
        end for
        c[j] = s * n2
    end for
    return c
end function
 
function evaluate(sequence c, atom cmin, cmax, x)
    atom x1 = (2*x - cmax - cmin) / (cmax - cmin),
         x2 = 2*x1,
         t = 0, s = 0
    for j=length(c) to 2 by -1 do
        {t, s} = {x2 * t - s + c[j], t}
    end for
    return x1 * t - s + c[1] / 2
end function
 
atom cmin = 0.0, cmax = 1.0
sequence c  = Cheb(cmin, cmax, 10, 10)
printf(1, "Coefficients:\n")
pp(c,{pp_Nest,1,pp_FltFmt,"%18.15f"})
printf(1,"\nx     computed    approximated    computed-approx\n")
constant n = 10
for i=0 to 10 do
    atom x = (cmin * (n - i) + cmax * i) / n,
         calc = cos(x),
         est = evaluate(c, cmin, cmax, x)
    printf(1,"%.1f %12.8f  %12.8f   %10.3e\n", {x, calc, est, calc-est})
end for
Output:
Coefficients:
{ 1.647169475390314,
 -0.232299371615172,
 -0.053715114622048,
  0.002458235266981,
  0.000282119057434,
 -0.000007722229156,
 -0.000000589855645,
  0.000000011521427,
  0.000000000659630,
 -0.000000000010022}

x     computed    approximated    computed-approx
0.0   1.00000000    1.00000000   -4.686e-13
0.1   0.99500417    0.99500417   -4.620e-13
0.2   0.98006658    0.98006658    4.600e-13
0.3   0.95533649    0.95533649   -2.604e-13
0.4   0.92106099    0.92106099   -1.970e-13
0.5   0.87758256    0.87758256    4.587e-13
0.6   0.82533561    0.82533561   -1.968e-13
0.7   0.76484219    0.76484219   -2.551e-13
0.8   0.69670671    0.69670671    4.470e-13
0.9   0.62160997    0.62160997   -4.450e-13
1.0   0.54030231    0.54030231   -4.477e-13

Python

Translation of: C++

<lang python>import math

def test_func(x): 

   return math.cos(x)

def mapper(x, min_x, max_x, min_to, max_to): 

   return (x - min_x) / (max_x - min_x) * (max_to - min_to) + min_to

def cheb_coef(func, n, min, max): 

   coef = [0.0] * n
   for i in xrange(n):
       f = func(mapper(math.cos(math.pi * (i + 0.5) / n), -1, 1, min, max)) * 2 / n
       for j in xrange(n):
           coef[j] += f * math.cos(math.pi * j * (i + 0.5) / n)
   return coef

def cheb_approx(x, n, min, max, coef): 

   a = 1
   b = mapper(x, min, max, -1, 1)
   c = float('nan')
   res = coef[0] / 2 + coef[1] * b

   x = 2 * b
   i = 2
   while i < n:
       c = x * b - a
       res = res + coef[i] * c
       (a, b) = (b, c)
       i += 1

   return res

def main(): 

   N = 10
   min = 0
   max = 1
   c = cheb_coef(test_func, N, min, max)

   print "Coefficients:"
   for i in xrange(N):
       print " % lg" % c[i]

   print "\n\nApproximation:\n    x      func(x)       approx      diff"
   for i in xrange(20):
       x = mapper(i, 0.0, 20.0, min, max)
       f = test_func(x)
       approx = cheb_approx(x, N, min, max, c)
       print "%1.3f %10.10f %10.10f % 4.2e" % (x, f, approx, approx - f)

   return None

main()</lang>

Output:
Coefficients:
  1.64717
 -0.232299
 -0.0537151
  0.00245824
  0.000282119
 -7.72223e-06
 -5.89856e-07
  1.15214e-08
  6.5963e-10
 -1.00219e-11


Approximation:
    x      func(x)       approx      diff
0.000 1.0000000000 1.0000000000  4.68e-13
0.050 0.9987502604 0.9987502604 -9.36e-14
0.100 0.9950041653 0.9950041653  4.62e-13
0.150 0.9887710779 0.9887710779 -4.73e-14
0.200 0.9800665778 0.9800665778 -4.60e-13
0.250 0.9689124217 0.9689124217 -2.32e-13
0.300 0.9553364891 0.9553364891  2.62e-13
0.350 0.9393727128 0.9393727128  4.61e-13
0.400 0.9210609940 0.9210609940  1.98e-13
0.450 0.9004471024 0.9004471024 -2.47e-13
0.500 0.8775825619 0.8775825619 -4.58e-13
0.550 0.8525245221 0.8525245221 -2.46e-13
0.600 0.8253356149 0.8253356149  1.96e-13
0.650 0.7960837985 0.7960837985  4.53e-13
0.700 0.7648421873 0.7648421873  2.54e-13
0.750 0.7316888689 0.7316888689 -2.28e-13
0.800 0.6967067093 0.6967067093 -4.47e-13
0.850 0.6599831459 0.6599831459 -4.37e-14
0.900 0.6216099683 0.6216099683  4.46e-13
0.950 0.5816830895 0.5816830895 -8.99e-14

Racket

Translation of: C

<lang racket>#lang typed/racket (: chebft (Real Real Nonnegative-Integer (Real -> Real) -> (Vectorof Real))) (define (chebft a b n func) 

 (define b-a/2 (/ (- b a) 2))
 (define b+a/2 (/ (+ b a) 2))
 (define pi/n (/ pi n))
 (define fac (/ 2 n))

 (define f (for/vector : (Vectorof Real)
             ((k : Nonnegative-Integer (in-range n)))
             (define y (cos (* pi/n (+ k 1/2))))
             (func (+ (* y b-a/2) b+a/2))))

 (for/vector : (Vectorof Real)
   ((j : Nonnegative-Integer (in-range n)))
   (define s (for/sum : Real
               ((k : Nonnegative-Integer (in-range n)))
               (* (vector-ref f k)
                  (cos (* pi/n j (+ k 1/2))))))
   (* fac s)))

(module+ test 

 (chebft 0 1 10 cos))
Tim Brown 2015</lang>
Output:
'#(1.6471694753903137
   -0.2322993716151719
   -0.05371511462204768
   0.0024582352669816343
   0.0002821190574339161
   -7.722229155637806e-006
   -5.898556451056081e-007
   1.1521427500937876e-008
   6.596299173544651e-010
   -1.0022016549982027e-011)

Raku

(formerly Perl 6)

Translation of: C

<lang perl6>sub chebft ( Code $func, Real \a, Real \b, Int \n ) { 

   my \bma = ½ × (b - a);
   my \bpa = ½ × (b + a);

   my @pi-n = ( (^n).list »+» ½ ) »×» (π/n);
   my @f    = ( @pi_n».cos »×» bma »+» bpa )».&$func;
   my @sums = (^n).map: { [+] @f »×« ( @pi-n »×» $_ )».cos };

   @sums »×» (2/n)

}

say chebft(&cos, 0, 1, 10)».fmt: '%+13.7e';</lang>

Output:
+1.6471695e+00
-2.3229937e-01
-5.3715115e-02
+2.4582353e-03
+2.8211906e-04
-7.7222292e-06
-5.8985565e-07
+1.1521427e-08
+6.5962992e-10
-1.0021994e-11

REXX

Translation of: C

This REXX program is a translation of the   C   program plus added optimizations. 

    Pafnuty Lvovich Chebysheff:   Chebyshev       [English  transliteration]
                                  Chebysheff      [   "           "        ]
                                  Chebyshov       [   "           "        ]
                                  Tchebychev      [French         "        ]
                                  Tchebysheff     [   "           "        ]
                                  Tschebyschow    [German         "        ]
                                  Tschebyschev    [   "           "        ]
                                  Tschebyschef    [   "           "        ]
                                  Tschebyscheff   [   "           "        ]


The numeric precision is dependent on the number of decimal digits specified in the value of pi. <lang rexx>/*REXX program calculates N Chebyshev coefficients for the range 0 ──► 1 (inclusive)*/ numeric digits length( pi() ) - length(.) /*DIGITS default is nine, but use 71. */ parse arg a b N . /*obtain optional arguments from the CL*/ if a== | a=="," then a= 0 /*A not specified? Then use default.*/ if b== | b=="," then b= 1 /*B " " " " " */ if N== | N=="," then N= 10 /*N " " " " " */ fac= 2 / N; pin= pi / N /*calculate a couple handy─dandy values*/ Dma= (b-a) / 2 /*calculate one─half of the difference.*/ Dpa= (b+a) / 2 /* " " " " sum. */ 

                    do k=0  for N;    f.k= cos( cos( pin * (k + .5) ) * Dma    +    Dpa)
                    end   /*k*/

    do j=0  for N;  z= pin * j                  /*The  LEFT(, ···) ────────►──────┐  */
    $= 0                                        /*  clause is used to align         │  */
                    do m=0  for N               /*  the non─negative values with    ↓  */
                    $= $ + f.m * cos(z*(m +.5)) /*  the     negative values.        │  */
                    end   /*m*/                 /*                     ┌─────◄──────┘  */
    cheby.j= fac * $                            /*                     ↓               */
    say right(j, length(N) +3)   " Chebyshev coefficient  is:"   left(, cheby.j >= 0),
        format(cheby.j, , 30)                   /*only show 30 decimal digits of coeff.*/
    end  /*j*/

exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ cos: procedure; parse arg x; numeric digits digits()+9; x=r2r(x); a=abs(x); numeric fuzz 5 

    if a=pi   then return -1;  if a=pi*.5 | a=pi*2  then return 0;    pit= pi/3;  z= 1
    if a=pit  then return .5;  if a=pit*2           then return -.5;  q= x*x;     _= 1
      do k=2  by 2  until p=z;  p=z;  _= -_ * q/(k*k - k);  z= z+_;   end;       return z

/*──────────────────────────────────────────────────────────────────────────────────────*/ pi: pi=3.1415926535897932384626433832795028841971693993751058209749445923078164;return pi r2r: return arg(1) // (pi() * 2) /*normalize radians ───► a unit circle.*/</lang>

output   when using the default inputs:
    0  Chebyshev coefficient  is:   1.647169475390313686961473816798
    1  Chebyshev coefficient  is:  -0.232299371615171942121038341178
    2  Chebyshev coefficient  is:  -0.053715114622047555071596203933
    3  Chebyshev coefficient  is:   0.002458235266981479866768882753
    4  Chebyshev coefficient  is:   0.000282119057434005702410217295
    5  Chebyshev coefficient  is:  -0.000007722229155810577892832847
    6  Chebyshev coefficient  is:  -5.898556452177103343296676960522E-7
    7  Chebyshev coefficient  is:   1.152142733310315857327524390711E-8
    8  Chebyshev coefficient  is:   6.596300035120132380676859918562E-10
    9  Chebyshev coefficient  is:  -1.002259170944625675156620531665E-11
output   when using the following input of:     ,   ,   20 
    0  Chebyshev coefficient  is:   1.647169475390313686961473816799
    1  Chebyshev coefficient  is:  -0.232299371615171942121038341150
    2  Chebyshev coefficient  is:  -0.053715114622047555071596207909
    3  Chebyshev coefficient  is:   0.002458235266981479866768726383
    4  Chebyshev coefficient  is:   0.000282119057434005702429677244
    5  Chebyshev coefficient  is:  -0.000007722229155810577212604038
    6  Chebyshev coefficient  is:  -5.898556452177850238987693546709E-7
    7  Chebyshev coefficient  is:   1.152142733081886533841160480101E-8
    8  Chebyshev coefficient  is:   6.596302208686010678189261798322E-10
    9  Chebyshev coefficient  is:  -1.001689435637395512060196156843E-11
   10  Chebyshev coefficient  is:  -4.586557765969596848147502951921E-13
   11  Chebyshev coefficient  is:   5.697353072301630964243748212466E-15
   12  Chebyshev coefficient  is:   2.173565878297512401879760404343E-16
   13  Chebyshev coefficient  is:  -2.284293234863639106096540267786E-18
   14  Chebyshev coefficient  is:  -7.468956910165861862760811388638E-20
   15  Chebyshev coefficient  is:   6.802288097339388765485830636223E-22
   16  Chebyshev coefficient  is:   1.945994872442404773393679283660E-23
   17  Chebyshev coefficient  is:  -1.563704507245591241161562138364E-25
   18  Chebyshev coefficient  is:  -3.976201538410589537318561880598E-27
   19  Chebyshev coefficient  is:   2.859065292763079576513213370136E-29

Ruby

<lang ruby>def mapp(x, min_x, max_x, min_to, max_to) 

   return (x - min_x) / (max_x - min_x) * (max_to - min_to) + min_to

end

def chebyshevCoef(func, min, max, coef) 

   n = coef.length

   for i in 0 .. n-1 do
       m = mapp(Math.cos(Math::PI * (i + 0.5) / n), -1, 1, min, max)
       f = func.call(m) * 2 / n

       for j in 0 .. n-1 do
           coef[j] = coef[j] + f * Math.cos(Math::PI * j * (i + 0.5) / n)
       end
   end

end

N = 10 def main 

   c = Array.new(N, 0)
   min = 0
   max = 1
   chebyshevCoef(lambda { |x| Math.cos(x) }, min, max, c)

   puts "Coefficients:"
   puts c

end

main()</lang>

Output:
Coefficients:
1.6471694753903139
-0.23229937161517178
-0.0537151146220477
0.002458235266981773
0.00028211905743405485
-7.722229156348348e-06
-5.898556456745974e-07
1.1521427756289171e-08
6.59630183807991e-10
-1.0021913854352249e-11

Scala

Output:

Best seen running in your browser either by ScalaFiddle (ES aka JavaScript, non JVM) or Scastie (remote JVM).

<lang Scala>import scala.math.{Pi, cos}

object ChebyshevCoefficients extends App { 

 final val N = 10
 final val (min, max) = (0, 1)
 val c = new Array[Double](N)

 def chebyshevCoef(func: Double => Double,
                   min: Double,
                   max: Double,
                   coef: Array[Double]): Unit =
   for (i <- coef.indices) {
     def map(x: Double,
             min_x: Double,
             max_x: Double,
             min_to: Double,
             max_to: Double): Double =
       (x - min_x) / (max_x - min_x) * (max_to - min_to) + min_to

     val m = map(cos(Pi * (i + 0.5f) / N), -1, 1, min, max)

     def f = func.apply(m) * 2 / N

     for (j <- coef.indices) coef(j) += f * cos(Pi * j * (i + 0.5f) / N)
   }

 chebyshevCoef((x: Double) => cos(x), min, max, c)
 println("Coefficients:")
 c.foreach(d => println(f"$d%23.16e"))

}</lang>

Sidef

Translation of: Raku

<lang ruby>func chebft (callback, a, b, n) { 

   var bma = (0.5 * b-a);
   var bpa = (0.5 * b+a);

   var pi_n = ((0..(n-1) »+» 0.5) »*» (Number.pi / n));
   var f = (pi_n »cos»() »*» bma »+» bpa «call« callback);
   var sums = (0..(n-1) «run« {|i| f »*« ((pi_n »*» i) »cos»()) «+» });

   sums »*» (2/n);

}

chebft(func(v){v.cos}, 0, 1, 10).each { |v| 

   say ("%+.10e" % v);

}</lang>

Output:
+1.6471694754e+00
-2.3229937162e-01
-5.3715114622e-02
+2.4582352670e-03
+2.8211905743e-04
-7.7222291558e-06
-5.8985564522e-07
+1.1521427333e-08
+6.5963000351e-10
-1.0022591709e-11

Swift

Translation of: Kotlin

<lang swift>import Foundation

typealias DFunc = (Double) -> Double

func mapRange(x: Double, min: Double, max: Double, minTo: Double, maxTo: Double) -> Double { 

 return (x - min) / (max - min) * (maxTo - minTo) + minTo

}

func chebCoeffs(fun: DFunc, n: Int, min: Double, max: Double) -> [Double] { 

 var res = [Double](repeating: 0, count: n)

 for i in 0..<n {
   let dI = Double(i)
   let dN = Double(n)
   let f = fun(mapRange(x: cos(.pi * (dI + 0.5) / dN), min: -1, max: 1, minTo: min, maxTo: max)) * 2.0 / dN

   for j in 0..<n {
     res[j] += f * cos(.pi * Double(j) * (dI + 0.5) / dN)
   }
 }

 return res

}

func chebApprox(x: Double, n: Int, min: Double, max: Double, coeffs: [Double]) -> Double { 

 var a = 1.0
 var b = mapRange(x: x, min: min, max: max, minTo: -1, maxTo: 1)
 var res = coeffs[0] / 2.0 + coeffs[1] * b
 let xx = 2 * b
 var i = 2

 while i < n {
   let c = xx * b - a
   res += coeffs[i] * c
   (a, b) = (b, c)
   i += 1
 }

 return res

}

let coeffs = chebCoeffs(fun: cos, n: 10, min: 0, max: 1)

print("Coefficients")

for coeff in coeffs { 

 print(String(format: "%+1.15g", coeff))

}

print("\nApproximations:\n x func(x) approx diff")

for i in stride(from: 0.0, through: 20, by: 1) { 

 let x = mapRange(x: i, min: 0, max: 20, minTo: 0, maxTo: 1)
 let f = cos(x)
 let approx = chebApprox(x: x, n: 10, min: 0, max: 1, coeffs: coeffs)

 print(String(format: "%1.3f  %1.8f  %1.8f  % 4.1e", x, f, approx, approx - f))

}</lang>

Output:
Coefficients
+1.64716947539031
-0.232299371615172
-0.0537151146220476
+0.00245823526698177
+0.000282119057434055
-7.72222915632059e-06
-5.89855645688475e-07
+1.15214277562892e-08
+6.59630204624673e-10
-1.0021858343201e-11

Approximations:
  x      func(x)     approx       diff
0.000  1.00000000  1.00000000   4.7e-13
0.050  0.99875026  0.99875026  -9.3e-14
0.100  0.99500417  0.99500417   4.6e-13
0.150  0.98877108  0.98877108  -4.7e-14
0.200  0.98006658  0.98006658  -4.6e-13
0.250  0.96891242  0.96891242  -2.3e-13
0.300  0.95533649  0.95533649   2.6e-13
0.350  0.93937271  0.93937271   4.6e-13
0.400  0.92106099  0.92106099   2.0e-13
0.450  0.90044710  0.90044710  -2.5e-13
0.500  0.87758256  0.87758256  -4.6e-13
0.550  0.85252452  0.85252452  -2.5e-13
0.600  0.82533561  0.82533561   2.0e-13
0.650  0.79608380  0.79608380   4.5e-13
0.700  0.76484219  0.76484219   2.5e-13
0.750  0.73168887  0.73168887  -2.3e-13
0.800  0.69670671  0.69670671  -4.5e-13
0.850  0.65998315  0.65998315  -4.4e-14
0.900  0.62160997  0.62160997   4.5e-13
0.950  0.58168309  0.58168309  -9.0e-14
1.000  0.54030231  0.54030231   4.5e-13

VBScript

Translation of: Microsoft Small Basic

To run in console mode with cscript. <lang vb>' N Chebyshev coefficients for the range 0 to 1 

 Dim coef(10),cheby(10)
 pi=4*Atn(1)
 a=0: b=1: n=10
 For i=0 To n-1
   coef(i)=Cos(Cos(pi/n*(i+1/2))*(b-a)/2+(b+a)/2)
 Next
 For i=0 To n-1
   w=0
   For j=0 To n-1
     w=w+coef(j)*Cos(pi/n*i*(j+1/2))
   Next
   cheby(i)=w*2/n
   If cheby(i)<=0 Then t="" Else t=" "
   WScript.StdOut.WriteLine i&" : "&t&cheby(i)
 Next</lang>
Output:
0 :  1,64716947539031
1 : -0,232299371615172
2 : -5,37151146220477E-02
3 :  2,45823526698163E-03
4 :  2,82119057433916E-04
5 : -7,72222915563781E-06
6 : -5,89855645105608E-07
7 :  1,15214275009379E-08
8 :  6,59629917354465E-10
9 : -1,0022016549982E-11

Visual Basic .NET

Translation of: C#

<lang vbnet>Module Module1 

   Structure ChebyshevApprox
       Public ReadOnly coeffs As List(Of Double)
       Public ReadOnly domain As Tuple(Of Double, Double)

       Public Sub New(func As Func(Of Double, Double), n As Integer, domain As Tuple(Of Double, Double))
           coeffs = ChebCoef(func, n, domain)
           Me.domain = domain
       End Sub

       Public Function Eval(x As Double) As Double
           Return ChebEval(coeffs, domain, x)
       End Function
   End Structure

   Function AffineRemap(from As Tuple(Of Double, Double), x As Double, t0 As Tuple(Of Double, Double)) As Double
       Return t0.Item1 + (x - from.Item1) * (t0.Item2 - t0.Item1) / (from.Item2 - from.Item1)
   End Function

   Function ChebCoef(fVals As List(Of Double)) As List(Of Double)
       Dim n = fVals.Count
       Dim theta = Math.PI / n
       Dim retval As New List(Of Double)
       For i = 1 To n
           retval.Add(0.0)
       Next
       For i = 1 To n
           Dim ii = i - 1
           Dim f = fVals(ii) * 2.0 / n
           Dim phi = (ii + 0.5) * theta
           Dim c1 = Math.Cos(phi)
           Dim s1 = Math.Sin(phi)
           Dim c = 1.0
           Dim s = 0.0
           For j = 1 To n
               Dim jj = j - 1
               retval(jj) += f * c
               ' update c -> cos(j*phi) for next value of j
               Dim cNext = c * c1 - s * s1
               s = c * s1 + s * c1
               c = cNext
           Next
       Next
       Return retval
   End Function

   Function ChebCoef(func As Func(Of Double, Double), n As Integer, domain As Tuple(Of Double, Double)) As List(Of Double)
       Dim Remap As Func(Of Double, Double)
       Remap = Function(x As Double)
                   Return AffineRemap(Tuple.Create(-1.0, 1.0), x, domain)
               End Function
       Dim theta = Math.PI / n
       Dim fVals As New List(Of Double)
       For i = 1 To n
           fVals.Add(0.0)
       Next
       For i = 1 To n
           Dim ii = i - 1
           fVals(ii) = func(Remap(Math.Cos((ii + 0.5) * theta)))
       Next
       Return ChebCoef(fVals)
   End Function

   Function ChebEval(coef As List(Of Double), x As Double) As Double
       Dim a = 1.0
       Dim b = x
       Dim c As Double
       Dim retval = 0.5 * coef(0) + b * coef(1)
       Dim it = coef.GetEnumerator
       it.MoveNext()
       it.MoveNext()
       While it.MoveNext
           Dim pc = it.Current
           c = 2.0 * b * x - a
           retval += pc * c
           a = b
           b = c
       End While
       Return retval
   End Function

   Function ChebEval(coef As List(Of Double), domain As Tuple(Of Double, Double), x As Double) As Double
       Return ChebEval(coef, AffineRemap(domain, x, Tuple.Create(-1.0, 1.0)))
   End Function

   Sub Main()
       Dim N = 10
       Dim fApprox As New ChebyshevApprox(AddressOf Math.Cos, N, Tuple.Create(0.0, 1.0))
       Console.WriteLine("Coefficients: ")
       For Each c In fApprox.coeffs
           Console.WriteLine(vbTab + "{0: 0.00000000000000;-0.00000000000000;zero}", c)
       Next

       Console.WriteLine(vbNewLine + "Approximation:" + vbNewLine + "    x       func(x)        approx      diff")
       Dim nX = 20.0
       Dim min = 0.0
       Dim max = 1.0
       For i = 1 To nX
           Dim x = AffineRemap(Tuple.Create(0.0, nX), i, Tuple.Create(min, max))
           Dim f = Math.Cos(x)
           Dim approx = fApprox.Eval(x)
           Console.WriteLine("{0:0.000} {1:0.00000000000000} {2:0.00000000000000} {3:E}", x, f, approx, approx - f)
       Next
   End Sub

End Module</lang>

Output:
Coefficients:
         1.64716947539031
        -0.23229937161517
        -0.05371511462205
         0.00245823526698
         0.00028211905743
        -0.00000772222916
        -0.00000058985565
         0.00000001152143
         0.00000000065963
        -0.00000000001002

Approximation:
    x       func(x)        approx      diff
0.050 0.99875026039497 0.99875026039487 -9.370282E-014
0.100 0.99500416527803 0.99500416527849 4.622969E-013
0.150 0.98877107793604 0.98877107793600 -4.662937E-014
0.200 0.98006657784124 0.98006657784078 -4.604095E-013
0.250 0.96891242171065 0.96891242171041 -2.322587E-013
0.300 0.95533648912561 0.95533648912587 2.609024E-013
0.350 0.93937271284738 0.93937271284784 4.606315E-013
0.400 0.92106099400289 0.92106099400308 1.980638E-013
0.450 0.90044710235268 0.90044710235243 -2.473577E-013
0.500 0.87758256189037 0.87758256188991 -4.586331E-013
0.550 0.85252452205951 0.85252452205926 -2.461364E-013
0.600 0.82533561490968 0.82533561490988 1.961764E-013
0.650 0.79608379854906 0.79608379854951 4.536371E-013
0.700 0.76484218728449 0.76484218728474 2.553513E-013
0.750 0.73168886887382 0.73168886887359 -2.267075E-013
0.800 0.69670670934717 0.69670670934672 -4.467537E-013
0.850 0.65998314588498 0.65998314588494 -4.485301E-014
0.900 0.62160996827066 0.62160996827111 4.444223E-013
0.950 0.58168308946388 0.58168308946379 -8.992806E-014
1.000 0.54030230586814 0.54030230586859 4.468648E-013

Wren

Translation of: Kotlin
Library: Wren-fmt

<lang ecmascript>import "/fmt" for Fmt

var mapRange = Fn.new { |x, min, max, minTo, maxTo| (x - min)/(max - min)*(maxTo - minTo) + minTo }

var chebCoeffs = Fn.new { |func, n, min, max| 

   var coeffs = List.filled(n, 0)
   for (i in 0...n) {
       var f = func.call(mapRange.call((Num.pi * (i + 0.5) / n).cos, -1, 1, min, max)) * 2 / n
       for (j in 0...n) coeffs[j] = coeffs[j] + f * (Num.pi * j * (i + 0.5) / n).cos
   }
   return coeffs

}

var chebApprox = Fn.new { |x, n, min, max, coeffs| 

   if (n < 2 || coeffs.count < 2) Fiber.abort("'n' can't be less than 2.")
   var a = 1
   var b = mapRange.call(x, min, max, -1, 1)
   var res = coeffs[0]/2 + coeffs[1]*b
   var xx = 2 * b
   var i = 2
   while (i < n) {
       var c = xx*b - a
       res = res + coeffs[i]*c
       a = b
       b = c
       i = i + 1
   }
   return res

}

var n = 10 var min = 0 var max = 1 var coeffs = chebCoeffs.call(Fn.new { |x| x.cos }, n, min, max) System.print("Coefficients:") for (coeff in coeffs) Fmt.print("$0s$1.15f", (coeff >= 0) ? " " : "", coeff) System.print("\nApproximations:\n x func(x) approx diff") for (i in 0..20) { 

   var x = mapRange.call(i, 0, 20, min, max)
   var f = x.cos
   var approx = chebApprox.call(x, n, min, max, coeffs)
   var diff = approx - f
   var diffStr = diff.toString
   var e = diffStr[-4..-1]
   diffStr = diffStr[0..-5]
   diffStr = (diff >= 0) ? " " + diffStr[0..3] : diffStr[0..4]
   Fmt.print("$1.3f  $1.8f $1.8f  $s", x, f, approx, diffStr + e)

}</lang>

Output:
Coefficients:
 1.64716947539031
-0.23229937161517
-0.05371511462205
 0.00245823526698
 0.00028211905743
-0.00000772222916
-0.00000058985565
 0.00000001152143
 0.00000000065963
-0.00000000001002

Approximations:
  x      func(x)    approx       diff
0.000  1.00000000 1.00000000   4.68e-13
0.050  0.99875026 0.99875026  -9.35e-14
0.100  0.99500417 0.99500417   4.61e-13
0.150  0.98877108 0.98877108  -4.72e-14
0.200  0.98006658 0.98006658  -4.60e-13
0.250  0.96891242 0.96891242  -2.31e-13
0.300  0.95533649 0.95533649   2.61e-13
0.350  0.93937271 0.93937271   4.61e-13
0.400  0.92106099 0.92106099   1.98e-13
0.450  0.90044710 0.90044710  -2.47e-13
0.500  0.87758256 0.87758256  -4.58e-13
0.550  0.85252452 0.85252452  -2.46e-13
0.600  0.82533561 0.82533561   1.95e-13
0.650  0.79608380 0.79608380   4.52e-13
0.700  0.76484219 0.76484219   2.54e-13
0.750  0.73168887 0.73168887  -2.27e-13
0.800  0.69670671 0.69670671  -4.47e-13
0.850  0.65998315 0.65998315  -4.37e-14
0.900  0.62160997 0.62160997   4.45e-13
0.950  0.58168309 0.58168309  -8.99e-14
1.000  0.54030231 0.54030231   4.47e-13

zkl

Translation of: C
Translation of: Perl

<lang zkl>var [const] PI=(1.0).pi; fcn chebft(a,b,n,func){ 

  bma,bpa,fac := 0.5*(b - a), 0.5*(b + a), 2.0/n;
  f:=n.pump(List,'wrap(k){ (PI*(0.5 + k)/n).cos():func(_*bma + bpa) });
  n.pump(List,'wrap(j){
     fac*n.reduce('wrap(sum,k){ sum + f[k]*(PI*j*(0.5 + k)/n).cos() },0.0);
  })

} chebft(0.0,1.0,10,fcn(x){ x.cos() }).enumerate().concat("\n").println();</lang>

Output:
L(0,1.64717)
L(1,-0.232299)
L(2,-0.0537151)
L(3,0.00245824)
L(4,0.000282119)
L(5,-7.72223e-06)
L(6,-5.89856e-07)
L(7,1.15214e-08)
L(8,6.5963e-10)
L(9,-1.00219e-11)
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