Chebyshev coefficients

Chebyshev coefficients are the basis of polynomial approximations of functions. Write a program to generate Chebyshev coefficients.

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.

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

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++

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

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

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

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

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

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

Perl 6

Works with: Rakudo version 2015.12

Translation of: C

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

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

   my @pi_n = ( (^$n).list »+» 0.5 ) »*» ( pi / $n );
   my @f    = ( @pi_n».cos »*» $bma »+» $bpa )».$func;
   my @sums = map { [+] @f »*« ( @pi_n »*» $_ )».cos }, ^$n;

   return @sums »*» ( 2 / $n );

}

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

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)

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() ) - 1 /*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
               if a=pit  then return .5;  if a=pit*2 then return -.5;   q=x*x;  z=1;  _=1
                     do ?=2  by 2  until p=z;  p=z;  _=-_*q/(?*(?-1));  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

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: Perl 6

<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

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

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)