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Chebyshev coefficients are the basis of polynomial approximations of functions.

Task

Write a program to generate Chebyshev coefficients.

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

11l

Translation of: Python

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)))

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

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

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

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

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

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

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

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

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.

#include <stdio.h>
#include <string.h>
#include <math.h>

#ifndef M_PI
#define M_PI 3.14159265358979323846
#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)
{
#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;
}

C#

Translation of: C++

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);
            }
        }
    }
}

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.

#include <iostream>
#include <iomanip>
#include <string>
#include <cmath>
#include <utility>
#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";
	}
}

D

This imperative code retains some of the style of the original C version.

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);
	}
}

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

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]
.

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.)

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]
}

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

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)
        }
    }
}

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]]

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
)

Calculate coefficients:

      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

Java

Partial translation of C via D

Works with: Java version 8

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);
    }
}

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

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

# Format a decimal number so that there are at least `left` characters
# to the left of the decimal point, and at most `right` characters to its right.
# No left-truncation occurs, so `left` can be specified as 0 to prevent left-padding.
# 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;

Chebyshev Coefficients

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

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

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)
# coefs
println("Coefficients:")
for x in c.c
    @printf("% .15f\n", x)
end
# 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

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

// 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)
    }
}

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

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()

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

' 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

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

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}"

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

use constant PI => 2 * atan2(1, 0);

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
}

printf "%+13.7e\n", $_ for chebft(sub{cos($_[0])}, 0, 1, 10);

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

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++

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()

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 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

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

sub chebft ( Code $func, Real \a, Real \b, Int \n ) {

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

    my @pi-n = ( ^n »+» 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';

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.

/*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.*/

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

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()

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).

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"))

}

Sidef

Translation of: Raku

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);
}

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

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))
}

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.

' 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

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#

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

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

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)
}

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

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();

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)