Chebyshev coefficients
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
C
C99. <lang C>#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; }</lang>
C#
<lang csharp>using System; using System.Collections.Generic; using System.Linq; using System.Text; using System.Threading.Tasks;
namespace Chebyshev {
class Program { struct ChebyshevApprox { public readonly List<double> coeffs; public readonly Tuple<double, double> domain;
public ChebyshevApprox(Func<double, double> func, int n, Tuple<double, double> domain) { coeffs = ChebCoef(func, n, domain); this.domain = domain; }
public double Call(double x) { return ChebEval(coeffs, domain, x); } }
static double AffineRemap(Tuple<double, double> from, double x, Tuple<double, double> to) { return to.Item1 + (x - from.Item1) * (to.Item2 - to.Item1) / (from.Item2 - from.Item1); }
static List<double> ChebCoef(List<double> fVals) { int n = fVals.Count; double theta = Math.PI / n; List<double> retval = new List<double>(); for (int i = 0; i < n; i++) { retval.Add(0.0); } for (int ii = 0; ii < n; ii++) { double f = fVals[ii] * 2.0 / n; double phi = (ii + 0.5) * theta; double c1 = Math.Cos(phi); double s1 = Math.Sin(phi); double c = 1.0; double s = 0.0; for (int j = 0; j < n; j++) { retval[j] += f * c; // update c -> cos(j*phi) for next value of j double cNext = c * c1 - s * s1; s = c * s1 + s * c1; c = cNext; } } return retval; }
static List<double> ChebCoef(Func<double, double> func, int n, Tuple<double, double> domain) { double remap(double x) { return AffineRemap(new Tuple<double, double>(-1.0, 1.0), x, domain); } double theta = Math.PI / n; List<double> fVals = new List<double>(); for (int i = 0; i < n; i++) { fVals.Add(0.0); } for (int ii = 0; ii < n; ii++) { fVals[ii] = func(remap(Math.Cos((ii + 0.5) * theta))); } return ChebCoef(fVals); }
static double ChebEval(List<double> coef, double x) { double a = 1.0; double b = x; double c; double retval = 0.5 * coef[0] + b * coef[1]; var it = coef.GetEnumerator(); it.MoveNext(); it.MoveNext(); while (it.MoveNext()) { double pc = it.Current; c = 2.0 * b * x - a; retval += pc * c; a = b; b = c; } return retval; }
static double ChebEval(List<double> coef, Tuple<double, double> domain, double x) { return ChebEval(coef, AffineRemap(domain, x, new Tuple<double, double>(-1.0, 1.0))); }
static void Main() { const int N = 10; ChebyshevApprox fApprox = new ChebyshevApprox(Math.Cos, N, new Tuple<double, double>(0.0, 1.0)); Console.WriteLine("Coefficients: "); foreach (var c in fApprox.coeffs) { Console.WriteLine("\t{0: 0.00000000000000;-0.00000000000000;zero}", c); }
Console.WriteLine("\nApproximation:\n x func(x) approx diff"); const int nX = 20; const int min = 0; const int max = 1; for (int i = 0; i < nX; i++) { double x = AffineRemap(new Tuple<double, double>(0, nX), i, new Tuple<double, double>(min, max)); double f = Math.Cos(x); double approx = fApprox.Call(x); Console.WriteLine("{0:0.000} {1:0.00000000000000} {2:0.00000000000000} {3:E}", x, f, approx, approx - f); } } }
}</lang>
- Output:
Coefficients: 1.64716947539031 -0.23229937161517 -0.05371511462205 0.00245823526698 0.00028211905743 -0.00000772222916 -0.00000058985565 0.00000001152143 0.00000000065963 -0.00000000001002 Approximation: x func(x) approx diff 0.000 1.00000000000000 1.00000000000047 4.689582E-013 0.050 0.99875026039497 0.99875026039487 -9.370282E-014 0.100 0.99500416527803 0.99500416527849 4.622969E-013 0.150 0.98877107793604 0.98877107793600 -4.662937E-014 0.200 0.98006657784124 0.98006657784078 -4.604095E-013 0.250 0.96891242171065 0.96891242171041 -2.322587E-013 0.300 0.95533648912561 0.95533648912587 2.609024E-013 0.350 0.93937271284738 0.93937271284784 4.606315E-013 0.400 0.92106099400289 0.92106099400308 1.980638E-013 0.450 0.90044710235268 0.90044710235243 -2.473577E-013 0.500 0.87758256189037 0.87758256188991 -4.586331E-013 0.550 0.85252452205951 0.85252452205926 -2.461364E-013 0.600 0.82533561490968 0.82533561490988 1.961764E-013 0.650 0.79608379854906 0.79608379854951 4.536371E-013 0.700 0.76484218728449 0.76484218728474 2.553513E-013 0.750 0.73168886887382 0.73168886887359 -2.267075E-013 0.800 0.69670670934717 0.69670670934672 -4.467537E-013 0.850 0.65998314588498 0.65998314588494 -4.485301E-014 0.900 0.62160996827066 0.62160996827111 4.444223E-013 0.950 0.58168308946388 0.58168308946379 -8.992806E-014
C++
Based on the C99 implementation above. The main improvement is that, because C++ containers handle memory for us, we can use a more functional style.
The two overloads of cheb_coef show a useful idiom for working with C++ templates; the non-template code, which does all the mathematical work, can be placed in a source file so that it is compiled only once (reducing code bloat from repeating substantial blocks of code). The template function is a minimal wrapper to call the non-template implementation.
The wrapper class ChebyshevApprox_ supports very terse user code.
<lang CPP>
- 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"; } } </lang>
D
This imperative code retains some of the style of the original C version. <lang d>import std.math: PI, cos;
/// Map x from range [min, max] to [min_to, max_to]. real map(in real x, in real min_x, in real max_x, in real min_to, in real max_to) pure nothrow @safe @nogc { return (x - min_x) / (max_x - min_x) * (max_to - min_to) + min_to; }
void chebyshevCoef(size_t N)(in real function(in real) pure nothrow @safe @nogc func,
in real min, in real max, ref real[N] coef)
pure nothrow @safe @nogc {
coef[] = 0.0; foreach (immutable i; 0 .. N) { immutable f = func(map(cos(PI * (i + 0.5f) / N), -1, 1, min, max)) * 2 / N; foreach (immutable j, ref cj; coef) cj += f * cos(PI * j * (i + 0.5f) / N);
} }
/// f(x) = sum_{k=0}^{n - 1} c_k T_k(x) - c_0/2
real chebyshevApprox(size_t N)(in real x, in real min, in real max, in ref real[N] coef)
pure nothrow @safe @nogc if (N >= 2) {
real a = 1.0L, b = map(x, min, max, -1, 1), result = coef[0] / 2 + coef[1] * b;
immutable x2 = 2 * b;
foreach (immutable ci; coef[2 .. $]) {
// T_{n+1} = 2x T_n - T_{n-1}
immutable c = x2 * b - a; result += ci * c; a = b; b = c; }
return result;
}
void main() @safe {
import std.stdio: writeln, writefln; enum uint N = 10;
real[N] c;
real min = 0, max = 1; static real test(in real x) pure nothrow @safe @nogc { return x.cos; }
chebyshevCoef(&test, min, max, c);
writefln("Coefficients:\n%( %+2.25g\n%)", c);
enum nX = 20;
writeln("\nApproximation:\n x func(x) approx diff");
foreach (immutable i; 0 .. nX) { immutable x = map(i, 0, nX, min, max);
immutable f = test(x); immutable approx = chebyshevApprox(x, min, max, c);
writefln("%1.3f % 10.10f % 10.10f % 4.2e", x, f, approx, approx - f); } }</lang>
- Output:
Coefficients: +1.6471694753903136868 -0.23229937161517194216 -0.053715114622047555044 +0.0024582352669814797779 +0.00028211905743400579387 -7.7222291558103533853e-06 -5.898556452178771968e-07 +1.1521427332860788728e-08 +6.5963000382704222411e-10 -1.0022591914390921452e-11 Approximation: x func(x) approx diff 0.000 1.00000000000000000000 1.00000000000046961190 4.70e-13 0.050 0.99875026039496624654 0.99875026039487216781 -9.41e-14 0.100 0.99500416527802576609 0.99500416527848803832 4.62e-13 0.150 0.98877107793604228670 0.98877107793599569749 -4.66e-14 0.200 0.98006657784124163110 0.98006657784078136889 -4.60e-13 0.250 0.96891242171064478408 0.96891242171041249593 -2.32e-13 0.300 0.95533648912560601967 0.95533648912586667367 2.61e-13 0.350 0.93937271284737892005 0.93937271284783928305 4.60e-13 0.400 0.92106099400288508277 0.92106099400308274515 1.98e-13 0.450 0.90044710235267692169 0.90044710235242891114 -2.48e-13 0.500 0.87758256189037271615 0.87758256188991362600 -4.59e-13 0.550 0.85252452205950574283 0.85252452205925896211 -2.47e-13 0.600 0.82533561490967829723 0.82533561490987400509 1.96e-13 0.650 0.79608379854905582896 0.79608379854950937939 4.54e-13 0.700 0.76484218728448842626 0.76484218728474395029 2.56e-13 0.750 0.73168886887382088633 0.73168886887359430061 -2.27e-13 0.800 0.69670670934716542091 0.69670670934671868322 -4.47e-13 0.850 0.65998314588498217039 0.65998314588493717370 -4.50e-14 0.900 0.62160996827066445648 0.62160996827110870299 4.44e-13 0.950 0.58168308946388349416 0.58168308946379353278 -9.00e-14
The same code, with N = 16:
Coefficients: +1.6471694753903136868 -0.23229937161517194214 -0.053715114622047555035 +0.0024582352669814797982 +0.00028211905743400571932 -7.722229155810705751e-06 -5.898556452177348953e-07 +1.1521427330794028337e-08 +6.5963022091481034181e-10 -1.0016894235462866363e-11 -4.5865582517937500406e-13 +5.6974586994888026802e-15 +2.1752822525027137867e-16 -2.3140940118987485263e-18 -1.0333801956502464137e-19 +2.5410988417629010172e-20 Approximation: x func(x) approx diff 0.000 1.00000000000000000000 1.00000000000000000030 3.25e-19 0.050 0.99875026039496624654 0.99875026039496624646 -1.08e-19 0.100 0.99500416527802576609 0.99500416527802576557 -5.42e-19 0.150 0.98877107793604228670 0.98877107793604228636 -3.79e-19 0.200 0.98006657784124163110 0.98006657784124163127 1.08e-19 0.250 0.96891242171064478408 0.96891242171064478451 3.79e-19 0.300 0.95533648912560601967 0.95533648912560601967 0.00e+00 0.350 0.93937271284737892005 0.93937271284737891962 -3.79e-19 0.400 0.92106099400288508277 0.92106099400288508260 -2.17e-19 0.450 0.90044710235267692169 0.90044710235267692169 5.42e-20 0.500 0.87758256189037271615 0.87758256189037271632 2.17e-19 0.550 0.85252452205950574283 0.85252452205950574274 -5.42e-20 0.600 0.82533561490967829723 0.82533561490967829697 -2.17e-19 0.650 0.79608379854905582896 0.79608379854905582861 -3.25e-19 0.700 0.76484218728448842626 0.76484218728448842630 5.42e-20 0.750 0.73168886887382088633 0.73168886887382088637 5.42e-20 0.800 0.69670670934716542091 0.69670670934716542087 -5.42e-20 0.850 0.65998314588498217039 0.65998314588498217022 -1.63e-19 0.900 0.62160996827066445648 0.62160996827066445674 2.71e-19 0.950 0.58168308946388349416 0.58168308946388349403 -1.63e-19
EasyLang
<lang>set_numfmt 18 15 a = 0 b = 1 n = 10 len coef[] n len cheby[] n for i range n
coef[i] = cos (180 / pi * (cos (180 / n * (i + 1 / 2)) * (b - a) / 2 + (b + a) / 2))
. for i range n
w = 0 for j range n w += coef[j] * cos (180 / n * i * (j + 1 / 2)) . cheby[i] = w * 2 / n print cheby[i]
.</lang>
Go
Wikipedia gives a formula for coefficients in a section "Example 1". Read past the bit about the inner product to where it gives the technique based on the discrete orthogonality condition. The N of the WP formulas is the parameter nNodes in the code here. It is not necessarily the same as n, the number of polynomial coefficients, the parameter nCoeff here.
The evaluation method is the Clenshaw algorithm.
Two variances here from the WP presentation and most mathematical presentations follow other examples on this page and so keep output directly comparable. One variance is that the Kronecker delta factor is dropped, which has the effect of doubling the first coefficient. This simplifies both coefficient generation and polynomial evaluation. A further variance is that there is no scaling for the range of function values. The result is that coefficients are not necessarily bounded by 1 (2 for the first coefficient) but by the maximum function value over the argument range from min to max (or twice that for the first coefficient.) <lang go>package main
import (
"fmt" "math"
)
type cheb struct {
c []float64 min, max float64
}
func main() {
fn := math.Cos c := newCheb(0, 1, 10, 10, fn) fmt.Println("coefficients:") for _, c := range c.c { fmt.Printf("% .15f\n", c) } fmt.Println("\nx computed approximated computed-approx") const n = 10 for i := 0.; i <= n; i++ { x := (c.min*(n-i) + c.max*i) / n computed := fn(x) approx := c.eval(x) fmt.Printf("%.1f %12.8f %12.8f % .3e\n", x, computed, approx, computed-approx) }
}
func newCheb(min, max float64, nCoeff, nNodes int, fn func(float64) float64) *cheb {
c := &cheb{ c: make([]float64, nCoeff), min: min, max: max, } f := make([]float64, nNodes) p := make([]float64, nNodes) z := .5 * (max + min) r := .5 * (max - min) for k := 0; k < nNodes; k++ { p[k] = math.Pi * (float64(k) + .5) / float64(nNodes) f[k] = fn(z + math.Cos(p[k])*r) } n2 := 2 / float64(nNodes) for j := 0; j < nCoeff; j++ { sum := 0. for k := 0; k < nNodes; k++ { sum += f[k] * math.Cos(float64(j)*p[k]) } c.c[j] = sum * n2 } return c
}
func (c *cheb) eval(x float64) float64 {
x1 := (2*x - c.min - c.max) / (c.max - c.min) x2 := 2 * x1 var s, t float64 for j := len(c.c) - 1; j >= 1; j-- { t, s = x2*t-s+c.c[j], t } return x1*t - s + .5*c.c[0]
}</lang>
- Output:
coefficients: 1.647169475390314 -0.232299371615172 -0.053715114622048 0.002458235266982 0.000282119057434 -0.000007722229156 -0.000000589855645 0.000000011521427 0.000000000659630 -0.000000000010022 x computed approximated computed-approx 0.0 1.00000000 1.00000000 -4.685e-13 0.1 0.99500417 0.99500417 -4.620e-13 0.2 0.98006658 0.98006658 4.601e-13 0.3 0.95533649 0.95533649 -2.607e-13 0.4 0.92106099 0.92106099 -1.972e-13 0.5 0.87758256 0.87758256 4.587e-13 0.6 0.82533561 0.82533561 -1.965e-13 0.7 0.76484219 0.76484219 -2.552e-13 0.8 0.69670671 0.69670671 4.470e-13 0.9 0.62160997 0.62160997 -4.449e-13 1.0 0.54030231 0.54030231 -4.476e-13
Groovy
<lang groovy>class ChebyshevCoefficients {
static double map(double x, double min_x, double max_x, double min_to, double max_to) { return (x - min_x) / (max_x - min_x) * (max_to - min_to) + min_to }
static void chebyshevCoef(Closure<Double> func, double min, double max, double[] coef) { final int N = coef.length for (int i = 0; i < N; i++) { double m = map(Math.cos(Math.PI * (i + 0.5f) / N), -1, 1, min, max) double f = func(m) * 2 / N
for (int j = 0; j < N; j++) { coef[j] += f * Math.cos(Math.PI * j * (i + 0.5f) / N) } } }
static void main(String[] args) { final int N = 10 double[] c = new double[N] double min = 0, max = 1 chebyshevCoef(Math.&cos, min, max, c)
println("Coefficients:") for (double d : c) { println(d) } }
}</lang>
- Output:
Coefficients: 1.6471694753903139 -0.23229937161517178 -0.0537151146220477 0.002458235266981773 2.8211905743405485E-4 -7.722229156320592E-6 -5.898556456745974E-7 1.1521427770166959E-8 6.59630183807991E-10 -1.0021913854352249E-11
J
From 'J for C Programmers: Calculating Chebyshev Coefficients [[1]] <lang J> chebft =: adverb define
f =. u 0.5 * (+/y) - (-/y) * 2 o. o. (0.5 + i. x) % x
(2 % x) * +/ f * 2 o. o. (0.5 + i. x) *"0 1 (i. x) % x
) </lang> Calculate coefficients: <lang J>
10 (2&o.) chebft 0 1
1.64717 _0.232299 _0.0537151 0.00245824 0.000282119 _7.72223e_6 _5.89856e_7 1.15214e_8 6.59629e_10 _1.00227e_11 </lang>
Java
Partial translation of C via D
<lang java>import static java.lang.Math.*; import java.util.function.Function;
public class ChebyshevCoefficients {
static double map(double x, double min_x, double max_x, double min_to, double max_to) { return (x - min_x) / (max_x - min_x) * (max_to - min_to) + min_to; }
static void chebyshevCoef(Function<Double, Double> func, double min, double max, double[] coef) {
int N = coef.length;
for (int i = 0; i < N; i++) {
double m = map(cos(PI * (i + 0.5f) / N), -1, 1, min, max); double f = func.apply(m) * 2 / N;
for (int j = 0; j < N; j++) { coef[j] += f * cos(PI * j * (i + 0.5f) / N); } } }
public static void main(String[] args) { final int N = 10; double[] c = new double[N]; double min = 0, max = 1; chebyshevCoef(x -> cos(x), min, max, c);
System.out.println("Coefficients:"); for (double d : c) System.out.println(d); }
}</lang>
Coefficients: 1.6471694753903139 -0.23229937161517178 -0.0537151146220477 0.002458235266981773 2.8211905743405485E-4 -7.722229156320592E-6 -5.898556456745974E-7 1.1521427770166959E-8 6.59630183807991E-10 -1.0021913854352249E-11
jq
Adapted from Wren
Works with gojq, the Go implementation of jq
Preliminaries <lang jq>def lpad($len): tostring | ($len - length) as $l | (" " * $l) + .; def rpad($len; $fill): tostring | ($len - length) as $l | . + ($fill * $l)[:$l];
- 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;</lang>
Chebyshev Coefficients <lang jq>def mapRange($x; $min; $max; $minTo; $maxTo):
(($x - $min)/($max - $min))*($maxTo - $minTo) + $minTo;
def chebCoeffs(func; n; min; max):
(1 | atan * 4) as $pi | reduce range(0;n) as $i ([]; # coeffs ((mapRange( ($pi * ($i + 0.5) / n)|cos; -1; 1; min; max) | func) * 2 / n) as $f | reduce range(0;n) as $j (.; .[$j] += $f * ($pi * $j * (($i + 0.5) / n)|cos)) );
def chebApprox(x; n; min; max; coeffs):
if n < 2 or (coeffs|length) < 2 then "'n' can't be less than 2." | error else { a: 1, b: mapRange(x; min; max; -1; 1) } |.res = coeffs[0]/2 + coeffs[1]*.b |.xx = 2 * .b | reduce range(2;n) as $i (.; (.xx * .b - .a) as $c | .res = (.res + (coeffs[$i]*$c)) | .a = .b | .b = $c) | .res end ;
def task:
[10, 0, 1] as [$n, $min, $max] | chebCoeffs(cos; $n; $min; $max) as $coeffs | "Coefficients:", ($coeffs[]|pp(2;14)), "\nApproximations:\n x func(x) approx diff", (range(0;21) as $i | mapRange($i; 0; 20; $min; $max) as $x | ($x|cos) as $f | chebApprox($x; $n; $min; $max; $coeffs) as $approx | ($approx - $f) as $diff | [ ($x|pp(0;3)|rpad( 4;"0")), ($f|pp(0;8)|rpad(10;"0")),
($approx|pp(0;8)),
($diff |pp(2;2)) ] | join(" ") );
task</lang>
- Output:
Coefficients: 1.64716947539031 -0.23229937161517 -0.05371511462204 0.00245823526698 0.00028211905743 -7.72222915562670e-06 -5.89855645688475e-07 1.15214280338449e-08 6.59629580124221e-10 -1.00220526322303e-11 Approximations: x func(x) approx diff 0.00 1.00000000 1.00000000 4.66e-13 0.05 0.99875026 0.99875026 -9.21e-14 0.10 0.99500416 0.99500416 4.62e-13 0.15 0.98877107 0.98877107 -4.74e-14 0.20 0.98006657 0.98006657 -4.60e-13 0.25 0.96891242 0.96891242 -2.32e-13 0.30 0.95533648 0.95533648 2.61e-13 0.35 0.93937271 0.93937271 4.60e-13 0.40 0.92106099 0.92106099 1.98e-13 0.45 0.90044710 0.90044710 -2.47e-13 0.50 0.87758256 0.87758256 -4.59e-13 0.55 0.85252452 0.85252452 -2.46e-13 0.60 0.82533561 0.82533561 1.95e-13 0.65 0.79608379 0.79608379 4.53e-13 0.70 0.76484218 0.76484218 2.55e-13 0.75 0.73168886 0.73168886 -2.26e-13 0.80 0.69670670 0.69670670 -4.46e-13 0.85 0.65998314 0.65998314 -4.45e-14 0.90 0.62160996 0.62160996 4.44e-13 0.95 0.58168308 0.58168308 -9.01e-14 1.00 0.54030230 0.54030230 4.47e-13
Julia
<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)
- 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</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
<lang scala>// version 1.1.2
typealias DFunc = (Double) -> Double
fun mapRange(x: Double, min: Double, max: Double, minTo: Double, maxTo:Double): Double {
return (x - min) / (max - min) * (maxTo - minTo) + minTo
}
fun chebCoeffs(func: DFunc, n: Int, min: Double, max: Double): DoubleArray {
val coeffs = DoubleArray(n) for (i in 0 until n) { val f = func(mapRange(Math.cos(Math.PI * (i + 0.5) / n), -1.0, 1.0, min, max)) * 2.0 / n for (j in 0 until n) coeffs[j] += f * Math.cos(Math.PI * j * (i + 0.5) / n) } return coeffs
}
fun chebApprox(x: Double, n: Int, min: Double, max: Double, coeffs: DoubleArray): Double {
require(n >= 2 && coeffs.size >= 2) var a = 1.0 var b = mapRange(x, min, max, -1.0, 1.0) var res = coeffs[0] / 2.0 + coeffs[1] * b val xx = 2 * b var i = 2 while (i < n) { val c = xx * b - a res += coeffs[i] * c a = b b = c i++ } return res
}
fun main(args: Array<String>) {
val n = 10 val min = 0.0 val max = 1.0 val coeffs = chebCoeffs(Math::cos, n, min, max) println("Coefficients:") for (coeff in coeffs) println("%+1.15g".format(coeff)) println("\nApproximations:\n x func(x) approx diff") for (i in 0..20) { val x = mapRange(i.toDouble(), 0.0, 20.0, min, max) val f = Math.cos(x) val approx = chebApprox(x, n, min, max, coeffs) System.out.printf("%1.3f %1.8f %1.8f % 4.1e\n", x, f, approx, approx - f) }
}</lang>
- Output:
Coefficients: +1.64716947539031 -0.232299371615172 -0.0537151146220477 +0.00245823526698177 +0.000282119057434055 -7.72222915632059e-06 -5.89855645674597e-07 +1.15214277701670e-08 +6.59630183807991e-10 -1.00219138543522e-11 Approximations: x func(x) approx diff 0.000 1.00000000 1.00000000 4.7e-13 0.050 0.99875026 0.99875026 -9.4e-14 0.100 0.99500417 0.99500417 4.6e-13 0.150 0.98877108 0.98877108 -4.7e-14 0.200 0.98006658 0.98006658 -4.6e-13 0.250 0.96891242 0.96891242 -2.3e-13 0.300 0.95533649 0.95533649 2.6e-13 0.350 0.93937271 0.93937271 4.6e-13 0.400 0.92106099 0.92106099 2.0e-13 0.450 0.90044710 0.90044710 -2.5e-13 0.500 0.87758256 0.87758256 -4.6e-13 0.550 0.85252452 0.85252452 -2.5e-13 0.600 0.82533561 0.82533561 2.0e-13 0.650 0.79608380 0.79608380 4.5e-13 0.700 0.76484219 0.76484219 2.5e-13 0.750 0.73168887 0.73168887 -2.3e-13 0.800 0.69670671 0.69670671 -4.5e-13 0.850 0.65998315 0.65998315 -4.4e-14 0.900 0.62160997 0.62160997 4.5e-13 0.950 0.58168309 0.58168309 -9.0e-14 1.000 0.54030231 0.54030231 4.5e-13
Lua
<lang lua>function map(x, min_x, max_x, min_to, max_to)
return (x - min_x) / (max_x - min_x) * (max_to - min_to) + min_to
end
function chebyshevCoef(func, minn, maxx, coef)
local N = table.getn(coef) for j=1,N do local i = j - 1 local m = map(math.cos(math.pi * (i + 0.5) / N), -1, 1, minn, maxx) local f = func(m) * 2 / N
for k=1,N do local p = k -1 coef[k] = coef[k] + f * math.cos(math.pi * p * (i + 0.5) / N) end end
end
function main()
local N = 10 local c = {} local minn = 0.0 local maxx = 1.0
for i=1,N do table.insert(c, 0) end
chebyshevCoef(function (x) return math.cos(x) end, minn, maxx, c)
print("Coefficients:") for i,d in pairs(c) do print(d) end
end
main() </lang>
- Output:
Coefficients: 1.6471694753903 -0.23229937161517 -0.053715114622048 0.0024582352669818 0.00028211905743405 -7.7222291563483e-006 -5.898556456746e-007 1.1521427756289e-008 6.5963018380799e-010 -1.0021913854352e-011
Microsoft Small Basic
<lang smallbasic>' N Chebyshev coefficients for the range 0 to 1 - 18/07/2018
pi=Math.pi a=0 b=1 n=10 For i=0 To n-1 coef[i]=Math.cos(Math.cos(pi/n*(i+1/2))*(b-a)/2+(b+a)/2) EndFor For i=0 To n-1 w=0 For j=0 To n-1 w=w+coef[j]*Math.cos(pi/n*i*(j+1/2)) EndFor cheby[i]=w*2/n t=" " If cheby[i]<=0 Then t="" EndIf TextWindow.WriteLine(i+" : "+t+cheby[i]) EndFor</lang>
- Output:
0 : 1,6471694753903136 1 : -0,2322993716151700684187787635 2 : -0,0537151146220494010749946688 3 : 0,0024582352669837594966069584 4 : 0,0002821190574317389206759282 5 : -0,0000077222291539069653168878 6 : -0,0000005898556481086082412444 7 : 0,0000000115214300591398939205 8 : 0,0000000006596278553822696656 9 : -0,0000000000100189955816952521
Nim
<lang Nim>import lenientops, math, strformat, sugar
type Cheb = object
c: seq[float] min, max: float
func initCheb(min, max: float; nCoeff, nNodes: int; fn: float -> float): Cheb =
result = Cheb(c: newSeq[float](nCoeff), min: min, max: max) var f, p = newSeq[float](nNodes) let z = 0.5 * (max + min) let r = 0.5 * (max - min) for k in 0..<nNodes: p[k] = PI * (k + 0.5) / nNodes f[k] = fn(z + cos(p[k]) * r)
let n2 = 2 / nNodes for j in 0..<nCoeff: var sum = 0.0 for k in 0..<nNodes: sum += f[k] * cos(j * p[k]) result.c[j] = sum * n2
func eval(cheb: Cheb; x: float): float =
let x1 = (2 * x - cheb.min - cheb.max) / (cheb.max - cheb.min) let x2 = 2 * x1 var s, t: float for j in countdown(cheb.c.high, 1): s = x2 * t - s + cheb.c[j] swap s, t result = x1 * t - s + 0.5 * cheb.c[0]
when isMainModule:
let fn: float -> float = cos let cheb = initCheb(0, 1, 10, 10, fn) echo "Coefficients:" for c in cheb.c: echo &"{c: .15f}"
echo "\n x computed approximated computed-approx" const N = 10 for i in 0..N: let x = (cheb.min * (N - i) + cheb.max * i) / N let computed = fn(x) let approx = cheb.eval(x) echo &"{x:.1f} {computed:12.8f} {approx:12.8f} {computed-approx: .3e}"</lang>
- Output:
Coefficients: 1.647169475390314 -0.232299371615172 -0.053715114622048 0.002458235266981 0.000282119057434 -0.000007722229156 -0.000000589855645 0.000000011521427 0.000000000659630 -0.000000000010022 x computed approximated computed-approx 0.0 1.00000000 1.00000000 -4.685e-13 0.1 0.99500417 0.99500417 -4.620e-13 0.2 0.98006658 0.98006658 4.601e-13 0.3 0.95533649 0.95533649 -2.605e-13 0.4 0.92106099 0.92106099 -1.970e-13 0.5 0.87758256 0.87758256 4.586e-13 0.6 0.82533561 0.82533561 -1.967e-13 0.7 0.76484219 0.76484219 -2.551e-13 0.8 0.69670671 0.69670671 4.470e-13 0.9 0.62160997 0.62160997 -4.450e-13 1.0 0.54030231 0.54030231 -4.476e-13
Perl
<lang perl>use constant PI => 3.141592653589793;
sub chebft {
my($func, $a, $b, $n) = @_; my($bma, $bpa) = ( 0.5*($b-$a), 0.5*($b+$a) );
my @pin = map { ($_ + 0.5) * (PI/$n) } 0..$n-1; my @f = map { $func->( cos($_) * $bma + $bpa ) } @pin; my @c = (0) x $n; for my $j (0 .. $n-1) { $c[$j] += $f[$_] * cos($j * $pin[$_]) for 0..$n-1; $c[$j] *= (2.0/$n); } @c;
}
print "$_\n" for chebft(sub{cos($_[0])}, 0, 1, 10);</lang>
- Output:
1.64716947539031 -0.232299371615172 -0.0537151146220477 0.00245823526698163 0.000282119057433938 -7.72222915566001e-06 -5.89855645105608e-07 1.15214274787334e-08 6.59629917354465e-10 -1.00219943455215e-11
Phix
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
<lang python>import math
def test_func(x):
return math.cos(x)
def mapper(x, min_x, max_x, min_to, max_to):
return (x - min_x) / (max_x - min_x) * (max_to - min_to) + min_to
def cheb_coef(func, n, min, max):
coef = [0.0] * n for i in xrange(n): f = func(mapper(math.cos(math.pi * (i + 0.5) / n), -1, 1, min, max)) * 2 / n for j in xrange(n): coef[j] += f * math.cos(math.pi * j * (i + 0.5) / n) return coef
def cheb_approx(x, n, min, max, coef):
a = 1 b = mapper(x, min, max, -1, 1) c = float('nan') res = coef[0] / 2 + coef[1] * b
x = 2 * b i = 2 while i < n: c = x * b - a res = res + coef[i] * c (a, b) = (b, c) i += 1
return res
def main():
N = 10 min = 0 max = 1 c = cheb_coef(test_func, N, min, max)
print "Coefficients:" for i in xrange(N): print " % lg" % c[i]
print "\n\nApproximation:\n x func(x) approx diff" for i in xrange(20): x = mapper(i, 0.0, 20.0, min, max) f = test_func(x) approx = cheb_approx(x, N, min, max, c) print "%1.3f %10.10f %10.10f % 4.2e" % (x, f, approx, approx - f)
return None
main()</lang>
- Output:
Coefficients: 1.64717 -0.232299 -0.0537151 0.00245824 0.000282119 -7.72223e-06 -5.89856e-07 1.15214e-08 6.5963e-10 -1.00219e-11 Approximation: x func(x) approx diff 0.000 1.0000000000 1.0000000000 4.68e-13 0.050 0.9987502604 0.9987502604 -9.36e-14 0.100 0.9950041653 0.9950041653 4.62e-13 0.150 0.9887710779 0.9887710779 -4.73e-14 0.200 0.9800665778 0.9800665778 -4.60e-13 0.250 0.9689124217 0.9689124217 -2.32e-13 0.300 0.9553364891 0.9553364891 2.62e-13 0.350 0.9393727128 0.9393727128 4.61e-13 0.400 0.9210609940 0.9210609940 1.98e-13 0.450 0.9004471024 0.9004471024 -2.47e-13 0.500 0.8775825619 0.8775825619 -4.58e-13 0.550 0.8525245221 0.8525245221 -2.46e-13 0.600 0.8253356149 0.8253356149 1.96e-13 0.650 0.7960837985 0.7960837985 4.53e-13 0.700 0.7648421873 0.7648421873 2.54e-13 0.750 0.7316888689 0.7316888689 -2.28e-13 0.800 0.6967067093 0.6967067093 -4.47e-13 0.850 0.6599831459 0.6599831459 -4.37e-14 0.900 0.6216099683 0.6216099683 4.46e-13 0.950 0.5816830895 0.5816830895 -8.99e-14
Racket
<lang racket>#lang typed/racket (: chebft (Real Real Nonnegative-Integer (Real -> Real) -> (Vectorof Real))) (define (chebft a b n func)
(define b-a/2 (/ (- b a) 2)) (define b+a/2 (/ (+ b a) 2)) (define pi/n (/ pi n)) (define fac (/ 2 n))
(define f (for/vector : (Vectorof Real) ((k : Nonnegative-Integer (in-range n))) (define y (cos (* pi/n (+ k 1/2)))) (func (+ (* y b-a/2) b+a/2))))
(for/vector : (Vectorof Real) ((j : Nonnegative-Integer (in-range n))) (define s (for/sum : Real ((k : Nonnegative-Integer (in-range n))) (* (vector-ref f k) (cos (* pi/n j (+ k 1/2)))))) (* fac s)))
(module+ test
(chebft 0 1 10 cos))
- Tim Brown 2015</lang>
- Output:
'#(1.6471694753903137 -0.2322993716151719 -0.05371511462204768 0.0024582352669816343 0.0002821190574339161 -7.722229155637806e-006 -5.898556451056081e-007 1.1521427500937876e-008 6.596299173544651e-010 -1.0022016549982027e-011)
Raku
(formerly Perl 6)
<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
REXX
This REXX program is a translation of the C program plus added optimizations.
Pafnuty Lvovich Chebysheff: Chebyshev [English transliteration] Chebysheff [ " " ] Chebyshov [ " " ] Tchebychev [French " ] Tchebysheff [ " " ] Tschebyschow [German " ] Tschebyschev [ " " ] Tschebyschef [ " " ] Tschebyscheff [ " " ]
The numeric precision is dependent on the number of decimal digits specified in the value of pi.
<lang rexx>/*REXX program calculates N Chebyshev coefficients for the range 0 ──► 1 (inclusive)*/
numeric digits length( pi() ) - length(.) /*DIGITS default is nine, but use 71. */
parse arg a b N . /*obtain optional arguments from the CL*/
if a== | a=="," then a= 0 /*A not specified? Then use default.*/
if b== | b=="," then b= 1 /*B " " " " " */
if N== | N=="," then N= 10 /*N " " " " " */
fac= 2 / N; pin= pi / N /*calculate a couple handy─dandy values*/
Dma= (b-a) / 2 /*calculate one─half of the difference.*/
Dpa= (b+a) / 2 /* " " " " sum. */
do k=0 for N; f.k= cos( cos( pin * (k + .5) ) * Dma + Dpa) end /*k*/
do j=0 for N; z= pin * j /*The LEFT(, ···) ────────►──────┐ */ $= 0 /* clause is used to align │ */ do m=0 for N /* the non─negative values with ↓ */ $= $ + f.m * cos(z*(m +.5)) /* the negative values. │ */ end /*m*/ /* ┌─────◄──────┘ */ cheby.j= fac * $ /* ↓ */ say right(j, length(N) +3) " Chebyshev coefficient is:" left(, cheby.j >= 0), format(cheby.j, , 30) /*only show 30 decimal digits of coeff.*/ end /*j*/
exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ cos: procedure; parse arg x; numeric digits digits()+9; x=r2r(x); a=abs(x); numeric fuzz 5
if a=pi then return -1; if a=pi*.5 | a=pi*2 then return 0; pit= pi/3; z= 1 if a=pit then return .5; if a=pit*2 then return -.5; q= x*x; _= 1 do k=2 by 2 until p=z; p=z; _= -_ * q/(k*k - k); z= z+_; end; return z
/*──────────────────────────────────────────────────────────────────────────────────────*/ pi: pi=3.1415926535897932384626433832795028841971693993751058209749445923078164;return pi r2r: return arg(1) // (pi() * 2) /*normalize radians ───► a unit circle.*/</lang>
- output when using the default inputs:
0 Chebyshev coefficient is: 1.647169475390313686961473816798 1 Chebyshev coefficient is: -0.232299371615171942121038341178 2 Chebyshev coefficient is: -0.053715114622047555071596203933 3 Chebyshev coefficient is: 0.002458235266981479866768882753 4 Chebyshev coefficient is: 0.000282119057434005702410217295 5 Chebyshev coefficient is: -0.000007722229155810577892832847 6 Chebyshev coefficient is: -5.898556452177103343296676960522E-7 7 Chebyshev coefficient is: 1.152142733310315857327524390711E-8 8 Chebyshev coefficient is: 6.596300035120132380676859918562E-10 9 Chebyshev coefficient is: -1.002259170944625675156620531665E-11
- output when using the following input of: , , 20
0 Chebyshev coefficient is: 1.647169475390313686961473816799 1 Chebyshev coefficient is: -0.232299371615171942121038341150 2 Chebyshev coefficient is: -0.053715114622047555071596207909 3 Chebyshev coefficient is: 0.002458235266981479866768726383 4 Chebyshev coefficient is: 0.000282119057434005702429677244 5 Chebyshev coefficient is: -0.000007722229155810577212604038 6 Chebyshev coefficient is: -5.898556452177850238987693546709E-7 7 Chebyshev coefficient is: 1.152142733081886533841160480101E-8 8 Chebyshev coefficient is: 6.596302208686010678189261798322E-10 9 Chebyshev coefficient is: -1.001689435637395512060196156843E-11 10 Chebyshev coefficient is: -4.586557765969596848147502951921E-13 11 Chebyshev coefficient is: 5.697353072301630964243748212466E-15 12 Chebyshev coefficient is: 2.173565878297512401879760404343E-16 13 Chebyshev coefficient is: -2.284293234863639106096540267786E-18 14 Chebyshev coefficient is: -7.468956910165861862760811388638E-20 15 Chebyshev coefficient is: 6.802288097339388765485830636223E-22 16 Chebyshev coefficient is: 1.945994872442404773393679283660E-23 17 Chebyshev coefficient is: -1.563704507245591241161562138364E-25 18 Chebyshev coefficient is: -3.976201538410589537318561880598E-27 19 Chebyshev coefficient is: 2.859065292763079576513213370136E-29
Ruby
<lang ruby>def mapp(x, min_x, max_x, min_to, max_to)
return (x - min_x) / (max_x - min_x) * (max_to - min_to) + min_to
end
def chebyshevCoef(func, min, max, coef)
n = coef.length
for i in 0 .. n-1 do m = mapp(Math.cos(Math::PI * (i + 0.5) / n), -1, 1, min, max) f = func.call(m) * 2 / n
for j in 0 .. n-1 do coef[j] = coef[j] + f * Math.cos(Math::PI * j * (i + 0.5) / n) end end
end
N = 10 def main
c = Array.new(N, 0) min = 0 max = 1 chebyshevCoef(lambda { |x| Math.cos(x) }, min, max, c)
puts "Coefficients:" puts c
end
main()</lang>
- Output:
Coefficients: 1.6471694753903139 -0.23229937161517178 -0.0537151146220477 0.002458235266981773 0.00028211905743405485 -7.722229156348348e-06 -5.898556456745974e-07 1.1521427756289171e-08 6.59630183807991e-10 -1.0021913854352249e-11
Scala
- Output:
Best seen running in your browser either by ScalaFiddle (ES aka JavaScript, non JVM) or Scastie (remote JVM).
<lang Scala>import scala.math.{Pi, cos}
object ChebyshevCoefficients extends App {
final val N = 10 final val (min, max) = (0, 1) val c = new Array[Double](N)
def chebyshevCoef(func: Double => Double, min: Double, max: Double, coef: Array[Double]): Unit = for (i <- coef.indices) { def map(x: Double, min_x: Double, max_x: Double, min_to: Double, max_to: Double): Double = (x - min_x) / (max_x - min_x) * (max_to - min_to) + min_to
val m = map(cos(Pi * (i + 0.5f) / N), -1, 1, min, max)
def f = func.apply(m) * 2 / N
for (j <- coef.indices) coef(j) += f * cos(Pi * j * (i + 0.5f) / N) }
chebyshevCoef((x: Double) => cos(x), min, max, c) println("Coefficients:") c.foreach(d => println(f"$d%23.16e"))
}</lang>
Sidef
<lang ruby>func chebft (callback, a, b, n) {
var bma = (0.5 * b-a); var bpa = (0.5 * b+a);
var pi_n = ((0..(n-1) »+» 0.5) »*» (Number.pi / n)); var f = (pi_n »cos»() »*» bma »+» bpa «call« callback); var sums = (0..(n-1) «run« {|i| f »*« ((pi_n »*» i) »cos»()) «+» });
sums »*» (2/n);
}
chebft(func(v){v.cos}, 0, 1, 10).each { |v|
say ("%+.10e" % v);
}</lang>
- Output:
+1.6471694754e+00 -2.3229937162e-01 -5.3715114622e-02 +2.4582352670e-03 +2.8211905743e-04 -7.7222291558e-06 -5.8985564522e-07 +1.1521427333e-08 +6.5963000351e-10 -1.0022591709e-11
Swift
<lang swift>import Foundation
typealias DFunc = (Double) -> Double
func mapRange(x: Double, min: Double, max: Double, minTo: Double, maxTo: Double) -> Double {
return (x - min) / (max - min) * (maxTo - minTo) + minTo
}
func chebCoeffs(fun: DFunc, n: Int, min: Double, max: Double) -> [Double] {
var res = [Double](repeating: 0, count: n)
for i in 0..<n { let dI = Double(i) let dN = Double(n) let f = fun(mapRange(x: cos(.pi * (dI + 0.5) / dN), min: -1, max: 1, minTo: min, maxTo: max)) * 2.0 / dN
for j in 0..<n { res[j] += f * cos(.pi * Double(j) * (dI + 0.5) / dN) } }
return res
}
func chebApprox(x: Double, n: Int, min: Double, max: Double, coeffs: [Double]) -> Double {
var a = 1.0 var b = mapRange(x: x, min: min, max: max, minTo: -1, maxTo: 1) var res = coeffs[0] / 2.0 + coeffs[1] * b let xx = 2 * b var i = 2
while i < n { let c = xx * b - a res += coeffs[i] * c (a, b) = (b, c) i += 1 }
return res
}
let coeffs = chebCoeffs(fun: cos, n: 10, min: 0, max: 1)
print("Coefficients")
for coeff in coeffs {
print(String(format: "%+1.15g", coeff))
}
print("\nApproximations:\n x func(x) approx diff")
for i in stride(from: 0.0, through: 20, by: 1) {
let x = mapRange(x: i, min: 0, max: 20, minTo: 0, maxTo: 1) let f = cos(x) let approx = chebApprox(x: x, n: 10, min: 0, max: 1, coeffs: coeffs)
print(String(format: "%1.3f %1.8f %1.8f % 4.1e", x, f, approx, approx - f))
}</lang>
- Output:
Coefficients +1.64716947539031 -0.232299371615172 -0.0537151146220476 +0.00245823526698177 +0.000282119057434055 -7.72222915632059e-06 -5.89855645688475e-07 +1.15214277562892e-08 +6.59630204624673e-10 -1.0021858343201e-11 Approximations: x func(x) approx diff 0.000 1.00000000 1.00000000 4.7e-13 0.050 0.99875026 0.99875026 -9.3e-14 0.100 0.99500417 0.99500417 4.6e-13 0.150 0.98877108 0.98877108 -4.7e-14 0.200 0.98006658 0.98006658 -4.6e-13 0.250 0.96891242 0.96891242 -2.3e-13 0.300 0.95533649 0.95533649 2.6e-13 0.350 0.93937271 0.93937271 4.6e-13 0.400 0.92106099 0.92106099 2.0e-13 0.450 0.90044710 0.90044710 -2.5e-13 0.500 0.87758256 0.87758256 -4.6e-13 0.550 0.85252452 0.85252452 -2.5e-13 0.600 0.82533561 0.82533561 2.0e-13 0.650 0.79608380 0.79608380 4.5e-13 0.700 0.76484219 0.76484219 2.5e-13 0.750 0.73168887 0.73168887 -2.3e-13 0.800 0.69670671 0.69670671 -4.5e-13 0.850 0.65998315 0.65998315 -4.4e-14 0.900 0.62160997 0.62160997 4.5e-13 0.950 0.58168309 0.58168309 -9.0e-14 1.000 0.54030231 0.54030231 4.5e-13
VBScript
To run in console mode with cscript. <lang vb>' N Chebyshev coefficients for the range 0 to 1
Dim coef(10),cheby(10) pi=4*Atn(1) a=0: b=1: n=10 For i=0 To n-1 coef(i)=Cos(Cos(pi/n*(i+1/2))*(b-a)/2+(b+a)/2) Next For i=0 To n-1 w=0 For j=0 To n-1 w=w+coef(j)*Cos(pi/n*i*(j+1/2)) Next cheby(i)=w*2/n If cheby(i)<=0 Then t="" Else t=" " WScript.StdOut.WriteLine i&" : "&t&cheby(i) Next</lang>
- Output:
0 : 1,64716947539031 1 : -0,232299371615172 2 : -5,37151146220477E-02 3 : 2,45823526698163E-03 4 : 2,82119057433916E-04 5 : -7,72222915563781E-06 6 : -5,89855645105608E-07 7 : 1,15214275009379E-08 8 : 6,59629917354465E-10 9 : -1,0022016549982E-11
Visual Basic .NET
<lang vbnet>Module Module1
Structure ChebyshevApprox Public ReadOnly coeffs As List(Of Double) Public ReadOnly domain As Tuple(Of Double, Double)
Public Sub New(func As Func(Of Double, Double), n As Integer, domain As Tuple(Of Double, Double)) coeffs = ChebCoef(func, n, domain) Me.domain = domain End Sub
Public Function Eval(x As Double) As Double Return ChebEval(coeffs, domain, x) End Function End Structure
Function AffineRemap(from As Tuple(Of Double, Double), x As Double, t0 As Tuple(Of Double, Double)) As Double Return t0.Item1 + (x - from.Item1) * (t0.Item2 - t0.Item1) / (from.Item2 - from.Item1) End Function
Function ChebCoef(fVals As List(Of Double)) As List(Of Double) Dim n = fVals.Count Dim theta = Math.PI / n Dim retval As New List(Of Double) For i = 1 To n retval.Add(0.0) Next For i = 1 To n Dim ii = i - 1 Dim f = fVals(ii) * 2.0 / n Dim phi = (ii + 0.5) * theta Dim c1 = Math.Cos(phi) Dim s1 = Math.Sin(phi) Dim c = 1.0 Dim s = 0.0 For j = 1 To n Dim jj = j - 1 retval(jj) += f * c ' update c -> cos(j*phi) for next value of j Dim cNext = c * c1 - s * s1 s = c * s1 + s * c1 c = cNext Next Next Return retval End Function
Function ChebCoef(func As Func(Of Double, Double), n As Integer, domain As Tuple(Of Double, Double)) As List(Of Double) Dim Remap As Func(Of Double, Double) Remap = Function(x As Double) Return AffineRemap(Tuple.Create(-1.0, 1.0), x, domain) End Function Dim theta = Math.PI / n Dim fVals As New List(Of Double) For i = 1 To n fVals.Add(0.0) Next For i = 1 To n Dim ii = i - 1 fVals(ii) = func(Remap(Math.Cos((ii + 0.5) * theta))) Next Return ChebCoef(fVals) End Function
Function ChebEval(coef As List(Of Double), x As Double) As Double Dim a = 1.0 Dim b = x Dim c As Double Dim retval = 0.5 * coef(0) + b * coef(1) Dim it = coef.GetEnumerator it.MoveNext() it.MoveNext() While it.MoveNext Dim pc = it.Current c = 2.0 * b * x - a retval += pc * c a = b b = c End While Return retval End Function
Function ChebEval(coef As List(Of Double), domain As Tuple(Of Double, Double), x As Double) As Double Return ChebEval(coef, AffineRemap(domain, x, Tuple.Create(-1.0, 1.0))) End Function
Sub Main() Dim N = 10 Dim fApprox As New ChebyshevApprox(AddressOf Math.Cos, N, Tuple.Create(0.0, 1.0)) Console.WriteLine("Coefficients: ") For Each c In fApprox.coeffs Console.WriteLine(vbTab + "{0: 0.00000000000000;-0.00000000000000;zero}", c) Next
Console.WriteLine(vbNewLine + "Approximation:" + vbNewLine + " x func(x) approx diff") Dim nX = 20.0 Dim min = 0.0 Dim max = 1.0 For i = 1 To nX Dim x = AffineRemap(Tuple.Create(0.0, nX), i, Tuple.Create(min, max)) Dim f = Math.Cos(x) Dim approx = fApprox.Eval(x) Console.WriteLine("{0:0.000} {1:0.00000000000000} {2:0.00000000000000} {3:E}", x, f, approx, approx - f) Next End Sub
End Module</lang>
- Output:
Coefficients: 1.64716947539031 -0.23229937161517 -0.05371511462205 0.00245823526698 0.00028211905743 -0.00000772222916 -0.00000058985565 0.00000001152143 0.00000000065963 -0.00000000001002 Approximation: x func(x) approx diff 0.050 0.99875026039497 0.99875026039487 -9.370282E-014 0.100 0.99500416527803 0.99500416527849 4.622969E-013 0.150 0.98877107793604 0.98877107793600 -4.662937E-014 0.200 0.98006657784124 0.98006657784078 -4.604095E-013 0.250 0.96891242171065 0.96891242171041 -2.322587E-013 0.300 0.95533648912561 0.95533648912587 2.609024E-013 0.350 0.93937271284738 0.93937271284784 4.606315E-013 0.400 0.92106099400289 0.92106099400308 1.980638E-013 0.450 0.90044710235268 0.90044710235243 -2.473577E-013 0.500 0.87758256189037 0.87758256188991 -4.586331E-013 0.550 0.85252452205951 0.85252452205926 -2.461364E-013 0.600 0.82533561490968 0.82533561490988 1.961764E-013 0.650 0.79608379854906 0.79608379854951 4.536371E-013 0.700 0.76484218728449 0.76484218728474 2.553513E-013 0.750 0.73168886887382 0.73168886887359 -2.267075E-013 0.800 0.69670670934717 0.69670670934672 -4.467537E-013 0.850 0.65998314588498 0.65998314588494 -4.485301E-014 0.900 0.62160996827066 0.62160996827111 4.444223E-013 0.950 0.58168308946388 0.58168308946379 -8.992806E-014 1.000 0.54030230586814 0.54030230586859 4.468648E-013
Wren
<lang ecmascript>import "/fmt" for Fmt
var mapRange = Fn.new { |x, min, max, minTo, maxTo| (x - min)/(max - min)*(maxTo - minTo) + minTo }
var chebCoeffs = Fn.new { |func, n, min, max|
var coeffs = List.filled(n, 0) for (i in 0...n) { var f = func.call(mapRange.call((Num.pi * (i + 0.5) / n).cos, -1, 1, min, max)) * 2 / n for (j in 0...n) coeffs[j] = coeffs[j] + f * (Num.pi * j * (i + 0.5) / n).cos } return coeffs
}
var chebApprox = Fn.new { |x, n, min, max, coeffs|
if (n < 2 || coeffs.count < 2) Fiber.abort("'n' can't be less than 2.") var a = 1 var b = mapRange.call(x, min, max, -1, 1) var res = coeffs[0]/2 + coeffs[1]*b var xx = 2 * b var i = 2 while (i < n) { var c = xx*b - a res = res + coeffs[i]*c a = b b = c i = i + 1 } return res
}
var n = 10 var min = 0 var max = 1 var coeffs = chebCoeffs.call(Fn.new { |x| x.cos }, n, min, max) System.print("Coefficients:") for (coeff in coeffs) Fmt.print("$0s$1.15f", (coeff >= 0) ? " " : "", coeff) System.print("\nApproximations:\n x func(x) approx diff") for (i in 0..20) {
var x = mapRange.call(i, 0, 20, min, max) var f = x.cos var approx = chebApprox.call(x, n, min, max, coeffs) var diff = approx - f var diffStr = diff.toString var e = diffStr[-4..-1] diffStr = diffStr[0..-5] diffStr = (diff >= 0) ? " " + diffStr[0..3] : diffStr[0..4] Fmt.print("$1.3f $1.8f $1.8f $s", x, f, approx, diffStr + e)
}</lang>
- Output:
Coefficients: 1.64716947539031 -0.23229937161517 -0.05371511462205 0.00245823526698 0.00028211905743 -0.00000772222916 -0.00000058985565 0.00000001152143 0.00000000065963 -0.00000000001002 Approximations: x func(x) approx diff 0.000 1.00000000 1.00000000 4.68e-13 0.050 0.99875026 0.99875026 -9.35e-14 0.100 0.99500417 0.99500417 4.61e-13 0.150 0.98877108 0.98877108 -4.72e-14 0.200 0.98006658 0.98006658 -4.60e-13 0.250 0.96891242 0.96891242 -2.31e-13 0.300 0.95533649 0.95533649 2.61e-13 0.350 0.93937271 0.93937271 4.61e-13 0.400 0.92106099 0.92106099 1.98e-13 0.450 0.90044710 0.90044710 -2.47e-13 0.500 0.87758256 0.87758256 -4.58e-13 0.550 0.85252452 0.85252452 -2.46e-13 0.600 0.82533561 0.82533561 1.95e-13 0.650 0.79608380 0.79608380 4.52e-13 0.700 0.76484219 0.76484219 2.54e-13 0.750 0.73168887 0.73168887 -2.27e-13 0.800 0.69670671 0.69670671 -4.47e-13 0.850 0.65998315 0.65998315 -4.37e-14 0.900 0.62160997 0.62160997 4.45e-13 0.950 0.58168309 0.58168309 -8.99e-14 1.000 0.54030231 0.54030231 4.47e-13
zkl
<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)