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{{draft task|Mathematics}}
Chebyshev coefficients are the basis of polynomial approximations of functions. Write a program to generate Chebyshev coefficients.
Calculate coefficients: cosine function, 10 coefficients, interval 0 1
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++
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); } }
{{out}}
Coefficients:
+1.6471694753903136868
-0.23229937161517194216
-0.053715114622047555044
+0.0024582352669814797779
+0.00028211905743400579387
-7.7222291558103533853e-06
-5.898556452178771968e-07
+1.1521427332860788728e-08
+6.5963000382704222411e-10
-1.0022591914390921452e-11
Approximation:
x func(x) approx diff
0.000 1.00000000000000000000 1.00000000000046961190 4.70e-13
0.050 0.99875026039496624654 0.99875026039487216781 -9.41e-14
0.100 0.99500416527802576609 0.99500416527848803832 4.62e-13
0.150 0.98877107793604228670 0.98877107793599569749 -4.66e-14
0.200 0.98006657784124163110 0.98006657784078136889 -4.60e-13
0.250 0.96891242171064478408 0.96891242171041249593 -2.32e-13
0.300 0.95533648912560601967 0.95533648912586667367 2.61e-13
0.350 0.93937271284737892005 0.93937271284783928305 4.60e-13
0.400 0.92106099400288508277 0.92106099400308274515 1.98e-13
0.450 0.90044710235267692169 0.90044710235242891114 -2.48e-13
0.500 0.87758256189037271615 0.87758256188991362600 -4.59e-13
0.550 0.85252452205950574283 0.85252452205925896211 -2.47e-13
0.600 0.82533561490967829723 0.82533561490987400509 1.96e-13
0.650 0.79608379854905582896 0.79608379854950937939 4.54e-13
0.700 0.76484218728448842626 0.76484218728474395029 2.56e-13
0.750 0.73168886887382088633 0.73168886887359430061 -2.27e-13
0.800 0.69670670934716542091 0.69670670934671868322 -4.47e-13
0.850 0.65998314588498217039 0.65998314588493717370 -4.50e-14
0.900 0.62160996827066445648 0.62160996827110870299 4.44e-13
0.950 0.58168308946388349416 0.58168308946379353278 -9.00e-14
The same code, with N = 16:
Coefficients:
+1.6471694753903136868
-0.23229937161517194214
-0.053715114622047555035
+0.0024582352669814797982
+0.00028211905743400571932
-7.722229155810705751e-06
-5.898556452177348953e-07
+1.1521427330794028337e-08
+6.5963022091481034181e-10
-1.0016894235462866363e-11
-4.5865582517937500406e-13
+5.6974586994888026802e-15
+2.1752822525027137867e-16
-2.3140940118987485263e-18
-1.0333801956502464137e-19
+2.5410988417629010172e-20
Approximation:
x func(x) approx diff
0.000 1.00000000000000000000 1.00000000000000000030 3.25e-19
0.050 0.99875026039496624654 0.99875026039496624646 -1.08e-19
0.100 0.99500416527802576609 0.99500416527802576557 -5.42e-19
0.150 0.98877107793604228670 0.98877107793604228636 -3.79e-19
0.200 0.98006657784124163110 0.98006657784124163127 1.08e-19
0.250 0.96891242171064478408 0.96891242171064478451 3.79e-19
0.300 0.95533648912560601967 0.95533648912560601967 0.00e+00
0.350 0.93937271284737892005 0.93937271284737891962 -3.79e-19
0.400 0.92106099400288508277 0.92106099400288508260 -2.17e-19
0.450 0.90044710235267692169 0.90044710235267692169 5.42e-20
0.500 0.87758256189037271615 0.87758256189037271632 2.17e-19
0.550 0.85252452205950574283 0.85252452205950574274 -5.42e-20
0.600 0.82533561490967829723 0.82533561490967829697 -2.17e-19
0.650 0.79608379854905582896 0.79608379854905582861 -3.25e-19
0.700 0.76484218728448842626 0.76484218728448842630 5.42e-20
0.750 0.73168886887382088633 0.73168886887382088637 5.42e-20
0.800 0.69670670934716542091 0.69670670934716542087 -5.42e-20
0.850 0.65998314588498217039 0.65998314588498217022 -1.63e-19
0.900 0.62160996827066445648 0.62160996827066445674 2.71e-19
0.950 0.58168308946388349416 0.58168308946388349403 -1.63e-19
Go
Wikipedia gives a formula for coefficients in a section [https://en.wikipedia.org/wiki/Chebyshev_polynomials#Example_1 "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 [https://en.wikipedia.org/wiki/Clenshaw_algorithm 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] }
{{out}}
coefficients:
1.647169475390314
-0.232299371615172
-0.053715114622048
0.002458235266982
0.000282119057434
-0.000007722229156
-0.000000589855645
0.000000011521427
0.000000000659630
-0.000000000010022
x computed approximated computed-approx
0.0 1.00000000 1.00000000 -4.685e-13
0.1 0.99500417 0.99500417 -4.620e-13
0.2 0.98006658 0.98006658 4.601e-13
0.3 0.95533649 0.95533649 -2.607e-13
0.4 0.92106099 0.92106099 -1.972e-13
0.5 0.87758256 0.87758256 4.587e-13
0.6 0.82533561 0.82533561 -1.965e-13
0.7 0.76484219 0.76484219 -2.552e-13
0.8 0.69670671 0.69670671 4.470e-13
0.9 0.62160997 0.62160997 -4.449e-13
1.0 0.54030231 0.54030231 -4.476e-13
J
From 'J for C Programmers: Calculating Chebyshev Coefficients [[http://www.jsoftware.com/learning/a_first_look_at_j_programs.htm#_Toc191734318]]
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 [[Chebyshev_coefficients#C|C]] via [[Chebyshev_coefficients#D|D]] {{works with|Java|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
Julia
{{works with|Julia|0.6}} {{trans|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
{{out}}
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
{{trans|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) } }
{{out}}
Coefficients:
+1.64716947539031
-0.232299371615172
-0.0537151146220477
+0.00245823526698177
+0.000282119057434055
-7.72222915632059e-06
-5.89855645674597e-07
+1.15214277701670e-08
+6.59630183807991e-10
-1.00219138543522e-11
Approximations:
x func(x) approx diff
0.000 1.00000000 1.00000000 4.7e-13
0.050 0.99875026 0.99875026 -9.4e-14
0.100 0.99500417 0.99500417 4.6e-13
0.150 0.98877108 0.98877108 -4.7e-14
0.200 0.98006658 0.98006658 -4.6e-13
0.250 0.96891242 0.96891242 -2.3e-13
0.300 0.95533649 0.95533649 2.6e-13
0.350 0.93937271 0.93937271 4.6e-13
0.400 0.92106099 0.92106099 2.0e-13
0.450 0.90044710 0.90044710 -2.5e-13
0.500 0.87758256 0.87758256 -4.6e-13
0.550 0.85252452 0.85252452 -2.5e-13
0.600 0.82533561 0.82533561 2.0e-13
0.650 0.79608380 0.79608380 4.5e-13
0.700 0.76484219 0.76484219 2.5e-13
0.750 0.73168887 0.73168887 -2.3e-13
0.800 0.69670671 0.69670671 -4.5e-13
0.850 0.65998315 0.65998315 -4.4e-14
0.900 0.62160997 0.62160997 4.5e-13
0.950 0.58168309 0.58168309 -9.0e-14
1.000 0.54030231 0.54030231 4.5e-13
Microsoft Small Basic
{{trans|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
{{out}}
0 : 1,6471694753903136
1 : -0,2322993716151700684187787635
2 : -0,0537151146220494010749946688
3 : 0,0024582352669837594966069584
4 : 0,0002821190574317389206759282
5 : -0,0000077222291539069653168878
6 : -0,0000005898556481086082412444
7 : 0,0000000115214300591398939205
8 : 0,0000000006596278553822696656
9 : -0,0000000000100189955816952521
Perl
{{trans|C}}
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);
{{out}}
1.64716947539031
-0.232299371615172
-0.0537151146220477
0.00245823526698163
0.000282119057433938
-7.72222915566001e-06
-5.89855645105608e-07
1.15214274787334e-08
6.59629917354465e-10
-1.00219943455215e-11
Perl 6
{{works with|Rakudo|2015.12}} {{trans|C}}
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;
{{out}}
+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
{{trans|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
{{out}}
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
Racket
{{trans|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
{{out}}
'#(1.6471694753903137
-0.2322993716151719
-0.05371511462204768
0.0024582352669816343
0.0002821190574339161
-7.722229155637806e-006
-5.898556451056081e-007
1.1521427500937876e-008
6.596299173544651e-010
-1.0022016549982027e-011)
REXX
{{trans|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() ) - 1 /*DIGITS default is nine, but use 71. */
parse arg a b N . /*obtain optional arguments from the CL*/
if a=='' | a=="," then a= 0 /*A not specified? Then use default.*/
if b=='' | b=="," then b= 1 /*B " " " " " */
if N=='' | N=="," then N=10 /*N " " " " " */
fac=2 / N; pin=pi / N /*calculate a couple handy─dandy values*/
Dma= (b-a) / 2 /*calculate one─half of the difference.*/
Dpa= (b+a) / 2 /* " " " " sum. */
do k=0 for N
f.k=cos( cos( pin * (k + .5) ) * Dma + Dpa)
end /*k*/
do j=0 for N; z=pin * j /*The LEFT('', ···) ──────►──────┐ */
$=0 /*clause is used to align │ */
do m=0 for N /*the non─negative values with ↓ */
$=$ + f.m * cos(z*(m + .5)) /*the negative values. │ */
end /*m*/ /* ┌─────◄──────┘ */
cheby.j=fac * $ /* ↓ */
say right(j, length(N) +3) " Chebyshev coefficient is:" left('', cheby.j >= 0),
format(cheby.j, , 30) /*only show 30 decimal digits of coeff.*/
end /*j*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
cos: procedure; parse arg x; numeric digits digits()+9; x=r2r(x); a=abs(x); numeric fuzz 5
if a=pi then return -1; if a=pi*.5 | a=pi*2 then return 0; pit= pi/3
if a=pit then return .5; if a=pit*2 then return -.5; q=x*x; z=1; _=1
do ?=2 by 2 until p=z; p=z; _=-_*q/(?*(?-1)); z=z+_; end /*?*/
return z
/*──────────────────────────────────────────────────────────────────────────────────────*/
pi: pi=3.1415926535897932384626433832795028841971693993751058209749445923078164;return pi
r2r: return arg(1) // (pi() * 2) /*normalize radians ───► a unit circle.*/
{{out|output|text= 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
{{out|output|text= when using the following input of: , , 20 }} 0 Chebyshev coefficient is: 1.647169475390313686961473816799 1 Chebyshev coefficient is: -0.232299371615171942121038341150 2 Chebyshev coefficient is: -0.053715114622047555071596207909 3 Chebyshev coefficient is: 0.002458235266981479866768726383 4 Chebyshev coefficient is: 0.000282119057434005702429677244 5 Chebyshev coefficient is: -0.000007722229155810577212604038 6 Chebyshev coefficient is: -5.898556452177850238987693546709E-7 7 Chebyshev coefficient is: 1.152142733081886533841160480101E-8 8 Chebyshev coefficient is: 6.596302208686010678189261798322E-10 9 Chebyshev coefficient is: -1.001689435637395512060196156843E-11 10 Chebyshev coefficient is: -4.586557765969596848147502951921E-13 11 Chebyshev coefficient is: 5.697353072301630964243748212466E-15 12 Chebyshev coefficient is: 2.173565878297512401879760404343E-16 13 Chebyshev coefficient is: -2.284293234863639106096540267786E-18 14 Chebyshev coefficient is: -7.468956910165861862760811388638E-20 15 Chebyshev coefficient is: 6.802288097339388765485830636223E-22 16 Chebyshev coefficient is: 1.945994872442404773393679283660E-23 17 Chebyshev coefficient is: -1.563704507245591241161562138364E-25 18 Chebyshev coefficient is: -3.976201538410589537318561880598E-27 19 Chebyshev coefficient is: 2.859065292763079576513213370136E-29
Scala
{{Out}}Best seen running in your browser either by [https://scalafiddle.io/sf/DqRNe2A/0 ScalaFiddle (ES aka JavaScript, non JVM)] or [https://scastie.scala-lang.org/M5Ye6h8ZRkmTCNzexUh3uw 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
{{trans|Perl 6}}
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); }
{{out}}
+1.6471694754e+00
-2.3229937162e-01
-5.3715114622e-02
+2.4582352670e-03
+2.8211905743e-04
-7.7222291558e-06
-5.8985564522e-07
+1.1521427333e-08
+6.5963000351e-10
-1.0022591709e-11
VBScript
{{trans|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
{{out}}
0 : 1,64716947539031
1 : -0,232299371615172
2 : -5,37151146220477E-02
3 : 2,45823526698163E-03
4 : 2,82119057433916E-04
5 : -7,72222915563781E-06
6 : -5,89855645105608E-07
7 : 1,15214275009379E-08
8 : 6,59629917354465E-10
9 : -1,0022016549982E-11
zkl
{{trans|C}}{{trans|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();
{{out}}
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)