# Chebyshev coefficients

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

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

## 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
```

## ALGOL 60

Works with: GNU Marst version Any - tested with release 2.7
Translation of: ALGOL W

...which is

Translation of: Java
```begin comment Chebyshev coefficients ;

real PI;

procedure chebyshevCoef( func, min, max, coef, N )
; value min, max,       N
; real procedure func
; real           min, max
; real     array coef
; integer        N
;
begin
real procedure map(       x, min x, max x, min to, max to )
; value x, min x, max x, min to, max to
; real  x, min x, max x, min to, max to
;
begin
map := ( x - min x ) / ( max x - min x ) * ( max to - min to ) + min to
end map ;

integer i, j;
for i := 0 step 1 until N - 1 do begin
real m, f;
m := map( cos( PI * ( i + 0.5 ) / N ), -1, 1, min, max );
f := func( m ) * 2 / N;
for j := 0 step 1 until N - 1 do begin
coef[ j ] := coef[ j ] + f * cos( PI * j * ( i + 0.5 ) / N )
end j
end i
end chebyshevCoef ;

PI := arctan( 1 ) * 4;
begin
integer N;
N := 10;
begin
real array c [ 0 : N - 1 ];
integer i;
chebyshevCoef( cos, 0, 1, c, N );
outstring( 1, "Coefficients:\n" );
for i := 0 step 1 until N - 1 do begin
if c[ i ] >= 0 then outstring( 1, " " );
outstring( 1, "  " );outreal( 1, c[ i ] );outstring( 1, "\n" )
end i
end
end
end```
Output:
```Coefficients:
1.64716947539
-0.232299371615
-0.053715114622
0.00245823526698
0.000282119057434
-7.72222915635e-006
-5.89855645675e-007
1.15214277563e-008
6.59630183808e-010
-1.00219138544e-011
```

## ALGOL 68

Translation of: Java

... using nested procedures and returning the coefficient array instead of using a reference parameter.

```BEGIN # Chebyshev Coefficients #

PROC chebyshev coef = ( PROC( REAL )REAL func, REAL min, max, INT n )[]REAL:
BEGIN

PROC map = ( REAL x, min x, max x, min to, max to )REAL:
( x - min x ) / ( max x - min x ) * ( max to - min to ) + min to;

[ 0 : n - 1 ]REAL coef; FOR i FROM LWB coef TO UPB coef DO coef[ i ] := 0 OD;
FOR i FROM 0 TO UPB coef DO
REAL m = map( cos( pi * ( i + 0.5 ) / n ), -1, 1, min, max );
REAL f = func( m ) * 2 / n;
FOR j FROM 0 TO UPB coef DO
coef[ j ] +:= f * cos( pi * j * ( i + 0.5 ) / n )
OD
OD;
coef
END # chebyshev coef # ;

BEGIN
INT n = 10;
REAL min := 0, max := 1;
[]REAL c = chebyshev coef( cos, min, max, n );
print( ( "Coefficients:", newline ) );
FOR i FROM LWB c TO UPB c DO
print( ( fixed( c[ i ], -18, 14 ), newline ) )
OD
END

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

## ALGOL W

Translation of: Java

... using nested procedures. In Algol W, procedures can't find the bounds of array parameters, so an extra parameter is reuired for the chebyshevCoef procedure.

```begin % Chebyshev coefficients %

procedure chebyshevCoef ( real procedure func
; real     value min, max
; real     array coef     ( * )
; integer  value N
) ;
begin
real procedure map  ( real value x, min_x, max_x, min_to, max_to ) ;
( x - min_x ) / ( max_x - min_x ) * ( max_to - min_to ) + min_to;

for i := 0 until N - 1 do begin
real m, f;
m := map( cos( PI * ( i + 0.5 ) / N ), -1, 1, min, max );
f := func( m ) * 2 / N;
for j := 0 until N - 1 do begin
coef( j ) := coef( j ) + f * cos( PI * j * ( i + 0.5 ) / N )
end for_j
end for_i
end chebyshevCoef ;

begin
integer N;
N := 10;
begin
real array c ( 0 :: N - 1 );
chebyshevCoef( cos, 0, 1, c, N );
write( "Coefficients:" );
for i := 0 until N - 1 do write( r_format := "S", r_w := 14, c( i ) )
end
end
end.```
Output:
```Coefficients:
1.6471694'+00
-2.3229937'-01
-5.3715114'-02
2.4582352'-03
2.8211905'-04
-7.7222291'-06
-5.8985564'-07
1.1521427'-08
6.5963014'-10
-1.0021983'-11
```

## BASIC

### Applesoft BASIC

The MSX-BASIC solution works without any changes.

### 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```

### Chipmunk Basic

Translation of: FreeBASIC
Works with: Chipmunk Basic version 3.6.4
```100 cls
110 rem pi = 4 * atn(1)
120 a = 0
130 b = 1
140 n = 10
150 dim cheby(n)
160 dim coef(n)
170 for i = 0 to n-1
180     coef(i) = cos(cos(pi/n*(i+1/2))*(b-a)/2+(b+a)/2)
190 next i
200 for i = 0 to n-1
210     w = 0
220     for j = 0 to n-1
230         w = w+coef(j)*cos(pi/n*i*(j+1/2))
240     next j
250     cheby(i) = w*2/n
260     print i;" : ";cheby(i)
270 next i
280 end
```
Output:
```0  : 1.647169
1  : -0.232299
2  : -0.053715
3  : 2.458235E-03
4  : 2.821191E-04
5  : -7.722229E-06
6  : -5.898556E-07
7  : 1.152143E-08
8  : 6.596304E-10
9  : -1.002234E-11```

### FreeBASIC

```Const pi As Double = 4 * Atn(1)
Dim As Integer i, j
Dim As Double w, a = 0, b = 1, n = 10
Dim As Double cheby(n), 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
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```

### Gambas

Translation of: FreeBASIC
```Public coef[10] As Float

Public Sub Main()
Dim i As Integer, j As Integer
Dim w As Float, a As Float = 0, b As Float = 1, n As Float = 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 += coef[j] * Cos(Pi / n * i * (j + 1 / 2))
Next
cheby[i] = w * 2 / n
Print i; " : "; cheby[i]
Next

End
```
Output:
```0 : 1,64716947539031
1 : -0,232299371615172
2 : -0,053715114622048
3 : 0,002458235266982
4 : 0,000282119057434
5 : -7,7222291556156E-6
6 : -5,89855645105608E-7
7 : 1,15214275009379E-8
8 : 6,59629906252235E-10
9 : -1,0022016549982E-11```

### GW-BASIC

Translation of: FreeBASIC
Works with: PC-BASIC version any
```100 CLS
110 PI# = 4 * ATN(1)
120 A# = 0
130 B# = 1
140 N# = 10
150 DIM CHEBY(N#)
160 DIM COEF(N#)
170 FOR I = 0 TO N#-1
180     COEF(I) = COS(COS(PI#/N#*(I+1/2))*(B#-A#)/2+(B#+A#)/2)
190 NEXT I
200 FOR I = 0 TO N#-1
210     W# = 0
220     FOR J = 0 TO N#-1
230         W# = W# + COEF(J) * COS(PI#/N#*I*(J+1/2))
240     NEXT J
250     CHEBY(I) = W# * 2 / N#
260     PRINT I; " : "; CHEBY(I)
270 NEXT I
280 END
```
Output:
```0  :  1.647169
1  : -.2322993
2  : -5.371515E-02
3  :  2.458321E-03
4  :  2.820671E-04
5  : -7.766486E-06
6  : -5.857175E-07
7  :  9.834766E-08
8  : -1.788139E-07
9  : -9.089708E-08```

### Minimal BASIC

Translation of: FreeBASIC
Works with: BASICA
```110 LET P = 4 * ATN(1)
120 LET A = 0
130 LET B = 1
140 LET N = 10
170 FOR I = 0 TO N-1
180   LET K(I) = COS(COS(P/N*(I+1/2))*(B-A)/2+(B+A)/2)
190 NEXT I
200 FOR I = 0 TO N-1
210   LET W = 0
220   FOR J = 0 TO N-1
230     LET W = W + K(J) * COS(P/N*I*(J+1/2))
240   NEXT J
250   LET C(I) = W * 2 / N
260   PRINT I; " : "; C(I)
270 NEXT I
280 END
```
Output:
``` 0  :  1.6471695
1  : -.23229937
2  : -5.3715115E-2
3  :  2.4582353E-3
4  :  2.8211906E-4
5  : -7.7222291E-6
6  : -5.8985565E-7
7  :  1.1521437E-8
8  :  6.5962449E-10
9  : -1.0018986E-11```

### MSX Basic

Translation of: FreeBASIC
Works with: MSX BASIC version any
```100 CLS : rem  10 HOME for Applesoft BASIC
110 PI = 4 * ATN(1)
120 A = 0
130 B = 1
140 N = 10
150 DIM CHEBY(N)
160 DIM COEF(N)
170 FOR I = 0 TO N-1
180   COEF(I) = COS(COS(PI/N*(I+1/2))*(B-A)/2+(B+A)/2)
190 NEXT I
200 FOR I = 0 TO N-1
210   W = 0
220   FOR J = 0 TO N-1
230     W = W + COEF(J) * COS(PI/N*I*(J+1/2))
240   NEXT J
250   CHEBY(I) = W * 2 / N
260   PRINT I; " : "; CHEBY(I)
270 NEXT I
280 END
```

### 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```

### Quite BASIC

Translation of: FreeBASIC
```100 cls
110 rem pi = 4 * atn(1)
120 let a = 0
130 let b = 1
140 let n = 10
150 array c
160 array k
170 for i = 0 to n-1
180   let k[i] = cos(cos(pi/n*(i+1/2))*(b-a)/2+(b+a)/2)
190 next i
200 for i = 0 to n-1
210   let w = 0
220   for j = 0 to n-1
230     let w = w + k[j] * cos(pi/n*i*(j+1/2))
240   next j
250   let c[i] = w * 2 / n
260   print i; " : "; c[i]
270 next i
280 end
```
Output:
```0 : 1.6471694753903137
1 : -0.23229937161517186
2 : -0.05371511462204768
3 : 0.0024582352669816343
4 : 0.0002821190574339161
5 : -0.0000077222291556156
6 : -5.898556451056081e-7
7 : 1.1521427500937876e-8
8 : 6.59629917354465e-10
9 : -1.0022016549982027e-11```

### Run BASIC

Translation of: FreeBASIC
Works with: Just BASIC
Works with: Liberty BASIC
```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 i; " : "; cheby(i)
next i
end
```

### 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;

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++) {
}
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++) {
}
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 12 0
a = 0
b = 1
n = 10
len coef[] n
len cheby[] n
for i = 0 to n - 1
coef[i + 1] = cos (180 / pi * (cos (180 / n * (i + 1 / 2)) * (b - a) / 2 + (b + a) / 2))
.
for i = 0 to n - 1
w = 0
for j = 0 to n - 1
w += coef[j + 1] * cos (180 / n * i * (j + 1 / 2))
.
cheby[i + 1] = w * 2 / n
print cheby[i + 1]
.
```

## 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

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 ;

[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")),
(\$approx|pp(0;8)),
(\$diff  |pp(2;2)) ]
| join("  ") );

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
```

## МК-61/52

Translation of: BASIC
```0   ПA  1    ПB  8   ПC  0   ПD  ИПC ИПD
-   x#0 44   пи  ИПC /   ИПD 1   ^   2
/   +   *    cos ИПB ИПA -   2   /   *
ИПB ИПA +    2   /   +   cos KПD ИПD 1
+   ПD  БП   08  0   ПD  ИПC ИПD -   x#0
95  0   ПB   ПE  ИПC ИПE -   x#0 83  пи
ИПC /   ИПD  *   ИПE 1   ^   2   /   +
*   cos KИПE *   ИПB +   ПB  ИПE 1   +
ПE  БП  54   ИПB 2   *   ИПC /   С/П ИПD
1   +   ПD   БП  46  С/П```

## 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 = ((^n »+» 0.5) »*» (Num.pi / n))
var f = (pi_n »cos()» »*» bma »+» bpa «call« callback)
var sums = (^n «run« {|i| f »*« ((pi_n »*» i) »cos()») «+» })

sums »*» (2/n)
}

for v in (chebft(func(v){v.cos}, 0, 1, 10)) {
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
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
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
```

## XPL0

Translation of: C
```include xpllib; \for Print and Pi

func real Map(X, MinX, MaxX, MinTo, MaxTo);
\Map X from range Min,Max to MinTo,MaxTo
real X, MinX, MaxX, MinTo, MaxTo;
return (X-MinX) / (MaxX-MinX) * (MaxTo-MinTo) + MinTo;

proc ChebCoef(N, Min, Max, Coef);
int  N;  real Min, Max, Coef;
int  I, J;
real F;
[for I:= 0 to N-1 do Coef(I):= 0.0;
for I:= 0 to N-1 do
[F:= Cos(Map(Cos(Pi*(float(I)+0.5)/float(N)), -1.0, 1.0, Min, Max)) *
2.0/float(N);
for J:= 0 to N-1 do
Coef(J):= Coef(J) + F*Cos(Pi*float(J) * (float(I)+0.5) / float(N));
];
];

func real ChebApprox(X, N, Min, Max, Coef);
real X;  int N;  real Min, Max, Coef;
real A, B, C, Res;
int  I;
[A:= 1.0;
B:= Map(X, Min, Max, -1.0, 1.0);
Res:= Coef(0)/2.0 + Coef(1)*B;
X:= 2.0*B;
for I:= 2 to N-1 do
[C:= X*B - A;
Res:= Res + Coef(I)*C;
A:= B;
B:= C;
];
return Res;
];

def  N=10, MinV=0.0, MaxV=1.0;
real C(N);
int  I;
real X, F, Approx;
[ChebCoef(N, MinV, MaxV, C);
Print("Coefficients:\n");
for I:= 0 to N-1 do
Print(" %2.15f\n", C(I));
Print("\nApproximation:\n   X     Cos(X)            Approx            Diff\n");
for I:= 0 to 20 do
[X:= Map(float(I), 0.0, 20.0, MinV, MaxV);
F:= Cos(X);
Approx:= ChebApprox(X, N, MinV, MaxV, C);
Print("%2.2f %2.14f %2.14f %0.1f\n", X, F, Approx, Approx-F);
];
]```
Output:
```Coefficients:
1.647169475390310
-0.232299371615172
-0.053715114622048
0.002458235266982
0.000282119057434
-0.000007722229156
-0.000000589855646
0.000000011521428
0.000000000659630
-0.000000000010022

Approximation:
X     Cos(X)            Approx            Diff
0.00  1.00000000000000  1.00000000000047  4.7E-013
0.05  0.99875026039497  0.99875026039487 -9.4E-014
0.10  0.99500416527803  0.99500416527849  4.6E-013
0.15  0.98877107793604  0.98877107793599 -4.7E-014
0.20  0.98006657784124  0.98006657784078 -4.6E-013
0.25  0.96891242171064  0.96891242171041 -2.3E-013
0.30  0.95533648912561  0.95533648912587  2.6E-013
0.35  0.93937271284738  0.93937271284784  4.6E-013
0.40  0.92106099400289  0.92106099400308  2.0E-013
0.45  0.90044710235268  0.90044710235243 -2.5E-013
0.50  0.87758256189037  0.87758256188991 -4.6E-013
0.55  0.85252452205951  0.85252452205926 -2.5E-013
0.60  0.82533561490968  0.82533561490987  2.0E-013
0.65  0.79608379854906  0.79608379854951  4.5E-013
0.70  0.76484218728449  0.76484218728474  2.5E-013
0.75  0.73168886887382  0.73168886887359 -2.3E-013
0.80  0.69670670934717  0.69670670934672 -4.5E-013
0.85  0.65998314588498  0.65998314588494 -4.4E-014
0.90  0.62160996827066  0.62160996827111  4.5E-013
0.95  0.58168308946388  0.58168308946379 -9.0E-014
1.00  0.54030230586814  0.54030230586859  4.5E-013
```

## 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)
```