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{{task|Mathematical operations|Matrices}} Find an approximating polynomial of known degree for a given data.

Example: For input data: x = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}; y = {1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321}; The approximating polynomial is: 3 x2 + 2 x + 1 Here, the polynomial's coefficients are (3, 2, 1).

This task is intended as a subtask for [[Measure relative performance of sorting algorithms implementations]].

```with Ada.Numerics.Real_Arrays;  use Ada.Numerics.Real_Arrays;

function Fit (X, Y : Real_Vector; N : Positive) return Real_Vector is
A : Real_Matrix (0..N, X'Range);  -- The plane
begin
for I in A'Range (2) loop
for J in A'Range (1) loop
A (J, I) := X (I)**J;
end loop;
end loop;
return Solve (A * Transpose (A), A * Y);
end Fit;
```

The function Fit implements least squares approximation of a function defined in the points as specified by the arrays ''x''''i'' and ''y''''i''. The basis φ''j'' is ''x''''j'', ''j''=0,1,..,''N''. The implementation is straightforward. First the plane matrix A is created. Aji''j''(''x''''i''). Then the linear problem AA''T''''c''=A''y'' is solved. The result ''c''''j'' are the coefficients. Constraint_Error is propagated when dimensions of X and Y differ or else when the problem is ill-defined.

### Example

```with Fit;

procedure Fitting is
C : constant Real_Vector :=
Fit
(  (0.0, 1.0,  2.0,  3.0,  4.0,  5.0,   6.0,   7.0,   8.0,   9.0,  10.0),
(1.0, 6.0, 17.0, 34.0, 57.0, 86.0, 121.0, 162.0, 209.0, 262.0, 321.0),
2
);
begin
Put (C (0), Aft => 3, Exp => 0);
Put (C (1), Aft => 3, Exp => 0);
Put (C (2), Aft => 3, Exp => 0);
end Fitting;
```

{{out}}

```
1.000 2.000 3.000

```

## ALGOL 68

{{works with|ALGOL 68|Standard - ''lu decomp'' and ''lu solve'' are from the [[:Category:Libgsl|GSL]] library}}

{{works with|ALGOL 68G|Any - tested with release mk15-0.8b.fc9.i386}}

```MODE FIELD = REAL;

MODE
VEC = [0]FIELD,
MAT = [0,0]FIELD;

PROC VOID raise index error := VOID: (
print(("stop", new line));
stop
);

COMMENT from http://rosettacode.org/wiki/Matrix_Transpose#ALGOL_68 END COMMENT
OP ZIP = ([,]FIELD in)[,]FIELD:(
[2 LWB in:2 UPB in,1 LWB in:1UPB in]FIELD out;
FOR i FROM LWB in TO UPB in DO
out[,i]:=in[i,]
OD;
out
);

COMMENT from http://rosettacode.org/wiki/Matrix_multiplication#ALGOL_68 END COMMENT
OP * = (VEC a,b)FIELD: ( # basically the dot product #
FIELD result:=0;
IF LWB a/=LWB b OR UPB a/=UPB b THEN raise index error FI;
FOR i FROM LWB a TO UPB a DO result+:= a[i]*b[i] OD;
result
);

OP * = (VEC a, MAT b)VEC: ( # overload vector times matrix #
[2 LWB b:2 UPB b]FIELD result;
IF LWB a/=LWB b OR UPB a/=UPB b THEN raise index error FI;
FOR j FROM 2 LWB b TO 2 UPB b DO result[j]:=a*b[,j] OD;
result
);

OP * = (MAT a, b)MAT: ( # overload matrix times matrix #
[LWB a:UPB a, 2 LWB b:2 UPB b]FIELD result;
IF 2 LWB a/=LWB b OR 2 UPB a/=UPB b THEN raise index error FI;
FOR k FROM LWB result TO UPB result DO result[k,]:=a[k,]*b OD;
result
);

COMMENT from http://rosettacode.org/wiki/Pyramid_of_numbers#ALGOL_68 END COMMENT
OP / = (VEC a, MAT b)VEC: ( # vector division #
[LWB a:UPB a,1]FIELD transpose a;
transpose a[,1]:=a;
(transpose a/b)[,1]
);

OP / = (MAT a, MAT b)MAT:( # matrix division #
[LWB b:UPB b]INT p ;
INT sign;
[,]FIELD lu = lu decomp(b, p, sign);
[LWB a:UPB a, 2 LWB a:2 UPB a]FIELD out;
FOR col FROM 2 LWB a TO 2 UPB a DO
out[,col] := lu solve(b, lu, p, a[,col]) [@LWB out[,col]]
OD;
out
);

FORMAT int repr = \$g(0)\$,
real repr = \$g(-7,4)\$;

PROC fit =  (VEC x, y, INT order)VEC:
BEGIN
[0:order, LWB x:UPB x]FIELD a;  # the plane #
FOR i FROM 2 LWB a TO 2 UPB a  DO
FOR j FROM LWB a TO UPB a DO
a [j, i] := x [i]**j
OD
OD;
( y * ZIP a ) / ( a * ZIP a )
END # fit #;

PROC print polynomial = (VEC x)VOID: (
BOOL empty := TRUE;
FOR i FROM UPB x BY -1 TO LWB x DO
IF x[i] NE 0 THEN
IF x[i] > 0 AND NOT empty THEN print ("+") FI;
empty := FALSE;
IF x[i] NE 1 OR i=0 THEN
IF ENTIER x[i] = x[i] THEN
printf((int repr, x[i]))
ELSE
printf((real repr, x[i]))
FI
FI;
CASE i+1 IN
SKIP,print(("x"))
OUT
printf((\$"x**"g(0)\$,i))
ESAC
FI
OD;
IF empty THEN print("0") FI;
print(new line)
);

fitting: BEGIN
VEC c =
fit
(  (0.0, 1.0,  2.0,  3.0,  4.0,  5.0,   6.0,   7.0,   8.0,   9.0,  10.0),
(1.0, 6.0, 17.0, 34.0, 57.0, 86.0, 121.0, 162.0, 209.0, 262.0, 321.0),
2
);
print polynomial(c);
VEC d =
fit
( (0, 1, 2, 3, 4, 5, 6, 7, 8, 9),
(2.7, 2.8, 31.4, 38.1, 58.0, 76.2, 100.5, 130.0, 149.3, 180.0),
2
);
print polynomial(d)
END # fitting #
```

{{out}}

```
3x**2+2x+1
1.0848x**2+10.3552x-0.6164

```

## BBC BASIC

{{works with|BBC BASIC for Windows}} The code listed below is good for up to 10000 data points and fits an order-5 polynomial, so the test data for this task is hardly challenging!

```      INSTALL @lib\$+"ARRAYLIB"

Max% = 10000
DIM vector(5), matrix(5,5)
DIM x(Max%), x2(Max%), x3(Max%), x4(Max%), x5(Max%)
DIM x6(Max%), x7(Max%), x8(Max%), x9(Max%), x10(Max%)
DIM y(Max%), xy(Max%), x2y(Max%), x3y(Max%), x4y(Max%), x5y(Max%)

npts% = 11
x() = 0,  1,  2,  3,  4,  5,  6,   7,   8,   9,   10
y() = 1,  6,  17, 34, 57, 86, 121, 162, 209, 262, 321

sum_x = SUM(x())
x2()  = x() * x()   : sum_x2  = SUM(x2())
x3()  = x() * x2()  : sum_x3  = SUM(x3())
x4()  = x2() * x2() : sum_x4  = SUM(x4())
x5()  = x2() * x3() : sum_x5  = SUM(x5())
x6()  = x3() * x3() : sum_x6  = SUM(x6())
x7()  = x3() * x4() : sum_x7  = SUM(x7())
x8()  = x4() * x4() : sum_x8  = SUM(x8())
x9()  = x4() * x5() : sum_x9  = SUM(x9())
x10() = x5() * x5() : sum_x10 = SUM(x10())

sum_y = SUM(y())
xy()  = x() * y()   : sum_xy  = SUM(xy())
x2y() = x2() * y()  : sum_x2y = SUM(x2y())
x3y() = x3() * y()  : sum_x3y = SUM(x3y())
x4y() = x4() * y()  : sum_x4y = SUM(x4y())
x5y() = x5() * y()  : sum_x5y = SUM(x5y())

matrix() = \
\ npts%,  sum_x,   sum_x2,  sum_x3,  sum_x4,  sum_x5, \
\ sum_x,  sum_x2,  sum_x3,  sum_x4,  sum_x5,  sum_x6, \
\ sum_x2, sum_x3,  sum_x4,  sum_x5,  sum_x6,  sum_x7, \
\ sum_x3, sum_x4,  sum_x5,  sum_x6,  sum_x7,  sum_x8, \
\ sum_x4, sum_x5,  sum_x6,  sum_x7,  sum_x8,  sum_x9, \
\ sum_x5, sum_x6,  sum_x7,  sum_x8,  sum_x9,  sum_x10

vector() = \
\ sum_y,  sum_xy,  sum_x2y, sum_x3y, sum_x4y, sum_x5y

PROC_invert(matrix())
vector() = matrix().vector()

@% = &2040A
PRINT "Polynomial coefficients = "
FOR term% = 5 TO 0 STEP -1
PRINT ;vector(term%) " * x^" STR\$(term%)
NEXT
```

{{out}}

```
Polynomial coefficients =
0.0000 * x^5
-0.0000 * x^4
0.0002 * x^3
2.9993 * x^2
2.0012 * x^1
0.9998 * x^0

```

## C

'''Include''' file (to make the code reusable easily) named polifitgsl.h

```#ifndef _POLIFITGSL_H
#define _POLIFITGSL_H
#include <gsl/gsl_multifit.h>
#include <stdbool.h>
#include <math.h>
bool polynomialfit(int obs, int degree,
double *dx, double *dy, double *store); /* n, p */
#endif
```

'''Implementation''' (the examples [http://www.gnu.org/software/gsl/manual/html_node/Fitting-Examples.html here] helped alot to code this quickly):

```#include "polifitgsl.h"

bool polynomialfit(int obs, int degree,
double *dx, double *dy, double *store) /* n, p */
{
gsl_multifit_linear_workspace *ws;
gsl_matrix *cov, *X;
gsl_vector *y, *c;
double chisq;

int i, j;

X = gsl_matrix_alloc(obs, degree);
y = gsl_vector_alloc(obs);
c = gsl_vector_alloc(degree);
cov = gsl_matrix_alloc(degree, degree);

for(i=0; i < obs; i++) {
for(j=0; j < degree; j++) {
gsl_matrix_set(X, i, j, pow(dx[i], j));
}
gsl_vector_set(y, i, dy[i]);
}

ws = gsl_multifit_linear_alloc(obs, degree);
gsl_multifit_linear(X, y, c, cov, &chisq, ws);

/* store result ... */
for(i=0; i < degree; i++)
{
store[i] = gsl_vector_get(c, i);
}

gsl_multifit_linear_free(ws);
gsl_matrix_free(X);
gsl_matrix_free(cov);
gsl_vector_free(y);
gsl_vector_free(c);
return true; /* we do not "analyse" the result (cov matrix mainly)
to know if the fit is "good" */
}
```

'''Testing''':

```#include <stdio.h>

#include "polifitgsl.h"

#define NP 11
double x[] = {0,  1,  2,  3,  4,  5,  6,   7,   8,   9,   10};
double y[] = {1,  6,  17, 34, 57, 86, 121, 162, 209, 262, 321};

#define DEGREE 3
double coeff[DEGREE];

int main()
{
int i;

polynomialfit(NP, DEGREE, x, y, coeff);
for(i=0; i < DEGREE; i++) {
printf("%lf\n", coeff[i]);
}
return 0;
}
```

{{out}}

```1.000000
2.000000
3.000000
```

## C++

{{trans|Java}}

```#include <algorithm>
#include <iostream>
#include <numeric>
#include <vector>

void polyRegression(const std::vector<int>& x, const std::vector<int>& y) {
int n = x.size();
std::vector<int> r(n);
std::iota(r.begin(), r.end(), 0);
double xm = std::accumulate(x.begin(), x.end(), 0.0) / x.size();
double ym = std::accumulate(y.begin(), y.end(), 0.0) / y.size();
double x2m = std::transform_reduce(r.begin(), r.end(), 0.0, std::plus<double>{}, [](double a) {return a * a; }) / r.size();
double x3m = std::transform_reduce(r.begin(), r.end(), 0.0, std::plus<double>{}, [](double a) {return a * a * a; }) / r.size();
double x4m = std::transform_reduce(r.begin(), r.end(), 0.0, std::plus<double>{}, [](double a) {return a * a * a * a; }) / r.size();

double xym = std::transform_reduce(x.begin(), x.end(), y.begin(), 0.0, std::plus<double>{}, std::multiplies<double>{});
xym /= fmin(x.size(), y.size());

double x2ym = std::transform_reduce(x.begin(), x.end(), y.begin(), 0.0, std::plus<double>{}, [](double a, double b) { return a * a * b; });
x2ym /= fmin(x.size(), y.size());

double sxx = x2m - xm * xm;
double sxy = xym - xm * ym;
double sxx2 = x3m - xm * x2m;
double sx2x2 = x4m - x2m * x2m;
double sx2y = x2ym - x2m * ym;

double b = (sxy * sx2x2 - sx2y * sxx2) / (sxx * sx2x2 - sxx2 * sxx2);
double c = (sx2y * sxx - sxy * sxx2) / (sxx * sx2x2 - sxx2 * sxx2);
double a = ym - b * xm - c * x2m;

auto abc = [a, b, c](int xx) {
return a + b * xx + c * xx*xx;
};

std::cout << "y = " << a << " + " << b << "x + " << c << "x^2" << std::endl;
std::cout << " Input  Approximation" << std::endl;
std::cout << " x   y     y1" << std::endl;

auto xit = x.cbegin();
auto xend = x.cend();
auto yit = y.cbegin();
auto yend = y.cend();
while (xit != xend && yit != yend) {
printf("%2d %3d  %5.1f\n", *xit, *yit, abc(*xit));
xit = std::next(xit);
yit = std::next(yit);
}
}

int main() {
using namespace std;

vector<int> x(11);
iota(x.begin(), x.end(), 0);

vector<int> y{ 1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321 };

polyRegression(x, y);

return 0;
}
```

{{out}}

```y = 1 + 2x + 3x^2
Input  Approximation
x   y     y1
0   1    1.0
1   6    6.0
2  17   17.0
3  34   34.0
4  57   57.0
5  86   86.0
6 121  121.0
7 162  162.0
8 209  209.0
9 262  262.0
10 321  321.0
```

## C#

```        public static double[] Polyfit(double[] x, double[] y, int degree)
{
// Vandermonde matrix
var v = new DenseMatrix(x.Length, degree + 1);
for (int i = 0; i < v.RowCount; i++)
for (int j = 0; j <= degree; j++) v[i, j] = Math.Pow(x[i], j);
var yv = new DenseVector(y).ToColumnMatrix();
QR qr = v.QR();
// Math.Net doesn't have an "economy" QR, so:
// cut R short to square upper triangle, then recompute Q
var r = qr.R.SubMatrix(0, degree + 1, 0, degree + 1);
var q = v.Multiply(r.Inverse());
var p = r.Inverse().Multiply(q.TransposeThisAndMultiply(yv));
return p.Column(0).ToArray();
}
```

Example:

```        static void Main(string[] args)
{
const int degree = 2;
var x = new[] {0.0, 1.0,  2.0,  3.0,  4.0,  5.0,   6.0,   7.0,   8.0,   9.0,  10.0};
var y = new[] {1.0, 6.0, 17.0, 34.0, 57.0, 86.0, 121.0, 162.0, 209.0, 262.0, 321.0};
var p = Polyfit(x, y, degree);
foreach (var d in p) Console.Write("{0} ",d);
Console.WriteLine();
for (int i = 0; i < x.Length; i++ )
Console.WriteLine("{0} => {1} diff {2}", x[i], Polyval(p,x[i]), y[i] - Polyval(p,x[i]));
}
```

## Common Lisp

Uses the routine (lsqr A b) from [[Multiple regression]] and (mtp A) from [[Matrix transposition]].

```;; Least square fit of a polynomial of order n the x-y-curve.
(defun polyfit (x y n)
(A (make-array `(,m ,(+ n 1)) :initial-element 0)))
(loop for i from 0 to (- m 1) do
(loop for j from 0 to n do
(setf (aref A i j)
(expt (aref x 0 i) j))))
(lsqr A (mtp y))))
```

Example:

```(let ((x (make-array '(1 11) :initial-contents '((0 1 2 3 4 5 6 7 8 9 10))))
(y (make-array '(1 11) :initial-contents '((1 6 17 34 57 86 121 162 209 262 321)))))
(polyfit x y 2))

#2A((0.9999999999999759d0) (2.000000000000005d0) (3.0d0))
```

## D

{{trans|Kotlin}}

```import std.algorithm;
import std.range;
import std.stdio;

auto average(R)(R r) {
auto t = r.fold!("a+b", "a+1")(0, 0);
return cast(double) t[0] / t[1];
}

void polyRegression(int[] x, int[] y) {
auto n = x.length;
auto r = iota(0, n).array;
auto xm = x.average();
auto ym = y.average();
auto x2m = r.map!"a*a".average();
auto x3m = r.map!"a*a*a".average();
auto x4m = r.map!"a*a*a*a".average();
auto xym = x.zip(y).map!"a[0]*a[1]".average();
auto x2ym = x.zip(y).map!"a[0]*a[0]*a[1]".average();

auto sxx = x2m - xm * xm;
auto sxy = xym - xm * ym;
auto sxx2 = x3m - xm * x2m;
auto sx2x2 = x4m - x2m * x2m;
auto sx2y = x2ym - x2m * ym;

auto b = (sxy * sx2x2 - sx2y * sxx2) / (sxx * sx2x2 - sxx2 * sxx2);
auto c = (sx2y * sxx - sxy * sxx2) / (sxx * sx2x2 - sxx2 * sxx2);
auto a = ym - b * xm - c * x2m;

real abc(int xx) {
return a + b * xx + c * xx * xx;
}

writeln("y = ", a, " + ", b, "x + ", c, "x^2");
writeln(" Input  Approximation");
writeln(" x   y     y1");
foreach (i; 0..n) {
writefln("%2d %3d  %5.1f", x[i], y[i], abc(x[i]));
}
}

void main() {
auto x = iota(0, 11).array;
auto y = [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321];
polyRegression(x, y);
}
```

{{out}}

```y = 1 + 2x + 3x^2
Input  Approximation
x   y     y1
0   1    1.0
1   6    6.0
2  17   17.0
3  34   34.0
4  57   57.0
5  86   86.0
6 121  121.0
7 162  162.0
8 209  209.0
9 262  262.0
10 321  321.0
```

## Emacs Lisp

Simple solution by Emacs Lisp and built-in Emacs Calc.

(setq x '[0 1 2 3 4 5 6 7 8 9 10]) (setq y '[1 6 17 34 57 86 121 162 209 262 321]) (calc-eval (format "fit(a*x^2+b*x+c,[x],[a,b,c],[%s %s])" x y))
```

{{out}}

```txt

"3. x^2 + 1.99999999996 x + 1.00000000006"

```

## Fortran

```module fitting
contains

function polyfit(vx, vy, d)
implicit none
integer, intent(in)                   :: d
integer, parameter                    :: dp = selected_real_kind(15, 307)
real(dp), dimension(d+1)              :: polyfit
real(dp), dimension(:), intent(in)    :: vx, vy

real(dp), dimension(:,:), allocatable :: X
real(dp), dimension(:,:), allocatable :: XT
real(dp), dimension(:,:), allocatable :: XTX

integer :: i, j

integer     :: n, lda, lwork
integer :: info
integer, dimension(:), allocatable :: ipiv
real(dp), dimension(:), allocatable :: work

n = d+1
lda = n
lwork = n

allocate(ipiv(n))
allocate(work(lwork))
allocate(XT(n, size(vx)))
allocate(X(size(vx), n))
allocate(XTX(n, n))

! prepare the matrix
do i = 0, d
do j = 1, size(vx)
X(j, i+1) = vx(j)**i
end do
end do

XT  = transpose(X)
XTX = matmul(XT, X)

! calls to LAPACK subs DGETRF and DGETRI
call DGETRF(n, n, XTX, lda, ipiv, info)
if ( info /= 0 ) then
print *, "problem"
return
end if
call DGETRI(n, XTX, lda, ipiv, work, lwork, info)
if ( info /= 0 ) then
print *, "problem"
return
end if

polyfit = matmul( matmul(XTX, XT), vy)

deallocate(ipiv)
deallocate(work)
deallocate(X)
deallocate(XT)
deallocate(XTX)

end function

end module
```

### Example

```program PolynomalFitting
use fitting
implicit none

! let us test it
integer, parameter      :: degree = 2
integer, parameter      :: dp = selected_real_kind(15, 307)
integer                 :: i
real(dp), dimension(11) :: x = (/ (i,i=0,10) /)
real(dp), dimension(11) :: y = (/ 1,   6,  17,  34, &
57,  86, 121, 162, &
209, 262, 321 /)
real(dp), dimension(degree+1) :: a

a = polyfit(x, y, degree)

write (*, '(F9.4)') a

end program
```

{{out}} (lower powers first, so this seems the opposite of the Python output):

```
1.0000
2.0000
3.0000

```

## FreeBASIC

```Sub GaussJordan(matrix() As Double,rhs() As Double,ans() As Double)
Dim As Integer n=Ubound(matrix,1)
Redim ans(0):Redim ans(1 To n)
Dim As Double b(1 To n,1 To n),r(1 To n)
For c As Integer=1 To n 'take copies
r(c)=rhs(c)
For d As Integer=1 To n
b(c,d)=matrix(c,d)
Next d
Next c
#macro pivot(num)
For p1 As Integer  = num To n - 1
For p2 As Integer  = p1 + 1 To n
If Abs(b(p1,num))<Abs(b(p2,num)) Then
Swap r(p1),r(p2)
For g As Integer=1 To n
Swap b(p1,g),b(p2,g)
Next g
End If
Next p2
Next p1
#endmacro
For k As Integer=1 To n-1
pivot(k)              'full pivoting
For row As Integer =k To n-1
If b(row+1,k)=0 Then Exit For
Var f=b(k,k)/b(row+1,k)
r(row+1)=r(row+1)*f-r(k)
For g As Integer=1 To n
b((row+1),g)=b((row+1),g)*f-b(k,g)
Next g
Next row
Next k
'back substitute
For z As Integer=n To 1 Step -1
ans(z)=r(z)/b(z,z)
For j As Integer = n To z+1 Step -1
ans(z)=ans(z)-(b(z,j)*ans(j)/b(z,z))
Next j
Next    z
End Sub

'Interpolate through points.
Sub Interpolate(x_values() As Double,y_values() As Double,p() As Double)
Var n=Ubound(x_values)
Redim p(0):Redim p(1 To n)
Dim As Double matrix(1 To n,1 To n),rhs(1 To n)
For a As Integer=1 To n
rhs(a)=y_values(a)
For b As Integer=1 To n
matrix(a,b)=x_values(a)^(b-1)
Next b
Next a
'Solve the linear equations
GaussJordan(matrix(),rhs(),p())
End Sub

'
### ===================== SET UP THE POINTS ============

Dim As Double x(1 To ...)={0,1,2,3,4,5,6,7,8,9,10}
Dim As Double y(1 To ...)={1,6,17,34,57,86,121,162,209,262,321}

Redim As Double Poly(0)
'Get the polynomial Poly()
Interpolate(x(),y(),Poly())

'print coefficients to console
print "Polynomial Coefficients:"
print
For z As Integer=1 To Ubound(Poly)
If z=1 Then
Print "constant term  ";tab(20);Poly(z)
Else
Print tab(8); "x^";z-1;" =  ";tab(20);Poly(z)
End If
Next z

sleep
```

{{out}}

```Polynomial Coefficients:

constant term       1
x^ 1 =       2
x^ 2 =       3
x^ 3 =       0
x^ 4 =       0
x^ 5 =       0
x^ 6 =       0
x^ 7 =       0
x^ 8 =       0
x^ 9 =       0
x^ 10 =      0
```

## GAP

```PolynomialRegression := function(x, y, n)
local a;
a := List([0 .. n], i -> List(x, s -> s^i));
return TransposedMat((a * TransposedMat(a))^-1 * a * TransposedMat([y]))[1];
end;

x := [0,  1,  2,  3,  4,  5,  6,   7,   8,   9,   10];
y := [1,  6,  17, 34, 57, 86, 121, 162, 209, 262, 321];

# Return coefficients in ascending degree order
PolynomialRegression(x, y, 2);
# [ 1, 2, 3 ]
```

## gnuplot

```# The polynomial approximation
f(x) = a*x**2 + b*x + c

# Initial values for parameters
a = 0.1
b = 0.1
c = 0.1

# Fit f to the following data by modifying the variables a, b, c
fit f(x) '-' via a, b, c
0   1
1   6
2  17
3  34
4  57
5  86
6 121
7 162
8 209
9 262
10 321
e

print sprintf("\n --- \n Polynomial fit: %.4f x^2 + %.4f x + %.4f\n", a, b, c)
```

## Go

### Library gonum/matrix

```package main

import (
"fmt"

"github.com/gonum/matrix/mat64"
)

var (
x = []float64{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
y = []float64{1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321}

degree = 2
)

func main() {
a := Vandermonde(x, degree)
b := mat64.NewDense(len(y), 1, y)
c := mat64.NewDense(degree+1, 1, nil)

qr := new(mat64.QR)
qr.Factorize(a)

err := c.SolveQR(qr, false, b)
if err != nil {
fmt.Println(err)
} else {
fmt.Printf("%.3f\n", mat64.Formatted(c))
}
}

func Vandermonde(a []float64, degree int) *mat64.Dense {
x := mat64.NewDense(len(a), degree+1, nil)
for i := range a {
for j, p := 0, 1.; j <= degree; j, p = j+1, p*a[i] {
x.Set(i, j, p)
}
}
return x
}
```

{{out}}

```
⎡1.000⎤
⎢2.000⎥
⎣3.000⎦

```

### Library go.matrix

Least squares solution using QR decomposition and package [http://github.com/skelterjohn/go.matrix go.matrix].

```package main

import (
"fmt"

"github.com/skelterjohn/go.matrix"
)

var xGiven = []float64{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
var yGiven = []float64{1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321}
var degree = 2

func main() {
m := len(yGiven)
n := degree + 1
y := matrix.MakeDenseMatrix(yGiven, m, 1)
x := matrix.Zeros(m, n)
for i := 0; i < m; i++ {
ip := float64(1)
for j := 0; j < n; j++ {
x.Set(i, j, ip)
ip *= xGiven[i]
}
}

q, r := x.QR()
qty, err := q.Transpose().Times(y)
if err != nil {
fmt.Println(err)
return
}
c := make([]float64, n)
for i := n - 1; i >= 0; i-- {
c[i] = qty.Get(i, 0)
for j := i + 1; j < n; j++ {
c[i] -= c[j] * r.Get(i, j)
}
c[i] /= r.Get(i, i)
}
fmt.Println(c)
}
```

{{out}} (lowest order coefficient first)

```
[0.9999999999999758 2.000000000000015 2.999999999999999]

```

Uses module Matrix.LU from [http://hackage.haskell.org/package/dsp hackageDB DSP]

```import Data.List
import Data.Array
import Control.Arrow
import Matrix.LU

ppoly p x = map (x**) p

polyfit d ry = elems \$ solve mat vec  where
mat = listArray ((1,1), (d,d)) \$ liftM2 concatMap ppoly id [0..fromIntegral \$ pred d]
vec = listArray (1,d) \$ take d ry
```

{{out}} in GHCi:

```*Main> polyfit 3 [1,6,17,34,57,86,121,162,209,262,321]
[1.0,2.0,3.0]
```

## HicEst

```REAL :: n=10, x(n), y(n), m=3, p(m)

x = (0,  1,  2,  3,  4,  5,  6,   7,   8,   9,   10)
y = (1,  6,  17, 34, 57, 86, 121, 162, 209, 262, 321)

p = 2 ! initial guess for the polynom's coefficients

SOLVE(NUL=Theory()-y(nr), Unknown=p, DataIdx=nr, Iters=iterations)

WRITE(ClipBoard, Name) p, iterations

FUNCTION Theory()
! called by the solver of the SOLVE function. All variables are global
Theory = p(1)*x(nr)^2 + p(2)*x(nr) + p(3)
END
```

{{out}}

```SOLVE performs a (nonlinear) least-square fit (Levenberg-Marquardt):
p(1)=2.997135145; p(2)=2.011348347; p(3)=0.9906627242; iterations=19;
```

## Hy

```(import [numpy [polyfit]])

(setv x (range 11))
(setv y [1 6 17 34 57 86 121 162 209 262 321])

(print (polyfit x y 2))
```

## J

```   Y=:1 6 17 34 57 86 121 162 209 262 321
(%. ^/~@x:@i.@#) Y
1 2 3 0 0 0 0 0 0 0 0
```

Note that this implementation does not use floating point numbers, so we do not introduce floating point errors. Using exact arithmetic has a speed penalty, but for small problems like this it is inconsequential.

The above solution fits a polynomial of order 11. If the order of the polynomial is known to be 3 (as is implied in the task description) then the following solution is probably preferable:

```   Y %. (i.3) ^/~ i.#Y
1 2 3
```

(note that this time we used floating point numbers, so that result is approximate rather than exact - it only looks exact because of how J displays floating point numbers (by default, J assumes six digits of accuracy) - changing (i.3) to (x:i.3) would give us an exact result, if that mattered.)

## Java

{{trans|D}} {{works with|Java|8}}

```import java.util.Arrays;
import java.util.function.IntToDoubleFunction;
import java.util.stream.IntStream;

public class PolynomialRegression {
private static void polyRegression(int[] x, int[] y) {
int n = x.length;
int[] r = IntStream.range(0, n).toArray();
double xm = Arrays.stream(x).average().orElse(Double.NaN);
double ym = Arrays.stream(y).average().orElse(Double.NaN);
double x2m = Arrays.stream(r).map(a -> a * a).average().orElse(Double.NaN);
double x3m = Arrays.stream(r).map(a -> a * a * a).average().orElse(Double.NaN);
double x4m = Arrays.stream(r).map(a -> a * a * a * a).average().orElse(Double.NaN);
double xym = 0.0;
for (int i = 0; i < x.length && i < y.length; ++i) {
xym += x[i] * y[i];
}
xym /= Math.min(x.length, y.length);
double x2ym = 0.0;
for (int i = 0; i < x.length && i < y.length; ++i) {
x2ym += x[i] * x[i] * y[i];
}
x2ym /= Math.min(x.length, y.length);

double sxx = x2m - xm * xm;
double sxy = xym - xm * ym;
double sxx2 = x3m - xm * x2m;
double sx2x2 = x4m - x2m * x2m;
double sx2y = x2ym - x2m * ym;

double b = (sxy * sx2x2 - sx2y * sxx2) / (sxx * sx2x2 - sxx2 * sxx2);
double c = (sx2y * sxx - sxy * sxx2) / (sxx * sx2x2 - sxx2 * sxx2);
double a = ym - b * xm - c * x2m;

IntToDoubleFunction abc = (int xx) -> a + b * xx + c * xx * xx;

System.out.println("y = " + a + " + " + b + "x + " + c + "x^2");
System.out.println(" Input  Approximation");
System.out.println(" x   y     y1");
for (int i = 0; i < n; ++i) {
System.out.printf("%2d %3d  %5.1f\n", x[i], y[i], abc.applyAsDouble(x[i]));
}
}

public static void main(String[] args) {
int[] x = IntStream.range(0, 11).toArray();
int[] y = new int[]{1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321};
polyRegression(x, y);
}
}
```

{{out}}

```y = 1.0 + 2.0x + 3.0x^2
Input  Approximation
x   y     y1
0   1    1.0
1   6    6.0
2  17   17.0
3  34   34.0
4  57   57.0
5  86   86.0
6 121  121.0
7 162  162.0
8 209  209.0
9 262  262.0
10 321  321.0
```

## Julia

{{works with|Julia|0.6}} The least-squares fit problem for a degree n can be solved with the built-in backslash operator (coefficients in increasing order of degree):

```polyfit(x::Vector, y::Vector, deg::Int) = collect(v ^ p for v in x, p in 0:deg) \ y

x = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
y = [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321]
@show polyfit(x, y, 2)
```

{{out}}

```polyfit(x, y, 2) = [1.0, 2.0, 3.0]
```

## Kotlin

{{trans|REXX}}

```// version 1.1.51

fun polyRegression(x: IntArray, y: IntArray) {
val n = x.size
val r = 0 until n
val xm = x.average()
val ym = y.average()
val x2m = r.map { it * it }.average()
val x3m = r.map { it * it * it }.average()
val x4m = r.map { it * it * it * it }.average()
val xym = x.zip(y).map { it.first * it.second }.average()
val x2ym = x.zip(y).map { it.first * it.first * it.second }.average()

val sxx = x2m - xm * xm
val sxy = xym - xm * ym
val sxx2 = x3m - xm * x2m
val sx2x2 = x4m - x2m * x2m
val sx2y = x2ym - x2m * ym

val b = (sxy * sx2x2 - sx2y * sxx2) / (sxx * sx2x2 - sxx2 * sxx2)
val c = (sx2y * sxx - sxy * sxx2) / (sxx * sx2x2 - sxx2 * sxx2)
val a = ym - b * xm - c * x2m

fun abc(xx: Int) = a + b * xx + c * xx * xx

println("y = \$a + \${b}x + \${c}x^2\n")
println(" Input  Approximation")
println(" x   y     y1")
for (i in 0 until n) {
System.out.printf("%2d %3d  %5.1f\n", x[i], y[i], abc(x[i]))
}
}

fun main(args: Array<String>) {
val x = IntArray(11) { it }
val y = intArrayOf(1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321)
polyRegression(x, y)
}
```

{{out}}

```
y = 1.0 + 2.0x + 3.0x^2

Input  Approximation
x   y     y1
0   1    1.0
1   6    6.0
2  17   17.0
3  34   34.0
4  57   57.0
5  86   86.0
6 121  121.0
7 162  162.0
8 209  209.0
9 262  262.0
10 321  321.0

```

## Lua

{{trans|Modula-2}}

```function eval(a,b,c,x)
return a + (b + c * x) * x
end

function regression(xa,ya)
local n = #xa

local xm = 0.0
local ym = 0.0
local x2m = 0.0
local x3m = 0.0
local x4m = 0.0
local xym = 0.0
local x2ym = 0.0

for i=1,n do
xm = xm + xa[i]
ym = ym + ya[i]
x2m = x2m + xa[i] * xa[i]
x3m = x3m + xa[i] * xa[i] * xa[i]
x4m = x4m + xa[i] * xa[i] * xa[i] * xa[i]
xym = xym + xa[i] * ya[i]
x2ym = x2ym + xa[i] * xa[i] * ya[i]
end
xm = xm / n
ym = ym / n
x2m = x2m / n
x3m = x3m / n
x4m = x4m / n
xym = xym / n
x2ym = x2ym / n

local sxx = x2m - xm * xm
local sxy = xym - xm * ym
local sxx2 = x3m - xm * x2m
local sx2x2 = x4m - x2m * x2m
local sx2y = x2ym - x2m * ym

local b = (sxy * sx2x2 - sx2y * sxx2) / (sxx * sx2x2 - sxx2 * sxx2)
local c = (sx2y * sxx - sxy * sxx2) / (sxx * sx2x2 - sxx2 * sxx2)
local a = ym - b * xm - c * x2m

print("y = "..a.." + "..b.."x + "..c.."x^2")

for i=1,n do
print(string.format("%2d %3d  %3d", xa[i], ya[i], eval(a, b, c, xa[i])))
end
end

local xa = {0, 1,  2,  3,  4,  5,   6,   7,   8,   9,  10}
local ya = {1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321}
regression(xa, ya)
```

{{out}}

```y = 1 + 2x + 3x^2
0   1    1
1   6    6
2  17   17
3  34   34
4  57   57
5  86   86
6 121  121
7 162  162
8 209  209
9 262  262
10 321  321
```

## Maple

```with(CurveFitting);
PolynomialInterpolation([[0, 1], [1, 6], [2, 17], [3, 34], [4, 57], [5, 86], [6, 121], [7, 162], [8, 209], [9, 262], [10, 321]], 'x');

```

Result:

```3*x^2+2*x+1
```

## Mathematica

Using the built-in "Fit" function.

```data = Transpose@{Range[0, 10], {1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321}};
Fit[data, {1, x, x^2}, x]
```

Second version: using built-in "InterpolatingPolynomial" function.

```Simplify@InterpolatingPolynomial[{{0, 1}, {1, 6}, {2, 17}, {3, 34}, {4, 57}, {5, 86}, {6, 121}, {7, 162}, {8, 209}, {9, 262}, {10, 321}}, x]
```

Result:

```1 + 2x + 3x^2
```

## MATLAB

Matlab has a built-in function "polyfit(x,y,n)" which performs this task. The arguments x and y are vectors which are parametrized by the index suck that $point_\left\{i\right\} = \left(x_\left\{i\right\},y_\left\{i\right\}\right)$ and the argument n is the order of the polynomial you want to fit. The output of this function is the coefficients of the polynomial which best fit these x,y value pairs.

``` x = [0,  1,  2,  3,  4,  5,  6,   7,   8,   9,   10];
>> y = [1,  6,  17, 34, 57, 86, 121, 162, 209, 262, 321];
>> polyfit(x,y,2)

ans =

2.999999999999998   2.000000000000019   0.999999999999956
```

=={{header|МК-61/52}}== Part 1: ПC С/П ПD ИП9 + П9 ИПC ИП5 + П5 ИПC x^2 П2 ИП6 + П6 ИП2 ИПC * ИП7 + П7 ИП2 x^2 ИП8 + П8 ИПC ИПD * ИПA + ПA ИП2 ИПD * ИПB + ПB ИПD КИП4 С/П БП 00

```

''Input'': В/О x<sub>1</sub> С/П y<sub>1</sub> С/П x<sub>2</sub> С/П y<sub>2</sub> С/П ...

Part 2:
<lang>ИП5	ПC	ИП6	ПD	П2	ИП7	П3	ИП4	ИПD	*
ИПC	ИП5	*	-	ПD	ИП4	ИП7	*	ИПC	ИП6
*	-	П7	ИП4	ИПA	*	ИПC	ИП9	*	-
ПA	ИП4	ИП3	*	ИП2	ИП5	*	-	П3	ИП4
ИП8	*	ИП2	ИП6	*	-	П8	ИП4	ИПB	*
ИП2	ИП9	*	-	ИПD	*	ИП3	ИПA	*	-
ИПD	ИП8	*	ИП7	ИП3	*	-	/	ПB	ИПA
ИПB	ИП7	*	-	ИПD	/	ПA	ИП9	ИПB	ИП6
*	-	ИПA	ИП5	*	-	ИП4	/	П9	С/П
```

''Result'': Р9 = a0, РA = a1, РB = a2.

```MODULE PolynomialRegression;
FROM FormatString IMPORT FormatString;
FROM RealStr IMPORT RealToStr;

PROCEDURE Eval(a,b,c,x : REAL) : REAL;
BEGIN
RETURN a + b*x + c*x*x;
END Eval;

PROCEDURE Regression(x,y : ARRAY OF INTEGER);
VAR
n,i : INTEGER;
xm,x2m,x3m,x4m : REAL;
ym : REAL;
xym,x2ym : REAL;
sxx,sxy,sxx2,sx2x2,sx2y : REAL;
a,b,c : REAL;
buf : ARRAY[0..63] OF CHAR;
BEGIN
n := SIZE(x)/SIZE(INTEGER);

xm := 0.0;
ym := 0.0;
x2m := 0.0;
x3m := 0.0;
x4m := 0.0;
xym := 0.0;
x2ym := 0.0;
FOR i:=0 TO n-1 DO
xm := xm + FLOAT(x[i]);
ym := ym + FLOAT(y[i]);
x2m := x2m + FLOAT(x[i]) * FLOAT(x[i]);
x3m := x3m + FLOAT(x[i]) * FLOAT(x[i]) * FLOAT(x[i]);
x4m := x4m + FLOAT(x[i]) * FLOAT(x[i]) * FLOAT(x[i]) * FLOAT(x[i]);
xym := xym + FLOAT(x[i]) * FLOAT(y[i]);
x2ym := x2ym + FLOAT(x[i]) * FLOAT(x[i]) * FLOAT(y[i]);
END;
xm := xm / FLOAT(n);
ym := ym / FLOAT(n);
x2m := x2m / FLOAT(n);
x3m := x3m / FLOAT(n);
x4m := x4m / FLOAT(n);
xym := xym / FLOAT(n);
x2ym := x2ym / FLOAT(n);

sxx := x2m - xm * xm;
sxy := xym - xm * ym;
sxx2 := x3m - xm * x2m;
sx2x2 := x4m - x2m * x2m;
sx2y := x2ym - x2m * ym;

b := (sxy * sx2x2 - sx2y * sxx2) / (sxx * sx2x2 - sxx2 * sxx2);
c := (sx2y * sxx - sxy * sxx2) / (sxx * sx2x2 - sxx2 * sxx2);
a := ym - b * xm - c * x2m;

WriteString("y = ");
RealToStr(a, buf);
WriteString(buf);
WriteString(" + ");
RealToStr(b, buf);
WriteString(buf);
WriteString("x + ");
RealToStr(c, buf);
WriteString(buf);
WriteString("x^2");
WriteLn;

FOR i:=0 TO n-1 DO
FormatString("%2i %3i  ", buf, x[i], y[i]);
WriteString(buf);
RealToStr(Eval(a,b,c,FLOAT(x[i])), buf);
WriteString(buf);
WriteLn;
END;
END Regression;

TYPE R = ARRAY[0..10] OF INTEGER;
VAR
x,y : R;
BEGIN
x := R{0,1,2,3,4,5,6,7,8,9,10};
y := R{1,6,17,34,57,86,121,162,209,262,321};
Regression(x,y);

END PolynomialRegression.
```

## Octave

```x = [0:10];
y = [1,   6,  17,  34,  57,  86, 121, 162, 209, 262, 321];
coeffs = polyfit(x, y, 2)
```

## PARI/GP

Lagrange interpolating polynomial:

```polinterpolate([0,1,2,3,4,5,6,7,8,9,10],[1,6,17,34,57,86,121,162,209,262,321])
```

In newer versions, this can be abbreviated:

```polinterpolate([0..10],[1,6,17,34,57,86,121,162,209,262,321])
```

{{out}}

```3*x^2 + 2*x + 1
```

Least-squares fit:

```V=[1,6,17,34,57,86,121,162,209,262,321]~;
M=matrix(#V,3,i,j,(i-1)^(j-1));Polrev(matsolve(M~*M,M~*V))
```

Code thanks to [http://pari.math.u-bordeaux.fr/archives/pari-users-1105/msg00006.html Bill Allombert] {{out}}

```3*x^2 + 2*x + 1
```

Least-squares polynomial fit in its own function:

```lsf(X,Y,n)=my(M=matrix(#X,n+1,i,j,X[i]^(j-1))); Polrev(matsolve(M~*M,M~*Y~))
lsf([0..10], [1,6,17,34,57,86,121,162,209,262,321], 2)
```

## Perl

This script depends on the Math::MatrixReal CPAN module to compute matrix determinants.

```use strict;
use warnings;
use feature 'say';

#This is a script to calculate an equation for a given set of coordinates.
#Input will be taken in sets of x and y. It can handle a grand total of 26 pairs.
#For matrix functions, we depend on the Math::MatrixReal package.
use Math::MatrixReal;

=pod

Step 1: Get each x coordinate all at once (delimited by " ") and each for y at once
on the next prompt in the same format (delimited by " ").
=cut

sub getPairs() {
my \$buffer = <STDIN>;
chomp(\$buffer);
return split(" ", \$buffer);
}
say("Please enter the values for the x coordinates, each delimited by a space. \(Ex: 0 1 2 3\)");
my @x = getPairs();
say("Please enter the values for the y coordinates, each delimited by a space. \(Ex: 0 1 2 3\)");
my @y = getPairs();
#This whole thing depends on the number of x's being the same as the number of y's
my \$pairs = scalar(@x);

=pod

Step 2: Devise the base equation of our polynomial using the following idea
There is some polynomial of degree n (n == number of pairs - 1) such that
f(x)=ax^n + bx^(n-1) + ... yx + z
=cut

#Create an array of coefficients and their degrees with the format ("coefficent degree")
my @alphabet;
my @degrees;
for(my \$alpha = "a", my \$degree = \$pairs - 1; \$degree >= 0; \$degree--, \$alpha++) {
push(@alphabet, "\$alpha");
push(@degrees, "\$degree");
}

=pod

Step 3: Using the array of coeffs and their degrees, set up individual equations solving for
each coordinate pair. Why put it in this format? It interfaces witht he Math::MatrixReal package better this way.
=cut

my @coeffs;
for(my \$count = 0; \$count < \$pairs; \$count++) {
my \$buffer = "[ ";
foreach (@degrees) {
\$buffer .= ((\$x[\$count] ** \$_) . " ");
}
push(@coeffs, (\$buffer . "]"));
}
my \$row;
foreach (@coeffs) {
\$row .= ("\$_\n");
}

=pod

Step 4: We now have rows of x's raised to powers. With this in mind, we create a coefficient matrix.
=cut

my \$matrix = Math::MatrixReal->new_from_string(\$row);
my \$buffMatrix = \$matrix->new_from_string(\$row);

=pod

Step 5: Now that we've gotten the matrix to do what we want it to do, we need to calculate the various determinants of the matrices
=cut

my \$coeffDet = \$matrix->det();

=pod

Step 6: Now that we have the determinant of the coefficient matrix, we need to find the determinants of the coefficient matrix with each column (1 at a time) replaced with the y values.
=cut

#NOTE: Unlike in Perl, matrix indices start at 1, not 0.
for(my \$rows = my \$column = 1; \$column <= \$pairs; \$column++) {
#Reassign the values in the current column to the y values
foreach (@y) {
\$buffMatrix->assign(\$rows, \$column, \$_);
\$rows++;
}
#Find the values for the variables a, b, ... y, z in the original polynomial
#To round the difference of the determinants, I had to get creative
my \$buffDet = \$buffMatrix->det() / \$coeffDet;
my \$tempDet = int(abs(\$buffDet) + .5);
\$alphabet[\$column - 1] = \$buffDet >= 0 ? \$tempDet : 0 - \$tempDet;
#Reset the buffer matrix and the row counter
\$buffMatrix = \$matrix->new_from_string(\$row);
\$rows = 1;
}

=pod

Step 7: Now that we've found the values of a, b, ... y, z of the original polynomial, it's time to form our polynomial!
=cut

my \$polynomial;
for(my \$i = 0; \$i < \$pairs-1; \$i++) {
if(\$alphabet[\$i] == 0) {
next;
}
if(\$alphabet[\$i] == 1) {
\$polynomial .= (\$degrees[\$i] . " + ");
}
if(\$degrees[\$i] == 1) {
\$polynomial .= (\$alphabet[\$i] . "x" . " + ");
}
else {
\$polynomial .= (\$alphabet[\$i] . "x^" . \$degrees[\$i] . " + ");
}
}
#Now for the last piece of the poly: the y-intercept.
\$polynomial .= \$alphabet[scalar(@alphabet)-1];

print("An approximating polynomial for your dataset is \$polynomial.\n");

```

{{output}}

```Please enter the values for the x coordinates, each delimited by a space. (Ex: 0 1 2 3)
0 1 2 3 4 5 6 7 8 9 10
Please enter the values for the y coordinates, each delimited by a space. (Ex: 0 1 2 3)
1 6 17 34 57 86 121 162 209 262 321
An approximating polynomial for your dataset is 3x^2 + 2x + 1.
```

## Perl 6

We'll use a Clifford algebra library.

```use Clifford;

constant @x1 = <0 1 2 3 4 5 6 7 8 9 10>;
constant @y = <1 6 17 34 57 86 121 162 209 262 321>;

constant \$x0 = [+] @e[^@x1];
constant \$x1 = [+] @x1 Z* @e;
constant \$x2 = [+] @x1 »**» 2  Z* @e;

constant \$y  = [+] @y Z* @e;

my \$J = \$x1 ∧ \$x2;
my \$I = \$x0 ∧ \$J;

my \$I2 = (\$I·\$I.reversion).Real;

.say for
((\$y ∧ \$J)·\$I.reversion)/\$I2,
((\$y ∧ (\$x2 ∧ \$x0))·\$I.reversion)/\$I2,
((\$y ∧ (\$x0 ∧ \$x1))·\$I.reversion)/\$I2;
```

{{out}}

```1
2
3

```

## Phix

{{trans|REXX}}

```constant x = {0,1,2,3,4,5,6,7,8,9,10}
constant y = {1,6,17,34,57,86,121,162,209,262,321}
constant n = length(x)

function regression()
atom {xm, ym, x2m, x3m, x4m, xym, x2ym} @= 0
for i=1 to n do
atom xi = x[i],
yi = y[i]
xm += xi
ym += yi
x2m += power(xi,2)
x3m += power(xi,3)
x4m += power(xi,4)
xym += xi*yi
x2ym += power(xi,2)*yi
end for
xm /= n
ym /= n
x2m /= n
x3m /= n
x4m /= n
xym /= n
x2ym /= n
atom Sxx = x2m-power(xm,2),
Sxy = xym-xm*ym,
Sxx2 = x3m-xm*x2m,
Sx2x2 = x4m-power(x2m,2),
Sx2y = x2ym-x2m*ym,
B = (Sxy*Sx2x2-Sx2y*Sxx2)/(Sxx*Sx2x2-power(Sxx2,2)),
C = (Sx2y*Sxx-Sxy*Sxx2)/(Sxx*Sx2x2-power(Sxx2,2)),
A = ym-B*xm-C*x2m
return {C,B,A}
end function

atom {a,b,c} = regression()

function f(atom x)
return a*x*x+b*x+c
end function

printf(1,"y=%gx^2+%gx+%g\n",{a,b,c})
printf(1,"\n  x   y  f(x)\n")
for i=1 to n do
printf(1," %2d %3d   %3g\n",{x[i],y[i],f(x[i])})
end for
```

{{out}}

```
y=3x^2+2x+1

x   y  f(x)
0   1     1
1   6     6
2  17    17
3  34    34
4  57    57
5  86    86
6 121   121
7 162   162
8 209   209
9 262   262
10 321   321

```

Alternatively, a simple plot, (as per [[Polynomial_regression#Racket|Racket]]): {{libheader|pGUI}}

```include pGUI.e

constant x = {0,1,2,3,4,5,6,7,8,9,10}
constant y = {1,6,17,34,57,86,121,162,209,262,321}

IupOpen()

Ihandle plot = IupPlot("GRID=YES, MARGINLEFT=50, MARGINBOTTOM=40")
-- (just add ", AXS_YSCALE=LOG10" for a nice log scale)
IupPlotBegin(plot, 0)
for i=1 to length(x) do
end for
{} = IupPlotEnd(plot)

Ihandle dlg = IupDialog(plot)
IupSetAttributes(dlg, "RASTERSIZE=%dx%d", {640, 480})
IupSetAttribute(dlg, "TITLE", "simple plot")
IupShow(dlg)

IupMainLoop()
IupClose()
```

## PowerShell

```
function qr([double[][]]\$A) {
\$m,\$n = \$A.count, \$A[0].count
\$pm,\$pn = (\$m-1), (\$n-1)
[double[][]]\$Q = 0..(\$m-1) | foreach{\$row = @(0) * \$m; \$row[\$_] = 1; ,\$row}
[double[][]]\$R = \$A | foreach{\$row = \$_; ,@(0..\$pn | foreach{\$row[\$_]})}
foreach (\$h in 0..\$pn) {
[double[]]\$u = \$R[\$h..\$pm] | foreach{\$_[\$h]}
[double]\$nu = \$u | foreach {[double]\$sq = 0} {\$sq += \$_*\$_} {[Math]::Sqrt(\$sq)}
\$u[0] -= if (\$u[0] -lt 1) {\$nu} else {-\$nu}
[double]\$nu = \$u | foreach {\$sq = 0} {\$sq += \$_*\$_} {[Math]::Sqrt(\$sq)}
[double[]]\$u = \$u | foreach { \$_/\$nu}
[double[][]]\$v = 0..(\$u.Count - 1) | foreach{\$i = \$_; ,(\$u | foreach{2*\$u[\$i]*\$_})}
[double[][]]\$CR = \$R | foreach{\$row = \$_; ,@(0..\$pn | foreach{\$row[\$_]})}
[double[][]]\$CQ = \$Q | foreach{\$row = \$_; ,@(0..\$pm | foreach{\$row[\$_]})}
foreach (\$i in  \$h..\$pm) {
foreach (\$j in  \$h..\$pn) {
\$R[\$i][\$j] -=  \$h..\$pm | foreach {[double]\$sum = 0} {\$sum += \$v[\$i-\$h][\$_-\$h]*\$CR[\$_][\$j]} {\$sum}
}
}
if (0 -eq \$h)  {
foreach (\$i in  \$h..\$pm) {
foreach (\$j in  \$h..\$pm) {
\$Q[\$i][\$j] -=  \$h..\$pm | foreach {\$sum = 0} {\$sum += \$v[\$i][\$_]*\$CQ[\$_][\$j]} {\$sum}
}
}
} else  {
\$p = \$h-1
foreach (\$i in  \$h..\$pm) {
foreach (\$j in  0..\$p) {
\$Q[\$i][\$j] -=  \$h..\$pm | foreach {\$sum = 0} {\$sum += \$v[\$i-\$h][\$_-\$h]*\$CQ[\$_][\$j]} {\$sum}
}
foreach (\$j in  \$h..\$pm) {
\$Q[\$i][\$j] -=  \$h..\$pm | foreach {\$sum = 0} {\$sum += \$v[\$i-\$h][\$_-\$h]*\$CQ[\$_][\$j]} {\$sum}
}
}
}
}
foreach (\$i in  0..\$pm) {
foreach (\$j in  \$i..\$pm) {\$Q[\$i][\$j],\$Q[\$j][\$i] = \$Q[\$j][\$i],\$Q[\$i][\$j]}
}
[PSCustomObject]@{"Q" = \$Q; "R" = \$R}
}

function leastsquares([Double[][]]\$A,[Double[]]\$y) {
\$QR = qr \$A
[Double[][]]\$Q = \$QR.Q
[Double[][]]\$R = \$QR.R
\$m,\$n = \$A.count, \$A[0].count
[Double[]]\$z = foreach (\$j in  0..(\$m-1)) {
0..(\$m-1) | foreach {\$sum = 0} {\$sum += \$Q[\$_][\$j]*\$y[\$_]} {\$sum}
}
[Double[]]\$x = @(0)*\$n
for (\$i = \$n-1; \$i -ge 0; \$i--) {
for (\$j = \$i+1; \$j -lt \$n; \$j++) {
\$z[\$i] -= \$x[\$j]*\$R[\$i][\$j]
}
\$x[\$i] = \$z[\$i]/\$R[\$i][\$i]
}
\$x
}

function polyfit([Double[]]\$x,[Double[]]\$y,\$n) {
\$m = \$x.Count
[Double[][]]\$A = 0..(\$m-1) | foreach{\$row = @(1) * (\$n+1); ,\$row}
for (\$i = 0; \$i -lt \$m; \$i++) {
for (\$j = \$n-1; 0 -le \$j; \$j--) {
\$A[\$i][\$j] = \$A[\$i][\$j+1]*\$x[\$i]
}
}
leastsquares \$A \$y
}

function show(\$m) {\$m | foreach {write-host "\$_"}}

\$A = @(@(12,-51,4), @(6,167,-68), @(-4,24,-41))
\$x = @(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10)
\$y = @(1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321)
"polyfit "
"X^2 X constant"
"\$(polyfit \$x \$y 2)"

```

{{out}}

```
polyfit
X^2 X constant
3 1.99999999999998 1.00000000000005

```

## Python

``` x = [0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10]
>>> y = [1,   6,  17,  34,  57,  86, 121, 162, 209, 262, 321]
>>> coeffs = numpy.polyfit(x,y,deg=2)
>>> coeffs
array([ 3.,  2.,  1.])
```

``` yf = numpy.polyval(numpy.poly1d(coeffs), x)
>>> yf
array([   1.,    6.,   17.,   34.,   57.,   86.,  121.,  162.,  209., 262.,  321.])
```

Find max absolute error:

``` '%.1g' % max(y-yf)
'1e-013'
```

### Example

For input arrays `x' and `y':

``` x = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
>>> y = [2.7, 2.8, 31.4, 38.1, 58.0, 76.2, 100.5, 130.0, 149.3, 180.0]
```
``` p = numpy.poly1d(numpy.polyfit(x, y, deg=2), variable='N')
>>> print p
2
1.085 N + 10.36 N - 0.6164
```

Thus we confirm once more that for already sorted sequences the considered quick sort implementation has quadratic dependence on sequence length (see [[Query Performance|'''Example''' section for Python language on ''Query Performance'' page]]).

## R

The easiest (and most robust) approach to solve this in R is to use the base package's ''lm'' function which will find the least squares solution via a QR decomposition:

```
x <- c(0,  1,  2,  3,  4,  5,  6,   7,   8,   9,   10)
y <- c(1,  6,  17, 34, 57, 86, 121, 162, 209, 262, 321)
coef(lm(y ~ x + I(x^2)))
```

{{out}}

```
(Intercept)           x      I(x^2)
1           2           3

```

Alternately, use poly:

```coef(lm(y ~ poly(x, 2, raw=T)))
```

{{out}}

```         (Intercept) poly(x, 2, raw = T)1 poly(x, 2, raw = T)2
1                    2                    3
```

## Racket

```
#lang racket
(require math plot)

(define xs '(0 1  2  3  4  5   6   7   8   9  10))
(define ys '(1 6 17 34 57 86 121 162 209 262 321))

(define (fit x y n)
(define Y (->col-matrix y))
(define V (vandermonde-matrix x (+ n 1)))
(define VT (matrix-transpose V))
(matrix->vector (matrix-solve (matrix* VT V) (matrix* VT Y))))

(define ((poly v) x)
(for/sum ([c v] [i (in-naturals)])
(* c (expt x i))))

(plot (list (points   (map vector xs ys))
(function (poly (fit xs ys 2)))))

```

{{out}} [[File:polyreg-racket.png]]

## REXX

```/* REXX ---------------------------------------------------------------
* Implementation of http://keisan.casio.com/exec/system/14059932254941
*--------------------------------------------------------------------*/
xl='0 1  2  3  4  5   6   7   8   9  10'
yl='1 6 17 34 57 86 121 162 209 262 321'
n=11
Do i=1 To n
Parse Var xl x.i xl
Parse Var yl y.i yl
End
xm=0
ym=0
x2m=0
x3m=0
x4m=0
xym=0
x2ym=0
Do i=1 To n
xm=xm+x.i
ym=ym+y.i
x2m=x2m+x.i**2
x3m=x3m+x.i**3
x4m=x4m+x.i**4
xym=xym+x.i*y.i
x2ym=x2ym+(x.i**2)*y.i
End
xm =xm /n
ym =ym /n
x2m=x2m/n
x3m=x3m/n
x4m=x4m/n
xym=xym/n
x2ym=x2ym/n
Sxx=x2m-xm**2
Sxy=xym-xm*ym
Sxx2=x3m-xm*x2m
Sx2x2=x4m-x2m**2
Sx2y=x2ym-x2m*ym
B=(Sxy*Sx2x2-Sx2y*Sxx2)/(Sxx*Sx2x2-Sxx2**2)
C=(Sx2y*Sxx-Sxy*Sxx2)/(Sxx*Sx2x2-Sxx2**2)
A=ym-B*xm-C*x2m
Say 'y='a'+'||b'*x+'c'*x**2'
Say ' Input  "Approximation"'
Say ' x   y     y1'
Do i=1 To 11
Say right(x.i,2) right(y.i,3) format(fun(x.i),5,3)
End
Exit
fun:
Parse Arg x
Return a+b*x+c*x**2
```

{{out}}

```y=1+2*x+3*x**2
Input  "Approximation"
x   y     y1
0   1     1.000
1   6     6.000
2  17    17.000
3  34    34.000
4  57    57.000
5  86    86.000
6 121   121.000
7 162   162.000
8 209   209.000
9 262   262.000
10 321   321.000
```

## Ruby

```require 'matrix'

def regress x, y, degree
x_data = x.map { |xi| (0..degree).map { |pow| (xi**pow).to_r } }

mx = Matrix[*x_data]
my = Matrix.column_vector(y)

((mx.t * mx).inv * mx.t * my).transpose.to_a[0].map(&:to_f)
end
```

'''Testing:'''

```p regress([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10],
[1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321],
2)
```

{{out}}

```[1.0, 2.0, 3.0]
```

## Scala

{{Out}}See it yourself by running in your browser [https://scastie.scala-lang.org/NklZH2LlScCpfsN4NSfFvA Scastie (remote JVM)]. {{libheader|Scala Math Polynomial}} {{libheader|Scastie qualified}} {{works with|Scala|2.13}}

```object PolynomialRegression extends App {
private def xy = Seq(1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321).zipWithIndex.map(_.swap)

private def polyRegression(xy: Seq[(Int, Int)]): Unit = {
val r = xy.indices

def average[U](ts: Iterable[U])(implicit num: Numeric[U]) = num.toDouble(ts.sum) / ts.size

def x3m: Double = average(r.map(a => a * a * a))
def x4m: Double = average(r.map(a => a * a * a * a))
def x2ym = xy.reduce((a, x) => (a._1 + x._1 * x._1 * x._2, 0))._1.toDouble / xy.size
def xym = xy.reduce((a, x) => (a._1 + x._1 * x._2, 0))._1.toDouble / xy.size

val x2m: Double = average(r.map(a => a * a))
val (xm, ym) = (average(xy.map(_._1)), average(xy.map(_._2)))
val (sxx, sxy) = (x2m - xm * xm, xym - xm * ym)
val sxx2: Double = x3m - xm * x2m
val sx2x2: Double = x4m - x2m * x2m
val sx2y: Double = x2ym - x2m * ym
val c: Double = (sx2y * sxx - sxy * sxx2) / (sxx * sx2x2 - sxx2 * sxx2)
val b: Double = (sxy * sx2x2 - sx2y * sxx2) / (sxx * sx2x2 - sxx2 * sxx2)
val a: Double = ym - b * xm - c * x2m

def abc(xx: Int) = a + b * xx + c * xx * xx

println(s"y = \$a + \${b}x + \${c}x^2")
println(" Input  Approximation")
println(" x   y     y1")
xy.foreach {el => println(f"\${el._1}%2d \${el._2}%3d  \${abc(el._1)}%5.1f")}
}

polyRegression(xy)

}
```

## Sidef

{{trans|Ruby}}

```func regress(x, y, degree) {
var A = Matrix.build(x.len, degree+1, {|i,j|
x[i]**j
})

var B = Matrix.column_vector(y...)
((A.transpose * A)**(-1) * A.transpose * B).transpose[0]
}

func poly(x) {
3*x**2 + 2*x + 1
}

var coeff = regress(
10.of { _ },
10.of { poly(_) },
2
)

say coeff
```

{{out}}

```[1, 2, 3]
```

## Stata

See '''[http://www.stata.com/help.cgi?fvvarlist Factor variables]''' in Stata help for explanations on the ''c.x##c.x'' syntax.

```. clear
. input x y
0 1
1 6
2 17
3 34
4 57
5 86
6 121
7 162
8 209
9 262
10 321
end

. regress y c.x##c.x

Source |       SS           df       MS      Number of obs   =        11
-------------+----------------------------------   F(2, 8)         =         .
Model |      120362         2       60181   Prob > F        =         .
Residual |           0         8           0   R-squared       =    1.0000
Total |      120362        10     12036.2   Root MSE        =         0

------------------------------------------------------------------------------
y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
x |          2          .        .       .            .           .
|
c.x#c.x |          3          .        .       .            .           .
|
_cons |          1          .        .       .            .           .
------------------------------------------------------------------------------
```

## Tcl

{{tcllib|math::linearalgebra}}

```package require math::linearalgebra

proc build.matrix {xvec degree} {
set sums [llength \$xvec]
for {set i 1} {\$i <= 2*\$degree} {incr i} {
set sum 0
foreach x \$xvec {
set sum [expr {\$sum + pow(\$x,\$i)}]
}
lappend sums \$sum
}

set order [expr {\$degree + 1}]
set A [math::linearalgebra::mkMatrix \$order \$order 0]
for {set i 0} {\$i <= \$degree} {incr i} {
set A [math::linearalgebra::setrow A \$i [lrange \$sums \$i \$i+\$degree]]
}
return \$A
}

proc build.vector {xvec yvec degree} {
set sums [list]
for {set i 0} {\$i <= \$degree} {incr i} {
set sum 0
foreach x \$xvec y \$yvec {
set sum [expr {\$sum + \$y * pow(\$x,\$i)}]
}
lappend sums \$sum
}

set x [math::linearalgebra::mkVector [expr {\$degree + 1}] 0]
for {set i 0} {\$i <= \$degree} {incr i} {
set x [math::linearalgebra::setelem x \$i [lindex \$sums \$i]]
}
return \$x
}

# Now, to solve the example from the top of this page
set x {0   1   2   3   4   5   6   7   8   9  10}
set y {1   6  17  34  57  86 121 162 209 262 321}

# build the system A.x=b
set degree 2
set A [build.matrix \$x \$degree]
set b [build.vector \$x \$y \$degree]
# solve it
set coeffs [math::linearalgebra::solveGauss \$A \$b]
# show results
puts \$coeffs
```

This will print: 1.0000000000000207 1.9999999999999958 3.0 which is a close approximation to the correct solution.

```DelVar x
seq(x,x,0,10) → xs
{1,6,17,34,57,86,121,162,209,262,321} → ys
Disp regeq(x)
```

`seq(''expr'',''var'',''low'',''high'')` evaluates ''expr'' with ''var'' bound to integers from ''low'' to ''high'' and returns a list of the results. ` →` is the assignment operator. `QuadReg`, "quadratic regression", does the fit and stores the details in a number of standard variables, including regeq, which receives the fitted quadratic (polynomial) function itself. We then apply that function to the (undefined as ensured by `DelVar`) variable x to obtain the expression in terms of x, and display it.

{{out}} `3.·x2 + 2.·x + 1.`

## Ursala

{{libheader|LAPACK}} The fit function defined below returns the coefficients of an nth-degree polynomial in order of descending degree fitting the lists of inputs x and outputs y. The real work is done by the dgelsd function from the lapack library. Ursala provides a simplified interface to this library whereby the data can be passed as lists rather than arrays, and all memory management is handled automatically.

```#import std
#import nat
#import flo

(fit "n") ("x","y") = ..dgelsd\"y" (gang \/*pow float*x iota successor "n")* "x"
```

test program:

```x = <0.,1.,2.,3.,4.,5.,6.,7.,8.,9.,10.>
y = <1.,6.,17.,34.,57.,86.,121.,162.,209.,262.,321.>

#cast %eL

example = fit2(x,y)
```

{{out}}

```<3.000000e+00,2.000000e+00,1.000000e+00>
```

## VBA

Excel VBA has built in capability for line estimation.

```Option Base 1
Private Function polynomial_regression(y As Variant, x As Variant, degree As Integer) As Variant
Dim a() As Double
ReDim a(UBound(x), 2)
For i = 1 To UBound(x)
For j = 1 To degree
a(i, j) = x(i) ^ j
Next j
Next i
polynomial_regression = WorksheetFunction.LinEst(WorksheetFunction.Transpose(y), a, True, True)
End Function
Public Sub main()
x = [{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}]
y = [{1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321}]
result = polynomial_regression(y, x, 2)
Debug.Print "coefficients   : ";
For i = UBound(result, 2) To 1 Step -1
Debug.Print Format(result(1, i), "0.#####"),
Next i
Debug.Print
Debug.Print "standard errors: ";
For i = UBound(result, 2) To 1 Step -1
Debug.Print Format(result(2, i), "0.#####"),
Next i
Debug.Print vbCrLf
Debug.Print "R^2 ="; result(3, 1)
Debug.Print "F   ="; result(4, 1)
Debug.Print "Degrees of freedom:"; result(4, 2)
Debug.Print "Standard error of y estimate:"; result(3, 2)
End Sub
```

{{out}}

```coefficients   : 1,         2,            3,
standard errors: 0,         0,            0,

R^2 = 1
F   = 7,70461300500498E+31
Degrees of freedom: 8
Standard error of y estimate: 2,79482284961344E-14
```

## zkl

Using the GNU Scientific Library

```var [const] GSL=Import("zklGSL");	// libGSL (GNU Scientific Library)
xs:=GSL.VectorFromData(0,  1,  2,  3,  4,  5,   6,   7,   8,   9,  10);
ys:=GSL.VectorFromData(1,  6, 17, 34, 57, 86, 121, 162, 209, 262, 321);
v :=GSL.polyFit(xs,ys,2);
v.format().println();
GSL.Helpers.polyString(v).println();
GSL.Helpers.polyEval(v,xs).format().println();
```

{{out}}

```
1.00,2.00,3.00
1 + 2x + 3x^2
1.00,6.00,17.00,34.00,57.00,86.00,121.00,162.00,209.00,262.00,321.00

```

Or, using lists: {{trans|Common Lisp}} Uses the code from [[Multiple regression#zkl]].

Example:

```polyfit(T(T(0.0,1.0,2.0,3.0,4.0,5.0,6.0,7.0,8.0,9.0,10.0)),
T(T(1.0,6.0,17.0,34.0,57.0,86.0,121.0,162.0,209.0,262.0,321.0)), 2)
.flatten().println();
```

{{out}}

```L(1,2,3)
```