# Polynomial regression

Polynomial regression
You are encouraged to solve this task according to the task description, using any language you may know.

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 planebegin   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 xi and yi. The basis φj is xj, j=0,1,..,N. The implementation is straightforward. First the plane matrix A is created. Ajij(xi). Then the linear problem AATc=Ay is solved. The result cj 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;with Ada.Float_Text_IO;  use Ada.Float_Text_IO; 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;`
Output:
``` 1.000 2.000 3.000
```

## ALGOL 68

Works with: ALGOL 68 version Standard - lu decomp and lu solve are from the GSL library

Works with: ALGOL 68G version 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 COMMENTOP 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 COMMENTOP * = (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 COMMENTOP / = (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 #`
Output:
```3x**2+2x+1
1.0848x**2+10.3552x-0.6164
```

## BBC BASIC

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`
Output:
```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 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 11double 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 3double 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;}`
Output:
```1.000000
2.000000
3.000000```

## C++

Translation of: 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;}`
Output:
```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#

Library: Math.Net
`        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]));            Console.ReadKey(true);        }`

## 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)  (let* ((m (cadr (array-dimensions x)))         (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

Translation of: 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);}`
Output:
```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)) `
Output:
```"3. x^2 + 1.99999999996 x + 1.00000000006"
```

## Fortran

Library: LAPACK
`module fittingcontains   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`
Output:
(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`
Output:
```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 orderPolynomialRegression(x, y, 2);# [ 1, 2, 3 ]`

## gnuplot

`# The polynomial approximationf(x) = a*x**2 + b*x + c # Initial values for parametersa = 0.1b = 0.1c = 0.1 # Fit f to the following data by modifying the variables a, b, cfit 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 321e 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}`
Output:
```⎡1.000⎤
⎢2.000⎥
⎣3.000⎦
```

### Library go.matrix

Least squares solution using QR decomposition and package 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)}`
Output:
(lowest order coefficient first)
```[0.9999999999999758 2.000000000000015 2.999999999999999]
```

Uses module Matrix.LU from hackageDB DSP

`import Data.Listimport Data.Arrayimport Control.Monadimport Control.Arrowimport 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`
Output:
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`
Output:
```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   (%. ^/[email protected]:@[email protected]#) Y1 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.#Y1 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

Translation of: D
Works with: Java version 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);    }}`
Output:
```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 version 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)`
Output:
`polyfit(x, y, 2) = [1.0, 2.0, 3.0]`

## Kotlin

Translation of: 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)}`
Output:
```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

Translation of: Modula-2
`function eval(a,b,c,x)    return a + (b + c * x) * xend 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])))    endend 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)`
Output:
```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 = [email protected]{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.

`[email protected][{{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 ${\displaystyle point_{i}=(x_{i},y_{i})}$ 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`

## МК-61/52

Part 1:

`ПC	С/П	ПD	ИП9	+	П9	ИПC	ИП5	+	П5ИПC	x^2	П2	ИП6	+	П6	ИП2	ИПC	*	ИП7+	П7	ИП2	x^2	ИП8	+	П8	ИПC	ИПD	*ИПA	+	ПA	ИП2	ИПD	*	ИПB	+	ПB	ИПDКИП4	С/П	БП	00`

Input: В/О x1 С/П y1 С/П x2 С/П y2 С/П ...

Part 2:

`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	ИП4ИП8	*	ИП2	ИП6	*	-	П8	ИП4	ИПB	*ИП2	ИП9	*	-	ИПD	*	ИП3	ИПA	*	-ИПD	ИП8	*	ИП7	ИП3	*	-	/	ПB	ИПAИПB	ИП7	*	-	ИПD	/	ПA	ИП9	ИПB	ИП6*	-	ИПA	ИП5	*	-	ИП4	/	П9	С/П`

Result: Р9 = a0, РA = a1, РB = a2.

## Modula-2

`MODULE PolynomialRegression;FROM FormatString IMPORT FormatString;FROM RealStr IMPORT RealToStr;FROM Terminal IMPORT WriteString,WriteLn,ReadChar; 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);     ReadChar;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])`
Output:
`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 Bill Allombert

Output:
`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 onceon 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'smy \$pairs = scalar(@x); =pod Step 2: Devise the base equation of our polynomial using the following ideaThere 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 foreach 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;`
Output:
```1
2
3
```

## Phix

Translation of: 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+cend 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`
Output:
```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 Racket):

Library: 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    IupPlotAdd(plot, x[i], y[i])end for{} = IupPlotEnd(plot) Ihandle dlg = IupDialog(plot)IupSetAttributes(dlg, "RASTERSIZE=%dx%d", {640, 480})IupSetAttribute(dlg, "TITLE", "simple plot")IupShow(dlg) IupMainLoop()IupClose()`

## Python

Library: NumPy
`>>> 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)>>> coeffsarray([ 3.,  2.,  1.])`

`>>> yf = numpy.polyval(numpy.poly1d(coeffs), x)>>> yfarray([   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       21.085 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 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)))`
Output:
```(Intercept)           x      I(x^2)
1           2           3
```

Alternately, use poly:

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

## 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=11Do i=1 To n  Parse Var xl x.i xl  Parse Var yl y.i yl  Endxm=0ym=0x2m=0x3m=0x4m=0xym=0x2ym=0Do 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  Endxm =xm /nym =ym /nx2m=x2m/nx3m=x3m/nx4m=x4m/nxym=xym/nx2ym=x2ym/nSxx=x2m-xm**2Sxy=xym-xm*ymSxx2=x3m-xm*x2mSx2x2=x4m-x2m**2Sx2y=x2ym-x2m*ymB=(Sxy*Sx2x2-Sx2y*Sxx2)/(Sxx*Sx2x2-Sxx2**2)C=(Sx2y*Sxx-Sxy*Sxx2)/(Sxx*Sx2x2-Sxx2**2)A=ym-B*xm-C*x2mSay '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)  EndExitfun:  Parse Arg x  Return a+b*x+c*x**2 `
Output:
```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)`
Output:
`[1.0, 2.0, 3.0]`

## Sidef

Translation of: 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`
Output:
```[1, 2, 3]
```

## Stata

See Factor variables in Stata help for explanations on the c.x##c.x syntax.

`. clear. input x y0 11 62 173 344 575 866 1217 1628 2099 26210 321end . 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-------------+----------------------------------   Adj 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

Library: Tcllib (Package: 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 pageset 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=bset degree 2set A [build.matrix \$x \$degree]set b [build.vector \$x \$y \$degree]# solve itset coeffs [math::linearalgebra::solveGauss \$A \$b]# show resultsputs \$coeffs`

This will print:

```1.0000000000000207 1.9999999999999958 3.0
```

which is a close approximation to the correct solution.

## TI-89 BASIC

`DelVar xseq(x,x,0,10) → xs{1,6,17,34,57,86,121,162,209,262,321} → ysQuadReg xs,ysDisp 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.

Output:

`3.·x2 + 2.·x + 1.`

## Ursala

Library: 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)`
Output:
`<3.000000e+00,2.000000e+00,1.000000e+00>`

## VBA

Excel VBA has built in capability for line estimation.

`Option Base 1Private 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 FunctionPublic 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`
Output:
```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();`
Output:
```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:

Translation of: 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();`
Output:
`L(1,2,3)`