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Polynomial regression

From Rosetta Code
(Redirected from Polynomial Fitting)
Task
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.

11l[edit]

Translation of: Swift
F average(arr)
R sum(arr) / Float(arr.len)
 
F poly_regression(x, y)
V xm = average(x)
V ym = average(y)
V x2m = average(x.map(i -> i * i))
V x3m = average(x.map(i -> i ^ 3))
V x4m = average(x.map(i -> i ^ 4))
V xym = average(zip(x, y).map((i, j) -> i * j))
V x2ym = average(zip(x, y).map((i, j) -> i * i * j))
V sxx = x2m - xm * xm
V sxy = xym - xm * ym
V sxx2 = x3m - xm * x2m
V sx2x2 = x4m - x2m * x2m
V sx2y = x2ym - x2m * ym
V b = (sxy * sx2x2 - sx2y * sxx2) / (sxx * sx2x2 - sxx2 * sxx2)
V c = (sx2y * sxx - sxy * sxx2) / (sxx * sx2x2 - sxx2 * sxx2)
V a = ym - b * xm - c * x2m
 
F abc(xx)
R (@a + @b * xx) + (@c * xx * xx)
 
print("y = #. + #.x + #.x^2\n".format(a, b, c))
print(‘ Input Approximation’)
print(‘ x y y1’)
 
L(i) 0 .< x.len
print(‘#2 #3 #3.1’.format(x[i], y[i], abc(i)))
 
V x = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
V y = [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321]
poly_regression(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

Ada[edit]

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 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[edit]

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[edit]

Translation of: Ada
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 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 #
Output:
3x**2+2x+1
 1.0848x**2+10.3552x-0.6164

BBC BASIC[edit]

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[edit]

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 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;
}
Output:
1.000000
2.000000
3.000000

C#[edit]

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<double> 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], Polynomial.Evaluate(x[i], p), y[i] - Polynomial.Evaluate(x[i], p));
Console.ReadKey(true);
}

C++[edit]

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

Common Lisp[edit]

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[edit]

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[edit]

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[edit]

Library: LAPACK
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[edit]

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[edit]

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[edit]

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[edit]

# 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[edit]

Library gonum/matrix[edit]

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[edit]

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]

Haskell[edit]

Uses module Matrix.LU from hackageDB DSP

import Data.List
import Data.Array
import Control.Monad
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
Output:
in GHCi:
*Main> polyfit 3 [1,6,17,34,57,86,121,162,209,262,321]
[1.0,2.0,3.0]

HicEst[edit]

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[edit]

(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[edit]

   Y=:1 6 17 34 57 86 121 162 209 262 321
(%. ^/[email protected]:@[email protected]#) 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[edit]

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[edit]

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[edit]

Translation of: REXX
// version 1.1.51
 
fun polyRegression(x: IntArray, y: IntArray) {
val xm = x.average()
val ym = y.average()
val x2m = x.map { it * it }.average()
val x3m = x.map { it * it * it }.average()
val x4m = x.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 ((xi, yi) in x zip y) {
System.out.printf("%2d %3d  %5.1f\n", xi, yi, abc(xi))
}
}
 
fun main() {
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[edit]

Translation of: 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)
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[edit]

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[edit]

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[edit]

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 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[edit]

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 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 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.

Modula-2[edit]

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.

Nim[edit]

Translation of: Kotlin
import lenientops, sequtils, stats, strformat
 
proc polyRegression(x, y: openArray[int]) =
 
let xm = mean(x)
let ym = mean(y)
let x2m = mean(x.mapIt(it * it))
let x3m = mean(x.mapIt(it * it * it))
let x4m = mean(x.mapIt(it * it * it * it))
let xym = mean(zip(x, y).mapIt(it[0] * it[1]))
let x2ym = mean(zip(x, y).mapIt(it[0] * it[0] * it[1]))
 
let sxx = x2m - xm * xm
let sxy = xym - xm * ym
let sxx2 = x3m - xm * x2m
let sx2x2 = x4m - x2m * x2m
let sx2y = x2ym - x2m * ym
 
let b = (sxy * sx2x2 - sx2y * sxx2) / (sxx * sx2x2 - sxx2 * sxx2)
let c = (sx2y * sxx - sxy * sxx2) / (sxx * sx2x2 - sxx2 * sxx2)
let a = ym - b * xm - c * x2m
 
func abc(x: int): float = a + b * x + c * x * x
 
echo &"y = {a} + {b}x + {c}x²\n"
echo " Input Approximation"
echo " x y y1"
for (xi, yi) in zip(x, y):
echo &"{xi:2} {yi:3} {abc(xi):5}"
 
 
let x = toSeq(0..10)
let y = [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321]
polyRegression(x, y)
Output:
y = 1.0 + 2.0x + 3.0x²

 Input  Approximation
 x   y     y1
 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

OCaml[edit]

Translation of: Kotlin
Library: Base
open Base
open Stdio
 
let mean fa =
let open Float in
(Array.reduce_exn fa ~f:(+)) / (of_int (Array.length fa))
 
let regression xs ys =
let open Float in
let xm = mean xs in
let ym = mean ys in
let x2m = Array.map xs ~f:(fun x -> x * x) |> mean in
let x3m = Array.map xs ~f:(fun x -> x * x * x) |> mean in
let x4m = Array.map xs ~f:(fun x -> let x2 = x * x in x2 * x2) |> mean in
let xzipy = Array.zip_exn xs ys in
let xym = Array.map xzipy ~f:(fun (x, y) -> x * y) |> mean in
let x2ym = Array.map xzipy ~f:(fun (x, y) -> x * x * y) |> mean in
 
let sxx = x2m - xm * xm in
let sxy = xym - xm * ym in
let sxx2 = x3m - xm * x2m in
let sx2x2 = x4m - x2m * x2m in
let sx2y = x2ym - x2m * ym in
 
let b = (sxy * sx2x2 - sx2y * sxx2) / (sxx * sx2x2 - sxx2 * sxx2) in
let c = (sx2y * sxx - sxy * sxx2) / (sxx * sx2x2 - sxx2 * sxx2) in
let a = ym - b * xm - c * x2m in
 
let abc xx = a + b * xx + c * xx * xx in
 
printf "y = %.1f + %.1fx + %.1fx^2\n\n" a b c;
printf " Input Approximation\n";
printf " x y y1\n";
Array.iter xzipy ~f:(fun (xi, yi) ->
printf "%2g %3g  %5.1f\n" xi yi (abc xi)
)
 
let () =
let x = Array.init 11 ~f:Float.of_int in
let y = [| 1.; 6.; 17.; 34.; 57.; 86.; 121.; 162.; 209.; 262.; 321. |] in
regression 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

Octave[edit]

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

PARI/GP[edit]

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[edit]

This code identical to that of Multiple regression task.

use strict;
use warnings;
use Statistics::Regression;
 
my @x = <0 1 2 3 4 5 6 7 8 9 10>;
my @y = <1 6 17 34 57 86 121 162 209 262 321>;
 
my @model = ('const', 'X', 'X**2');
 
my $reg = Statistics::Regression->new( '', [@model] );
$reg->include( $y[$_], [ 1.0, $x[$_], $x[$_]**2 ]) for 0..@y-1;
my @coeff = $reg->theta();
 
printf "%-6s %8.3f\n", $model[$_], $coeff[$_] for 0..@model-1;
Output:
const     1.000
X         2.000
X**2      3.000

PDL Alternative:

#!/usr/bin/perl -w
use strict;
 
use PDL;
use PDL::Math;
use PDL::Fit::Polynomial;
 
my $x = float [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10];
my $y = float [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321];
# above will output: 3.00000037788248 * $x**2 + 1.99999750988868 * $x + 1.00000180493936
 
# $x = float [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9];
# $y = float [ 2.7, 2.8, 31.4, 38.1, 58.0, 76.2, 100.5, 130.0, 149.3, 180.0];
# above correctly returns: " 1.08484845125187 * $x**2 + 10.3551513321297 * $x-0.616363852007752 "
 
my ($yfit, $coeffs) = fitpoly1d $x, $y, 3; # 3rd degree
 
foreach (reverse(0..$coeffs->dim(0)-1)) {
print " +" unless(($coeffs->at($_) <0) || $_==$coeffs->dim(0)-1); # let the unary minus replace the + operator
print " ";
print $coeffs->at($_);
print " * \$x" if($_);
print "**$_" if($_>1);
print "\n" unless($_)
}
 
Output:
 3.00000037788248 * $x**2 + 1.99999750988868 * $x + 1.00000180493936

Phix[edit]

Translation of: REXX
constant x = {0,1,2,3,4,5,6,7,8,9,10},
         y = {1,6,17,34,57,86,121,162,209,262,321},
         n = length(x)
 
function regression()
    atom xm = 0, ym = 0, x2m = 0, x3m = 0, x4m = 0, xym = 0, 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
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

plot[edit]

A simple plot, as per Racket:

Library: Phix/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,`TITLE="simple plot", RASTERSIZE=640x480`)
IupShow(dlg)
if platform()!=JS then 
    IupMainLoop()
    IupClose()
end if

PowerShell[edit]

 
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)"
 
Output:
polyfit 
X^2 X constant
3 1.99999999999998 1.00000000000005

Python[edit]

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)
>>> coeffs
array([ 3., 2., 1.])

Substitute back received coefficients.

>>> 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[edit]

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 Example section for Python language on Query Performance page).

R[edit]

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[edit]

 
#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:

Polyreg-racket.png

Raku[edit]

(formerly 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

REXX[edit]

/* 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
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[edit]

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]

Scala[edit]

Output:
See it yourself by running in your browser Scastie (remote JVM).
Works with: Scala version 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[edit]

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[edit]

See 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
-------------+---------------------------------- 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 . . . . .
------------------------------------------------------------------------------

Swift[edit]

Translation of: Kotlin
 
 
let x = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
let y = [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321]
 
func average(_ input: [Int]) -> Int {
return input.reduce(0, +) / input.count
}
 
func polyRegression(x: [Int], y: [Int]) {
let xm = average(x)
let ym = average(y)
let x2m = average(x.map { $0 * $0 })
let x3m = average(x.map { $0 * $0 * $0 })
let x4m = average(x.map { $0 * $0 * $0 * $0 })
let xym = average(zip(x,y).map { $0 * $1 })
let x2ym = average(zip(x,y).map { $0 * $0 * $1 })
 
let sxx = x2m - xm * xm
let sxy = xym - xm * ym
let sxx2 = x3m - xm * x2m
let sx2x2 = x4m - x2m * x2m
let sx2y = x2ym - x2m * ym
 
let b = (sxy * sx2x2 - sx2y * sxx2) / (sxx * sx2x2 - sxx2 * sxx2)
let c = (sx2y * sxx - sxy * sxx2) / (sxx * sx2x2 - sxx2 * sxx2)
let a = ym - b * xm - c * x2m
 
func abc(xx: Int) -> Int {
return (a + b * xx) + (c * xx * xx)
}
 
print("y = \(a) + \(b)x + \(c)x^2\n")
print(" Input Approximation")
print(" x y y1")
 
for i in 0 ..< x.count {
let result = Double(abc(xx: i))
print(String(format: "%2d %3d  %5.1f", x[i], y[i], result))
}
}
 
polyRegression(x: x, y: 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

Tcl[edit]

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 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.

TI-83 BASIC[edit]

DelVar X
seq(X,X,0,10) → L1
{1,6,17,34,57,86,121,162,209,262,321} → L2
QuadReg L1,L2
Output:
y=ax²+bx+c
a=3
b=2
c=1


TI-89 BASIC[edit]

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

Output:

3.·x2 + 2.·x + 1.

Ursala[edit]

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[edit]

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
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 

Wren[edit]

Translation of: REXX
Library: Wren-math
Library: Wren-seq
Library: Wren-fmt
import "/math" for Nums
import "/seq" for Lst
import "/fmt" for Fmt
 
var polynomialRegression = Fn.new { |x, y|
var xm = Nums.mean(x)
var ym = Nums.mean(y)
var x2m = Nums.mean(x.map { |e| e * e })
var x3m = Nums.mean(x.map { |e| e * e * e })
var x4m = Nums.mean(x.map { |e| e * e * e * e })
var z = Lst.zip(x, y)
var xym = Nums.mean(z.map { |p| p[0] * p[1] })
var x2ym = Nums.mean(z.map { |p| p[0] * p[0] * p[1] })
 
var sxx = x2m - xm * xm
var sxy = xym - xm * ym
var sxx2 = x3m - xm * x2m
var sx2x2 = x4m - x2m * x2m
var sx2y = x2ym - x2m * ym
 
var b = (sxy * sx2x2 - sx2y * sxx2) / (sxx * sx2x2 - sxx2 * sxx2)
var c = (sx2y * sxx - sxy * sxx2) / (sxx * sx2x2 - sxx2 * sxx2)
var a = ym - b * xm - c * x2m
 
var abc = Fn.new { |xx| a + b * xx + c * xx * xx }
 
System.print("y = %(a) + %(b)x + %(c)x^2\n")
System.print(" Input Approximation")
System.print(" x y y1")
for (p in z) Fmt.print("$2d $3d $5.1f", p[0], p[1], abc.call(p[0]))
}
 
var x = List.filled(11, 0)
for (i in 1..10) x[i] = i
var y = [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321]
polynomialRegression.call(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

zkl[edit]

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)