polynomial fit.cpp

//Polynomial Fit
#include<iostream>
#include<iomanip>
#include<cmath>
using namespace std;
int main()
{
    int i,j,k,n,N;
    cout.precision(4);                        //set precision
    cout.setf(ios::fixed);
    cout<<"\nEnter the no. of data pairs to be entered:\n";        //To find the size of arrays that will store x,y, and z values
    cin>>N;
    double x[N],y[N];
    cout<<"\nEnter the x-axis values:\n";                //Input x-values
    for (i=0;i<N;i++)
        cin>>x[i];
    cout<<"\nEnter the y-axis values:\n";                //Input y-values
    for (i=0;i<N;i++)
        cin>>y[i];
    cout<<"\nWhat degree of Polynomial do you want to use for the fit?\n";
    cin>>n;                                // n is the degree of Polynomial
    double X[2*n+1];                        //Array that will store the values of sigma(xi),sigma(xi^2),sigma(xi^3)....sigma(xi^2n)
    for (i=0;i<2*n+1;i++)
    {
        X[i]=0;
        for (j=0;j<N;j++)
            X[i]=X[i]+pow(x[j],i);        //consecutive positions of the array will store N,sigma(xi),sigma(xi^2),sigma(xi^3)....sigma(xi^2n)
    }
    double B[n+1][n+2],a[n+1];            //B is the Normal matrix(augmented) that will store the equations, 'a' is for value of the final coefficients
    for (i=0;i<=n;i++)
        for (j=0;j<=n;j++)
            B[i][j]=X[i+j];            //Build the Normal matrix by storing the corresponding coefficients at the right positions except the last column of the matrix
    double Y[n+1];                    //Array to store the values of sigma(yi),sigma(xi*yi),sigma(xi^2*yi)...sigma(xi^n*yi)
    for (i=0;i<n+1;i++)
    {
        Y[i]=0;
        for (j=0;j<N;j++)
        Y[i]=Y[i]+pow(x[j],i)*y[j];        //consecutive positions will store sigma(yi),sigma(xi*yi),sigma(xi^2*yi)...sigma(xi^n*yi)
    }
    for (i=0;i<=n;i++)
        B[i][n+1]=Y[i];                //load the values of Y as the last column of B(Normal Matrix but augmented)
    n=n+1;                //n is made n+1 because the Gaussian Elimination part below was for n equations, but here n is the degree of polynomial and for n degree we get n+1 equations
    cout<<"\nThe Normal(Augmented Matrix) is as follows:\n";
    for (i=0;i<n;i++)            //print the Normal-augmented matrix
    {
        for (j=0;j<=n;j++)
            cout<<B[i][j]<<setw(16);
        cout<<"\n";
    }
    for (i=0;i<n;i++)                    //From now Gaussian Elimination starts(can be ignored) to solve the set of linear equations (Pivotisation)
        for (k=i+1;k<n;k++)
            if (B[i][i]<B[k][i])
                for (j=0;j<=n;j++)
                {
                    double temp=B[i][j];
                    B[i][j]=B[k][j];
                    B[k][j]=temp;
                }

    for (i=0;i<n-1;i++)            //loop to perform the gauss elimination
        for (k=i+1;k<n;k++)
            {
                double t=B[k][i]/B[i][i];
                for (j=0;j<=n;j++)
                    B[k][j]=B[k][j]-t*B[i][j];    //make the elements below the pivot elements equal to zero or elimnate the variables
            }
    for (i=n-1;i>=0;i--)                //back-substitution
    {                        //x is an array whose values correspond to the values of x,y,z..
        a[i]=B[i][n];                //make the variable to be calculated equal to the rhs of the last equation
        for (j=0;j<n;j++)
            if (j!=i)            //then subtract all the lhs values except the coefficient of the variable whose value                                   is being calculated
                a[i]=a[i]-B[i][j]*a[j];
        a[i]=a[i]/B[i][i];            //now finally divide the rhs by the coefficient of the variable to be calculated
    }
    cout<<"\nThe values of the coefficients are as follows:\n";
    for (i=0;i<n;i++)
        cout<<"x^"<<i<<"="<<a[i]<<endl;            // Print the values of x^0,x^1,x^2,x^3,....
    cout<<"\nHence the fitted Polynomial is given by:\ny=";
    for (i=0;i<n;i++)
        cout<<" + ("<<a[i]<<")"<<"x^"<<i;
    cout<<"\n";
    return 0;
}//output attached as .jpg