C program to compute the polynomial regression algorithm

Polynomial regression is a form of regression analysis used to model the relationship between an independent variable x and dependent variable y as an nth degree polynomial. This technique extends linear regression to capture non-linear relationships in data.

Syntax

y = a0 + a1*x + a2*x² + a3*x³ + ... + an*x?

Where a0, a1, a2, ..., an are coefficients to be determined using least squares method.

Algorithm

The polynomial regression algorithm works by −

• Setting up a system of normal equations using least squares method
• Creating coefficient matrix from input data points
• Solving the system using Gaussian elimination
• Obtaining polynomial coefficients

Example

Following is the C program to compute polynomial regression coefficients −

#include <stdio.h>
#include <math.h>

int main() {
    int i, j, k, m, n;
    float x[20], y[20], u, a[10], c[20][20], power, r;
    
    printf("Enter degree of polynomial (m) and number of data points (n): ");
    scanf("%d %d", &m, &n);
    
    printf("Enter x and y values:<br>");
    for(i = 1; i <= n; i++) {
        printf("x[%d], y[%d]: ", i, i);
        scanf("%f %f", &x[i], &y[i]);
    }
    
    /* Form coefficient matrix */
    for(j = 1; j <= m + 1; j++) {
        for(k = 1; k <= m + 1; k++) {
            c[j][k] = 0;
            for(i = 1; i <= n; i++) {
                power = pow(x[i], j + k - 2);
                c[j][k] = c[j][k] + power;
            }
        }
    }
    
    /* Form constant terms */
    for(j = 1; j <= m + 1; j++) {
        c[j][m + 2] = 0;
        for(i = 1; i <= n; i++) {
            r = pow(x[i], j - 1);
            c[j][m + 2] = c[j][m + 2] + y[i] * r;
        }
    }
    
    printf("\nAugmented matrix:<br>");
    for(i = 1; i <= m + 1; i++) {
        for(j = 1; j <= m + 2; j++) {
            printf("%.2f\t", c[i][j]);
        }
        printf("<br>");
    }
    
    /* Gaussian elimination */
    for(k = 1; k <= m + 1; k++) {
        for(i = 1; i <= m + 1; i++) {
            if(i != k) {
                u = c[i][k] / c[k][k];
                for(j = k; j <= m + 2; j++) {
                    c[i][j] = c[i][j] - u * c[k][j];
                }
            }
        }
    }
    
    printf("\nPolynomial coefficients:<br>");
    for(i = 1; i <= m + 1; i++) {
        a[i] = c[i][m + 2] / c[i][i];
        printf("a[%d] = %.6f<br>", i - 1, a[i]);
    }
    
    return 0;
}

Output

Enter degree of polynomial (m) and number of data points (n): 2 3
Enter x and y values:
x[1], y[1]: 1 1
x[2], y[2]: 2 4
x[3], y[3]: 3 9

Augmented matrix:
3.00    6.00    14.00   14.00
6.00    14.00   36.00   36.00
14.00   36.00   98.00   98.00

Polynomial coefficients:
a[0] = 0.000000
a[1] = 0.000000
a[2] = 1.000000

How It Works

The algorithm creates normal equations by minimizing the sum of squared errors. For a polynomial of degree m with n data points, it solves the matrix equation to find coefficients that best fit the given data points.

Key Points

• The degree of polynomial should be less than the number of data points
• Higher degree polynomials may lead to overfitting
• The algorithm uses Gaussian elimination to solve the linear system

Conclusion

Polynomial regression extends linear regression to model non-linear relationships. The C implementation uses matrix operations and Gaussian elimination to compute polynomial coefficients that minimize squared error.