According to Sharma’s article, any polynomial equations could fit to a given set of data points in this way.

To fit a qudratic equation to the data points below, we may try to use Sharma’s method with N = 5 (We have 5 data-point pairs). After Gaussian elimination, we will get a0 = 0.55, a1 = -1.07 and a2 = 1.42.

Using excel curve fitting function (Polynomial & Order = 2), we will obtain identical result.

A simple C program for fitting a quadratic equation to data points using the Least Squares Method

```
#include <stdio.h>
#include <math.h>
#define array_size 5
float x_data[array_size] = {0, 1, 2, 3, 4};
float y_data[array_size] = {1, 1.8, 1.3, 2.5, 6.3};
float x[9] = {0}; //The number of elements of a 3 x 3 matrix (LHS) = 9
float y[3] = {0}; //The number of elements of a 3 x 1 matrix (RHS) = 3
int main()
{
//%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
//%% Least Squares Approximation Method
//%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
float x1 = 0;
float x2 = 0;
float x3 = 0;
float x4 = 0;
float xY = 0;
float xY2 = 0;
float sumY = 0;
for(int i = 0; i<=array_size; i++)
{
float temp = x_data[i];
float temp2 = y_data[i];
x1 += temp;
x2 += pow(temp,2);
x3 += pow(temp,3);
x4 += pow(temp,4);
xY += temp * temp2;
xY2 += pow(temp,2) * temp2;
sumY += temp2;
}
x[0] = array_size;
x[1] = x1;
x[2] = x2;
x[3] = x1;
x[4] = x2;
x[5] = x3;
x[6] = x2;
x[7] = x3;
x[8] = x4;
y[0] = sumY;
y[1] = xY;
y[2] = xY2;
//%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
//%% Gaussian elimiation
//%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
float common = x[3]/x[0];
x[4] -= x[1] * common;
x[5] -= x[2] * common;
y[1] -= y[0] * common;
float common2 = x[6]/x[0];
x[7] -= x[1] * common2;
x[8] -= x[2] * common2;
y[2] -= y[0] * common2;
float common3 = x[7]/x[4];
x[8] -= x[5] * common3;
y[2] -= y[1] * common3;
// %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
// %% Back substitution
// %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
float a0 = y[2]/x[8];
float a1 = (y[1] - a0 * x[5])/x[4];
float a2 = (y[0] - a0 * x[2] - a1 * x[1])/x[0];
printf("%.2fx^2%+.2fx%+.2f",a0,a1,a2);
return 0;
}
```

A simple Python program for fitting a quadratic equation to data points using the Least Squares Method

```
def array_size():
return 5
x_data = [0, 1, 2, 3, 4]
y_data = [1, 1.8, 1.3, 2.5, 6.3]
x = [0]*9 #The number of elements of a 3 x 3 matrix (LHS) = 9
y = [0]*3 #The number of elements of a 3 x 1 matrix (RHS) = 3
if __name__ == "__main__":
#%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
#%% Least Squares Approximation Method
#%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
x1 = 0
x2 = 0
x3 = 0
x4 = 0
xY = 0
xY2 = 0
sumY = 0
i = 0
while(i<array_size()):
temp = x_data[i]
temp2 = y_data[i]
x1 += temp
x2 += pow(temp,2)
x3 += pow(temp,3)
x4 += pow(temp,4)
xY += temp * temp2
xY2 += pow(temp,2) * temp2
sumY += temp2
i+=1
x[0] = array_size()
x[1] = x1
x[2] = x2
x[3] = x1
x[4] = x2
x[5] = x3
x[6] = x2
x[7] = x3
x[8] = x4
y[0] = sumY
y[1] = xY
y[2] = xY2
#%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
#%% Gaussian elimiation
#%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
common = x[3]/x[0]
x[4] -= x[1] * common
x[5] -= x[2] * common
y[1] -= y[0] * common
common2 = x[6]/x[0]
x[7] -= x[1] * common2
x[8] -= x[2] * common2
y[2] -= y[0] * common2
common3 = x[7]/x[4]
x[8] -= x[5] * common3
y[2] -= y[1] * common3
#%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
#%% Back substitution
#%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
a0 = y[2]/x[8]
a1 = (y[1] - a0 * x[5])/x[4]
a2 = (y[0] - a0 * x[2] - a1 * x[1])/x[0]
print("%.2fx^2%+.2fx%+.2f"%(a0,a1,a2))
```

For more complicated polynomial fitting using the Least Squares Method, please refer to Sharma’s original article.