//example of least squares curve fitting


//example data
//x = { -3, -2, 0, 3, 4 }
//y = { 18, 10, 2, 2, 5 }


public class leastsqs
{
    
    
    public static void main ( String args[] ) 
    {
	
	
	//raw data
	int values = 5;
	int x[] = { -3, -2, 0, 3, 4 };
	int y[] = { 18, 10, 2, 2, 5 };
	
	System.out.println ();
	System.out.println ( "Raw data " );
	System.out.println ( "x = -3, -2, 0, 3, 4 " );
	System.out.println ( "y = 18, 10, 2, 2, 5 " );
	
	//substitue data into equations
	System.out.println();
	System.out.println ( "Substituting values into equation ax^2 + bx + c = y " );
	
	for ( int i=0; i<values; i++){
	    System.out.println ( x[i]*x[i] + "a  " + x[i] + "b  " + "c  =  " + y[i] );
	}
    
	
	//need 3 equations for 3 unknowns... so 
	//multiply each equation by the coefficent of a in that equation and sum
	//and repeat for b and c
	
	System.out.println ();
	System.out.println ( "Need 3 equations for 3 unknowns so multiply each equation by the " );
	System.out.println ( " coefficent of a in that equation and sum, then do the same for b and c " );
	
	//a
	int tempeq[][] = new int[values][4]; 
	for ( int i=0; i<values; i++){
	    tempeq[i][0] = x[i]*x[i] * x[i]*x[i]; 
	    tempeq[i][1] = x[i]*x[i] * x[i];
	    tempeq[i][2] = x[i]*x[i];
	    tempeq[i][3] = x[i]*x[i] * y[i];
	}
	
	int a1 = 0;	
	int a2 = 0;
	int a3 = 0;
	int a4 = 0;
	
	for ( int i=0; i<values; i++){
	    a1 += tempeq[i][0];
	    a2 += tempeq[i][1];
	    a3 += tempeq[i][2];
	    a4 += tempeq[i][3];
	}
	
	System.out.println ( "First equation:  " + a1 + "a   + " + a2 + "b  + " + a3 + " = " + a4 );
	
	
	
	//b
	for ( int i=0; i<values; i++){
	    for ( int j=0; j<3; j++){
		tempeq[i][j] = 0;
	    }
	}
	for ( int i=0; i<values; i++){
	    tempeq[i][0] = x[i] * x[i]*x[i]; 
	    tempeq[i][1] = x[i] * x[i];
	    tempeq[i][2] = x[i];
	    tempeq[i][3] = x[i] * y[i];
	}
	
	int b1 = 0;	
	int b2 = 0;
	int b3 = 0;
	int b4 = 0;
	
	for ( int i=0; i<values; i++){
	    b1 += tempeq[i][0];
	    b2 += tempeq[i][1];
	    b3 += tempeq[i][2];
	    b4 += tempeq[i][3];
	}
	
	System.out.println ( "Second equation:  " + b1 + "a   + " + b2 + "b  + " + b3 + " = " + b4 );
	
	
	//c  ( c=1 )
	for ( int i=0; i<values; i++){
	    for ( int j=0; j<3; j++){
		tempeq[i][j] = 0;
	    }
	}
	
	for ( int i=0; i<values; i++){
	    tempeq[i][0] = x[i]*x[i]; 
	    tempeq[i][1] = x[i];
	    tempeq[i][2] = 1;
	    tempeq[i][3] = y[i];
	}
	
	int c1 = 0;	
	int c2 = 0;
	int c3 = 0;
	int c4 = 0;
	
	for ( int i=0; i<values; i++){
	    c1 += tempeq[i][0];
	    c2 += tempeq[i][1];
	    c3 += tempeq[i][2];
	    c4 += tempeq[i][3];
	}
	
	System.out.println ( "Third equation:  " + c1 + "a   + " + c2 + "b  + " + c3 + " = " + c4 );
	
	//now solve the 3 unknowns
	System.out.println();
	System.out.println ( "Now solve for the 3 unknowns, a, b, c " );
	
	double c = (  (b1*c2-c1*b2)*(b1*a4-a1*b4) - (b1*a2-b2*a1)*(b1*c4-c1*b4) ) / ((b1*c2-c1*b2)*(b1*a3-a1*b3)-(b1*a2-b2*a1)*(b1*c3-c1*b3)) ;
	double b = (b1*c4-b4*c1-(b1*c3-b3*c1)*c)/(b1*c2-c1*b2);
	double a = (a4-(a2*b)-(a3*c))/a1;
	
	System.out.println ( "a= " + a + "  b= " + b + "  c= " + c );
	System.out.println ( "So the curve equation ~= " + a + "x^2 + " + b + "x + " + c );
	
    }
    
    
    
}