How to get coefficients for second degree polynomial with only two given boundary conditions (means with two equations but with three coefficients)?

I have Energy displacement numeric data in which Energy follows second degree polynomial (Ax^2+Bx+C) as given below;

x (disp) = [ 0 1 2 3]; E (energy) = [0 1 4.5 10.5];

Consider first point x = 0 and 1, gives

0 = C1;

1 = A1+B1+C1;

Consider second point x = 1 and 2, gives

1 = A2+B2+C2;

4.5 = 4A2+2B2+C2;

But I have two equations with three coefficients, is there anyway numeric approximation method in matlab to solve for three coefficients with two equations?

Answer 1

[David Goodmanson]

Hi Ankit,

x = [ 0 1 2 3];
E = [0 1 4.5 10.5]; 
c = polyfit(x,E,2)    % fit a quadratic
E1 = polyval(c,x)
figure(1)
plot(x,E,'o-',x,E1)
grid on

c =
1.2500   -0.2500   -0.0000

polyfit fits a quadratic (resulting coefficients are in c) to the four points in a least squared error sense. It so happens that the four points exactly fit the resulting quadratic.