I need help figuring out the mistake in my function approximation

I am doing function approximation but i'm not getting a smooth function at all. I've tried least squares method and backslash for the same question. Could someone please point out my mistake and help me? Thank you.

Answer 1

[the cyclist]

Your 5th-degree polynomial is smooth (but not unique, because you only have 4 data points). You just don't plot on a fine enough grid to see it. Here, I plot just your 2nd and 5th-order polynomials, on a finer grid.

x = [0.1 0.3 0.4 0.6 0.9];
y = [0 2.2 1 0.6 3.7];


x0 = 0 : 0.02 : 0.9;

for m = [2 5]

    P = polyfit(x,y,m);
    yaprox = polyval(P,x);
    yaprox0 = polyval(P,x0);

    figure
    hold on
    plot(x,y,'o','MarkerSize',8);
    plot(x0,yaprox0,'ro-','MarkerSize',2);
    xlabel('x')
    ylabel('y')
    legend('data','fit')

end

Warning: Polynomial is not unique; degree >= number of data points.

[Matlab1]

thank you for your reply!

Answer 2

[John D'Errico]

Avoid the use of the normal equations as you used, i.e., this crapola that you called the least squares method. Backslash is just as much a least squares method!

% A = [sum(x.^2) sum(x.^3)
%     sum(x.^3) sum(x.^4)];
% b = [sum(y.*x);sum(y.*x.^2)];
% 
% % solution of the system
% c = inv(A)*b;

Properly solve for the coefficients using backslash.

x = [0.1 0.4 0.9 1.6 1.8]';
y = [0.4 1.0 1.8 4.1 5.5]';

Your model is: f(x) = c1x + c2x^2

A = [x,x.^2];
C12 = A\y;

plot(x,y,'o');
hold on
yfun = @(X) C12(1)*X + C12(2)*X.^2;
fplot(yfun,[min(x),max(x)])

Note my use of fplot there. Your evaluation of the points on the curve

% xx = x(1):0.1:x(end);

was a bad idea. Why? How many points did it generate? LOOK CAREFULLY! Just because 0.1 seems like a small number, IS IT REALLY????? What is x again?

x = [0.1 0.4 0.9 1.6 1.8]';

Alternatively, you might have used linspace to generate that vector. I would suggest your real problem was in just the use of a stride of 0.1 between points to evaluate the curve at.