Polynomial Approximation, is it possible in matlab?

I have four points on the graph with the following coordinates.

x1 - 1

y1 - 3.5

x2 - 2

y2 - 14/3

x3 - 3

y3 - 14

x4 - 4

y4 - 28

Is it possible using the Lagrange approximation polynomial coefficient calculation method to find the polynomial / function given by the four points?

I don't know the algorithm very well and I don't have the strongest matlab knowledge. I'm learning.

Thank you.

MATLAB Version: 8.5.0.197613 (R2015a)

Answer 1

[Luna]

Hi,

Try below (it uses least squares):

For lagrange you can look at that link:

lagrangepoly

x1 = 1;
y1 = 3.5;

x2 = 2;
y2 = 14/3;

x3 = 3;
y3 = 14;

x4 = 4;
y4 = 28;

x1Array = [x1,x2,x3,x4];
y1Array = [y1,y2,y3,y4];
n = 1; % polynomial degree (you can change it as you wish)
p = polyfit(x1Array,y1Array,n); % p is coefficient of your polynomial: P(X) = P(1)*X^N + P(2)*X^(N-1) +...+ P(N)*X + P(N+1) descending order.
newY = polyval(p,x1Array); % function results

plot(x1Array,y1Array, 'bo',x1Array,newY,'-r');
grid;
legend('Data','Fitted Data');

[Milky Way]

Ok. But where is the function, the polynomial. When I run the program I only show a graph.

[madhan ravi]

reduce n less than 4

[Milky Way]

Okay. But how can I find the polynomial? Simple form: aX^n + bX^(n-1) .... the polynomial itself.

[Steven Lord]

See the comment on line 12.

[Milky Way]

@Steven Lord Ok. n=3 => P(1)*X^3 + P(2)*X^2 + P(3)*X + P(4)

And how those coefficients can be calculated? (P(1), P(2), P(3) and P(4)). I can't figure it out.

[Torsten]

p(x)=(x-x2)*(x-x3)*(x-x4)/((x1-x2)*(x1-x3)*(x1-x4))*y1+...
     (x-x1)*(x-x3)*(x-x4)/((x2-x1)*(x2-x3)*(x2-x4))*y2+...
     (x-x1)*(x-x2)*(x-x4)/((x3-x1)*(x3-x2)*(x3-x4))*y3+...
     (x-x1)*(x-x2)*(x-x3)/((x4-x1)*(x4-x2)*(x4-x3))*y4
 

Multiply out and order according to powers of x.

[Milky Way]

And can't be solved in Matlab? I wonder if it can be solved in Matlab to get rid of "manual" work.

[Torsten]

Yes, the coefficients can be taken from the line

p = polyfit(x1Array,y1Array,n); % p is coefficient of your polynomial: P(X) = P(1)*X^N + P(2)*X^(N-1) +...+ P(N)*X + P(N+1) descending order.

of your above code.

[Milky Way]

Yes. Thank you very much, I'm at the beginning, but as I said, I'll learn. :)

[Torsten]

But you know that using the interpolating spline method to calculate the coefficients is not what you are supposed to do in your homework (it says something about "Lagrange interpolation polynomial", doesn't it) ??

[Luna]

Yes, it says lagrange that's why I gave the lagrangepoly link from fileexchange in my answer.

Please check the file and use. It does the same thing with only Lagrange method.

lagrangepoly

It also explains with examples.