matlabmathnumerical-methodsnumerical-analysistaylor-series

My code for Taylor series graphs cos(2x) instead of cos(x) in Matlab

I'm writing a function that calculates the Taylor series of any Function.

syms  x
y=cos(x);
y0=0;
a=0;
for i=0:25
      diff(y,i);                                %%Gives the derivative formula
      y0=y0+diff(y,i)*((x-a)^i)/factorial(i);   %%sums every new element of the series
end

x=0:0.1:2*pi;
res = subs(y0,x);
plot(x,res,x,cos(x))

This is the Matlab code.

My problem is that it graphs cos(2x) instead of cos(x), similarly it graphs ln(2x) instead of ln(x) and so on.

I have checked the factorials and they appear to be correct. What could be the problem, have I messed up the series or have I made a Matlab mistake?

Solution

  • You are constructing the Taylor polynomial around the point x with incrementx-a, that is, you are computing an approximation for

    f(x+(x-a))=f(2*x-a)

    Now as a=0, this means that as observed, you get f(2*x).

    You would need to evaluate the derivatives at a to get the correct coefficients. 

    y0=y0+subs(diff(y,i),a)*((x-a)^i)/factorial(i);   %%sums every new element of the series