atan Taylor Polynomial using polyval function.

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Jim Oste
Jim Oste 2015-2-17
回答： Evelia Coss 2021-2-18
I need to write a function that computes the Taylor polynomial P_n(x) for a general odd number n>=1 using the built-in Matlab function polyval. I have the following function so far for the Taylor polynomial:
[ y ] = ost_arctanTaylor(n, x)
%Computes the Taylor polynomial for f(x) = atan(x)
for i = 1:2:n
y = ((-1).^(i)).*((x.^(2.*i-1))./(2.*i-1));
end
end
I am confused on how to integrate the polyval function into this code. Any help?

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John D'Errico
John D'Errico 2015-2-17
You are trying to evaluate the polynomial itself in a loop, although the loop as you wrote it will not do what you think.
The requirement was for polyval to evaluate it, NOT you. So you need to build the coefficients of the polynomial, and pass them into polyval. Let it do the work.
  2 个评论
Jim Oste
Jim Oste 2015-2-18
Do you know how I would go about coding the coefficients? I know that polyval evaluates the function given the coefficients of descending powers but with the Taylor series for arctan(x) I only need odd powers. How would the polyval function know not to evaluated anything for even powers?
John D'Errico
John D'Errico 2015-2-18
Polyval does not care if you give it zero coefficients. Make the even term coefficients zero. It WILL use those zero coefficients, effectively multiplying by zero, but who cares? You will just create a vector of coefficients, with the HIGHEST order coefficient first. So, for example, the atan series looks like
x - x^3/3 + x^5/5 ...
then the polynomial as polyval would care about it would be:
P = [1/5 0 -1./3 0 1 0];
Don't forget that zero'th order (constant) term at the end. It is important.

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Evelia Coss
Evelia Coss 2021-2-18
x = input('Number that you want to analyze');
i = input('Number of iterations:');
% Create an cumulative variable
y = 0;
for n = 0:i
arctang = ((-1).^n).*(x.^(2.*n+1)./(2.*n+1));
y = y + arctang;
end
disp('Arctangent value of x is:');
disp(y);
I used the integral of arctang, see the link: https://math.stackexchange.com/questions/29649/why-is-arctanx-x-x3-3x5-5-x7-7-dots

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