Taylor Series cosx function

Use the Taylor Series cos x = 1-x^2/2!+x^4/4!-x^6-6!+x^8/8!+..... to write a program to compute cos x correct to four decimal places (x is in radians). See how many terms are needed to get 4-figure agreement with the MATLAB build-in function cos x.

[James Tursa]

Please make an attempt at coding this yourself, then post your code and ask specific questions about it and we can help.

http://www.mathworks.com/matlabcentral/answers/6200-tutorial-how-to-ask-a-question-on-answers-and-get-a-fast-answer

[John D'Errico]

I would add that it very much depends on the value of x. Thus, for tiny x, say 0.00001, 4 figure agreement arises from just the constant term in that series. However, were you to try to compute cos(x) from that series for x = 1000, expect to need a huge number of terms.

[James Tursa]

Adding to John's comment, I would note that this series can become useless fairly quickly as x gets "large" because the large intermediate sums amass huge cancellation errors, yielding garbage for results. At the very least, one would need to rely on the knowledge that cos is periodic and adjust the input with some type of mod operation before feeding it into this series. However, a good method for this up-front mod operation is not at all trivial. Using a naive mod method can result in a lousy input value which will yield a garbage result compared to MATLAB.

MATLAB has a very sophisticated method of doing this mod operation internally that basically preserves all of the significant digits down to 1e-17 even if the input is relatively large. E.g., an extreme example:

>> x = 1e16  % relatively large input
x =
     1.000000000000000e+16
>> cos(x)
ans =
  -0.626168198133086
>> xm = mod(x,2*pi)  % naive mod operation
xm =
   2.637242432414304
>> cos(xm)
ans =
  -0.875488657479139  % garbage result

>> digits 50  % set up for extra digits in arithmetic
>> vx = vpa('1e16')
vx =
10000000000000000.0
>> v2pi = 2*vpa('pi')
v2pi =
6.2831853071795864769252867665590057683943387987502
>> k = floor(vx/v2pi)
k =
1591549430918953
>> vxm = vx - k * v2pi  % extended digit "mod" equivalent operation
vxm =
2.2474252491623665482579555800417389331267475864972
>> cos(double(vxm))
ans =
  -0.626168198133086  % good result

As you can see from the above, the naive mod when used with large inputs gave garbage results when compared to MATLAB's internal method. And MATLAB's internal method, whatever it is, basically mimics all of this extra precision arithmetic shown above to get its result.

Based on this, one simple way to do a mod(x,2*pi) operation in MATLAB using the same background methods that MATLAB uses is as follows:

function y = mod2pi(x) % Result is in range -pi to +pi
y = atan2(sin(x),cos(x));
% To force the result in 0 to 2pi, uncomment the following lines
% g = y < 0;
% y(g) = y(g) + 2*pi;
end

[John D'Errico]

This is usually called range reduction, reducing the set of values x for which the series must be expected to operate over.