Why are these sums not the same?

Question

Steve 2015-1-16

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https://ww2.mathworks.cn/matlabcentral/answers/170238-why-are-these-sums-not-the-same

编辑： Roger Stafford 2015-1-16

采纳的回答： Roger Stafford

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Hello together,

What is the difference between these two:

1:

            >>sum(cos(0.5*pi)*(0:99))
            ans =
                3.0310e-13

and 2:

            >>S=0;
            >> for J=0:99
                S=cos(0.5*pi)*j;
               end;
            >> S =
               6.0620e-15

Why is there a different result?

Thank you for any help!

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Answer 1

Roger Stafford 2015-1-16

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https://ww2.mathworks.cn/matlabcentral/answers/170238-why-are-these-sums-not-the-same#answer_165077

编辑：Roger Stafford 2015-1-16

在 MATLAB Online 中打开

There is no reason they should be the same! In the for-loop method you are not doing a summation, as is done in the first method. All you obtain is the value of the last term, namely cos(0.5*pi)*99.

If you change the last method to

S = S + cos(0.5*pi)*j;

any difference between that value and the one for the first method will be due to differences in round off errors.

Ideally cos(0.5*pi) should be an exact zero, but both because pi cannot be represented exactly and because of errors in the cosine computation, it actually gives 6.1232e-17 as its value.