3.3 Errors with power series

Source: Series-summation ideas from Computational Physics (Landau, Páez, Bordeianu), §3.3. Rewritten in original wording. Continues 3.2 Experimental error investigation.

A standard job in computational physics is to turn a series into a number. The sine expansion is exact as an infinite sum:

$$\sin x = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + \cdots$$

Your task is to evaluate $\sin x$ from this series alone for some $x<2\pi$ and some $x>2\pi$, with a relative error under one part in $10^{8}$. An infinite sum is mathematics, not an algorithm: the machine has to stop. The algorithm is the finite recipe

$$\sin x \simeq \sum_{n=1}^{N} \frac{(-1)^{n-1} x^{2n-1}}{(2n-1)!}.$$

The remaining question is when $N$ is enough. Do not stop because the printout matches a table or math.sin — that is grading the answer with the answer. A usable test is: the next term is already smaller than the error you are willing to live with. For large $|x|$ that forces a large $N$; the terms may even grow for a long stretch before they start to shrink and the whole sum settles.

Do not form $x^{2n-1}/(2n-1)!$ from scratch

Computing a huge power and a huge factorial, then dividing, is a bad loop. Neighbouring terms are cheaper — one multiply per step — and the last term is a usable stop proxy while round-off is small. Walk the finite sum, the recursive ratio, and $|t_n/S_n|<\varepsilon$ — pick the matching equation at each step.

Theorem (term as remainder). If round-off is not yet in charge, the first omitted term is a usable estimate of the truncation error. Stop when that term is a small fraction of the running sum.

term ← x,  sum ← x,  n ← 1,  ε ← 1e-8
repeat
    n ← n + 1
    term ← term * (-x*x) / ((2n-1)*(2n-2))
    sum ← sum + term
until |term/sum| < ε

Slide $x$ past $2\pi$. The ratio $|t_n/S_n|$ stays large until $n$ has outrun $|x|$; then it drops through the dashed $\varepsilon$ line.

Quiz

Power-series summation

1 / 4

Why is the infinite sine series not yet an algorithm?

Pick an answer to see if you are right.

Lab: $\sin x$ to $10^{-8}$

Implement the recurrence and the $|t_n/S_n|$ stop. Do not call math.sin or math.factorial. The wrapper prints eight digits after the decimal. One test uses $x<2\pi$ and one uses $x>2\pi$.

Activity: Sine by recurrence

Input: x eps. Print my_sin(x, eps) with :.8f. Start from term = sum = x. Each new term is the previous times -x*x/((2*n-1)*(2*n-2)) with n counting terms (n starts at 1, then 2, 3, …).

If $\varepsilon$ is tighter than the machine can represent, the ratio $|t_n/S_n|$ will bottom out around $\epsilon_m$ and the loop may spin. Keep the requested tolerance above machine precision, as in 3.2’s trough: more terms are not always more digits.

Next: 3.3b Series labs — table, false convergence, cancellation dump, $2\pi$ reduction, bad factorial form, and short write-ups.