Horner's scheme for sine Tailor's series computatioin stability

[Touchkin]

Hi all! Please help me answer these questions:
1. Why is the standard Horner's scheme for the computation of Taylor's series for sine unstabil?
The standard scheme is sin(x) = x(1 + x^2(-1/3! + x^2(1/5! + x^2(-1/7! + ... x^2(-1/(2n-1)! + x^2/(2n+1)!)...)

2. How can we modify the scheme to make it stabil?

I think instability is somehow connected with the changing sign of the terms in the Taylor's series, but I am not sure how. If you have any ideas ar you know the answer it will be interesting to hear.

 

[Aaron Bex]

The instability of the standard Horner's scheme for the computation of Taylor's series for sine is due to the fact that the terms in the series switch between positive and negative values. To make the scheme more stable, it can be modified by multiplying each term in the series with the sign of its previous term. This way, all terms in the series will have the same sign, which makes the scheme more stable.

 

[piercas]

Hello! I can provide some insight on the stability of Horner's scheme for the computation of Taylor's series for sine. The instability in this scheme is caused by the alternating signs of the terms in the series. This can lead to a large accumulation of round-off errors, especially for larger values of x. This can result in inaccurate or even divergent results.

To modify the scheme and make it more stable, we can use an alternating series summation method. This involves rearranging the terms in the series so that the alternating signs are eliminated. This can be done by grouping the terms in pairs and using the identity sin(x) = sin(x/2)cos(x/2) to simplify the terms. This results in a new series that converges more quickly and reduces the accumulation of round-off errors.

Another approach is to use a more accurate representation of the sine function, such as a Chebyshev series or a Taylor's series with higher order terms. This can also help to reduce the effects of round-off errors and improve the stability of the computation.

In summary, the instability in Horner's scheme for the computation of Taylor's series for sine can be addressed by modifying the scheme itself or by using more accurate representations of the sine function. Hope this helps!