R help - Aug 2023 - Numerical stability of: 1/(1 - cos(x)) - 2/x^2

Dear R-Users,

I tried to compute the following limit:
x = 1E-3;
(-log(1 - cos(x)) - 1/(cos(x)-1)) / 2 - 1/(x^2) + log(x)
# 0.4299226
log(2)/2 + 1/12
# 0.4299069

However, the result diverges as x decreases:
x = 1E-4
(-log(1 - cos(x)) - 1/(cos(x)-1)) / 2 - 1/(x^2) + log(x)
# 0.9543207
# correct: 0.4299069

I expected the precision to remain good with x = 1E-4 or x = 1E-5.

This part blows up - probably some significant loss of precision of 
cos(x) when x -> 0:
1/(cos(x) - 1) - 2/x^2

Maybe there is some room for improvement.

Sincerely,

Leonard
=========The limit was part of the integral:
up = pi/5;
integrate(function(x) 1 / sin(x)^3 - 1/x^3 - 1/(2*x), 0, up)
(log( (1 - cos(up)) / (1 + cos(up)) ) +
 ?? ?+ 1/(cos(up) - 1) + 1/(cos(up) + 1) + 2*log(2) - 1/3) / 4 +
 ?? ?+ (1/(2*up^2) - log(up)/2);

# see:
https://github.com/discoleo/R/blob/master/Math/Integrals.Trig.Fractions.Poly.R

You could rewrite

1 - cos(x)

as

2 * sin(x/2)^2

and that might give you more precision?

On Wed, Aug 16, 2023, 01:50 Leonard Mada via R-help <r-help at
r-project.org>
wrote:
> Dear R-Users,
>
> I tried to compute the following limit:
> x = 1E-3;
> (-log(1 - cos(x)) - 1/(cos(x)-1)) / 2 - 1/(x^2) + log(x)
> # 0.4299226
> log(2)/2 + 1/12
> # 0.4299069
>
> However, the result diverges as x decreases:
> x = 1E-4
> (-log(1 - cos(x)) - 1/(cos(x)-1)) / 2 - 1/(x^2) + log(x)
> # 0.9543207
> # correct: 0.4299069
>
> I expected the precision to remain good with x = 1E-4 or x = 1E-5.
>
> This part blows up - probably some significant loss of precision of
> cos(x) when x -> 0:
> 1/(cos(x) - 1) - 2/x^2
>
> Maybe there is some room for improvement.
>
> Sincerely,
>
> Leonard
> =========> The limit was part of the integral:
> up = pi/5;
> integrate(function(x) 1 / sin(x)^3 - 1/x^3 - 1/(2*x), 0, up)
> (log( (1 - cos(up)) / (1 + cos(up)) ) +
>      + 1/(cos(up) - 1) + 1/(cos(up) + 1) + 2*log(2) - 1/3) / 4 +
>      + (1/(2*up^2) - log(up)/2);
>
> # see:
>
>
https://github.com/discoleo/R/blob/master/Math/Integrals.Trig.Fractions.Poly.R
>
> ______________________________________________
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