Dear Martin,
Thank you very much for your analysis.
I add only a small comment:
- the purpose of the modified formula was to apply l'Hospital;
- there are other ways to transform the formula; although applying
l'Hospital once is probably more robust than simple transformations (but
the computations are also more tedious and error prone);
After more careful thinking, I believe that it is a limitation due to
floating points:
x = 1E-4
1/(-x^2/2 - x^4/24) + 2/x^2
1/6
y = 1 - x^2/2 - x^4/24;
1/(cos(x) - 1) + 2/x^2
1/(y - 1) + 2/x^2
# -1.215494
# correct: 1/6
We need the 3rd term for the correct computation of cos(x) in this
problem: but this is x^4 / 24, which for 1E-4 requires precision at
least up to 1E-16 / 24, or ~ 1E-18. I did not thought initially about
that. The trigonometric functions skip one term, and are therefore much
uglier than the log. The problem really stems from the representation of
1 - x^2/2 as shown below:
x = 1E-4
print(1 - x^2/2, digits=20)
print(0.999999995, digits=20) # fails
# 0.99999999500000003039
Maybe some functions of type cos1p and cos1n would be handy for such
computations (to replace the manual series expansion):
cos1p(x) = 1 + cos(x)
cos1n(x) = 1 - cos(x)
Though, I do not have yet the big picture.
Sincerely,
Leonard
On 8/17/2023 1:57 PM, Martin Maechler wrote:
Leonard Mada
on Wed, 16 Aug 2023 20:50:52 +0300 writes:
> Dear Iris,
> Dear Martin,
> Thank you very much for your replies. I add a few comments.
> 1.) Correct formula
> The formula in the Subject Title was correct. A small glitch swept into
> the last formula:
> - 1/(cos(x) - 1) - 2/x^2
> or
> 1/(1 - cos(x)) - 2/x^2 # as in the subject title;
> 2.) log1p
> Actually, the log-part behaves much better. And when it fails, it fails
> completely (which is easy to spot!).
> x = 1E-6
> log(x) -log(1 - cos(x))/2
> # 0.3465291
> x = 1E-8
> log(x) -log(1 - cos(x))/2
> # Inf
> log(x) - log1p(- cos(x))/2
> # Inf => fails as well!
> # although using only log1p(cos(x)) seems to do the trick;
> log1p(cos(x)); log(2)/2;
> 3.) 1/(1 - cos(x)) - 2/x^2
> It is possible to convert the formula to one which is numerically more
> stable. It is also possible to compute it manually, but it involves much
> more work and is also error prone:
> (x^2 - 2 + 2*cos(x)) / (x^2 * (1 - cos(x)))
> And applying L'Hospital:
> (2*x - 2*sin(x)) / (2*x * (1 - cos(x)) + x^2*sin(x))
> # and a 2nd & 3rd & 4th time
> 1/6
> The big problem was that I did not expect it to fail for x = 1E-4. I
> thought it is more robust and works maybe until 1E-5.
> x = 1E-5
> 2/x^2 - 2E+10
> # -3.814697e-06
> This is the reason why I believe that there is room for improvement.
> Sincerely,
> Leonard
Thank you, Leonard.
Yes, I agree that it is amazing how much your formula suffers from
(a generalization of) "cancellation" --- leading you to think
there was a problem with cos() or log() or .. in R.
But really R uses the system builtin libmath library, and the
problem is really the inherent instability of your formula.
Indeed your first approximation was not really much more stable:
## 3.) 1/(1 - cos(x)) - 2/x^2
## It is possible to convert the formula to one which is numerically more
## stable. It is also possible to compute it manually, but it involves much
## more work and is also error prone:
## (x^2 - 2 + 2*cos(x)) / (x^2 * (1 - cos(x)))
## MM: but actually, that approximation does not seem better (close to the
breakdown region):
f1 <- \(x) 1/(1 - cos(x)) - 2/x^2
f2 <- \(x) (x^2 - 2 + 2*cos(x)) / (x^2 * (1 - cos(x)))
curve(f1, 1e-8, 1e-1, log="xy" n=2^10)
curve(f2, add = TRUE, col=2, n=2^10)
## Zoom in:
curve(f1, 1e-4, 1e-1, log="xy",n=2^9)
curve(f2, add = TRUE, col=2, n=2^9)
## Zoom in much more in y-direction:
yl <- 1/6 + c(-5, 20)/100000
curve(f1, 1e-4, 1e-1, log="x", ylim=yl, n=2^9)
abline(h = 1/6, lty=3, col="gray")
curve(f2, add = TRUE, n=2^9, col=adjustcolor(2, 1/2))
Now, you can use the Rmpfr package (interface to the GNU MPFR
multiple-precision C library) to find out more :
if(!requireNamespace("Rmpfr")) install.packages("Rmpfr")
M <- function(x, precBits=128) Rmpfr::mpfr(x, precBits)
(xM <- M(1e-8))# yes, only ~ 16 dig accurate
## 1.000000000000000020922560830128472675327e-8
M(10, 128)^-8 # would of course be more accurate,
## but we want the calculation for the double precision number 1e-8
## Now you can draw "the truth" into the above plots:
curve(f1, 1e-4, 1e-1, log="xy",n=2^9)
curve(f2, add = TRUE, col=2, n=2^9)
## correct:
curve(f1(M(x, 256)), add = TRUE, col=4, lwd=2, n=2^9)
abline(h = 1/6, lty=3, col="gray")
But, indeed we take note how much it is the formula instability:
Also MPFR needs a lot of extra bits precision before it gets to
the correct numbers:
xM <- c(M(1e-8, 80), M(1e-8, 96), M(1e-8, 112),
M(1e-8, 128), M(1e-8, 180), M(1e-8, 256))
## to and round back to 70 bits for display:
R <- \(x) Rmpfr::roundMpfr(x, 70)
R(f1(xM))
R(f2(xM))
## [1] 0 0
0.15407439555097885670915
## [4] 0.16666746653133802175779 0.16666666666666666749979
0.16666666666666666750001
## 1. f1() is even worse than f2() {here at x=1e-8}
## 2. Indeed, even 96 bits precision is *not* sufficient at all, ...
## which is amazing to me as well !!
Best regards,
Martin
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