[VA] SRC #010 - Pi Day 2022 Special

[Valentin Albillo]

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Hi, all,

(03-18-2022 04:22 PM)Albert Chan Wrote:  Lets recover true PN, and compare errors of products vs exp(sum of logs) [...] Note that ln(C) is odd function. Rewrite ln(C) as polynomial of 1/N, we have:

\(\displaystyle
\ln(C) = \frac{1}{N}
+ \frac{5/9}{N^3}
+ \frac{13/45}{N^5}
+ \frac{127/315}{N^7}
- \frac{89/135}{N^9}
\;+\; ... \) [...]

I must point out that this formal series of correction factors is asymptotic and divergent, i.e., its coefficients might be small and even decreasing for a while but eventually they grow bigger and bigger, both numerators and denominators, and thus can't be used to obtain arbitrary precision, as I explained in another case in post #27 of my Short & Sweet Math Challenge #24. Quoting myself from that post:

Quote:

The coefficients of the formal series for cin(x) and tin(x) can be obtained in a number of ways [...] but it's important to be aware that both formal series do not converge. In fact, their radius of convergence is 0 and thus they behave like asymptotic series, so you can't get arbitrarily accurate results by taking more and more terms, you must instead truncate the series after a certain number of terms to get the most accurate results. Using further terms only worsens the accuracy.
  
Although at first sight the coefficients of the formal series for cin(x) and tin(x) seem to (slowly) get smaller and smaller, matter of fact they tend to grow ever bigger after a while, tending to infinity. For instance, for tin(x) we find that the smallest coefficient in absolute value is:

Coeff₃₇ = -0.000000000594338574503
but afterwards we have, e.g.:

Coeff₁₀₁ = 0.0833756228055
Coeff₁₅₁ = 388536047335.239
Coeff₂₀₁ = 6555423874651256623811186991.51
Coeff₂₅₁ = -35365220492708296140377087748804440170254492009.57

The same happens in the present case: you can use a certain number of coefficients to improve accuracy up to the "sweet point" of maximum accuracy, but after that the accuracy quickly degrades and thus using more coefficients is useless and to be avoided.

Quote:PI * PI + → 3.141608361513791562872866895754895      // true PN

VA (products for PN) errors = 15,684,238,090 ULP             // O(n^2) error ?
JFG (log sum for PN) errors = 106 ULP

Regrettably, presently I have no software available to compute the product for n = 2 to n = 100,000 with high accuracy (say, to 100 digits) so I can't check for sure, but I find it somewhat hard to believe that my computation using the 34 digits afforded by Free42 Decimal would lose 11 digits in the process, I'd rather expect 6-7 digits lost at most.

Likewise, Jean-François Garnier computation of said product using logarithms performs about 100,000 multiplications, divisions (1/x) and logarithms (LN1+X) but only loses 3 digits ? Really ?

To settle down the question, if someone with access to Mathematica or some other arbitrary-precision software can compute the product for N=100,000 using 100 digits, say, or as many as necessary to ensure full 34 correct digits or more, and post here the resulting value I'd appreciate it. Thanks in advance.

V.