On the Rapid Computation of Various Polylogarithmic Constants

Abstract

We give algorithms for the computation of the d-th digit of certain transcendental numbers in various bases. These algorithms can be easily implemented (multiple precision arithmetic is not needed), require virtually no memory, and feature run times that scale nearly linearly with the order of the digit desired. They make it feasible to compute, for example, the billionth binary digit of log(2) or _ on a modest workstation in a few hours run time.We demonstrate this technique by computing the ten billionth hexadecimal digit of it, the billionth hexadecimal digits of _'_, log(2) and log2(2), and the ten billionth decimal digit of log(9/10).These calculations rest on the observation that very special types of identities exist for certain numbers like n, it 2, log(2) and 1og2(2). These are essentially polylogarithmic ladders in an integer base. A number of these identities that we derive in this work appear to be new, for example the critical identity for 7z:

Parallel integer relation detection: Techniques and applications

“…Remove corner on H diagonal: If m ≤ n − 2 then set t 0 := H 2 mm + H 2 m,m+1 , t 1 := H mm /t 0 and t 2 := H m,m+1 /t 0 ; for i := m to n: set t 3 := H im , t 4 := H i,m+1 , H im := t 1 t 3 + t 2 t 4 and H i,m+1 := −t 2 t 3 + t 1 t 4 ; endfor; endif. 4. Reduce H: For i := m + 1 to n: for j := min(i − 1, m + 1) to 1 step −1: set t := nint(H ij /H jj ) and y j := y j + ty i ; for k := 1 to j: set H ik := H ik − tH jk ; endfor; for k := 1 to n: set A ik := A ik − tA jk and B kj := B kj + tB ki ; endfor; endfor; endfor.…”

“…Through the centuries mathematicians have assumed that there is no shortcut to computing just the n-th digit of π. Thus, it came as no small surprise when such an algorithm was recently discovered [4]. In particular, this simple scheme allows one to compute the n-th hexadecimal (or binary) digit of π without computing any of the first n − 1 digits, without using multiple-precision arithmetic software, and at the expense of very little computer memory.…”

“…It is likely the first instance in history of a significant new formula for π discovered by computer. Further base-2 results are given in [4,14]. In [13] base-3 results were obtained, including…”

Reconfigurable Hardware Implementation of the Successive Overrelaxation Method

“…The SOR rate of convergence strongly depends on the choice of the relaxation factor, [3]. Extensive work has been done on finding a good estimate of this factor in the [0, 2] interval [3,23].…”

“…As mathematical problems become more complex, it might be still possible to find their solutions by means of available computing devices. However, there are several mathematical problems whose solutions are difficult to be realized using available computing power [3][4][5]. Examples of such problems are factoring very large numbers (RSA depends on this problem's computational difficulty) [5], finding the solution to partial differential equations [6], and deciding whether a knot in 3-dimensional Euclidean space is unknotted (topological problem) [7].…”