On the rapid computation of various polylogarithmic constants

@article{Bailey1997OnTR,
  title={On the rapid computation of various polylogarithmic constants},
  author={David H. Bailey and Peter B. Borwein and Simon Plouffe},
  journal={Math. Comput.},
  year={1997},
  volume={66},
  pages={903-913}
}
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 π on a modest work station in a few hours run time. 
On the computation of the n^th decimal digit of various transcendental numbers
A method for computing the ௧௛ decimal digit of ߨ in ( ଷ ݋() ଷ ) time and with very little memory is presented here. The computation is based on the recently discovered Bailey-Borwein-PlouffeExpand
Computation of the nth decimal digit of π with low memory
This paper presents an algorithm that computes directly the n-th decimal digit of π in sub-quadratic time and very low memory. It improves previous results of Simon Plouffe, later refined by FabriceExpand
N-th digit computation
Xavier Gourdon and Pascal Sebah February 12, 2003 Is it possible to compute directly the n-th digit of π without computing all its first n digits ? At first sight, the problem seems of the sameExpand
Birth, growth and computation of pi to ten trillion digits
The universal real constant pi, the ratio of the circumference of any circle and its diameter, has no exact numerical representation in a finite number of digits in any number/radix system. It hasExpand
Recognizing Numerical Constants
The advent of inexpensive, high-performance computers and new efficient algorithms have made possible the automatic recognition of numerically computed constants. In other words, techniques now existExpand
Memory Efficient Arithmetic
TLDR
The algorithm is highly parallelizable, although if one uses M processors, to get an M-fold reduction in running time, the memory requirements will increase by a factor of M. Expand
Finding new mathematical identities via numerical computations
TLDR
A remarkable new formula forπ permits one to directly compute the n-th hexadecimal digit of π, without computing the first n - 1 digits, and without the need of multiple-precision arithmetic software. Expand
Construction of binomial sums for and polylogarithmic constants inspired by BBP formulas.
We present new sums involving binomial coecients for and various logarithms and polylogarithms constants. These sums are a generalization of BBP formulas first introduced by D. Bailey, P. Borwein andExpand
History of the formulas and algorithms for pi
Throughout more than two millennia many formulas have been obtained, some of them beautiful, to calculate the number pi. Among them, we can find series, infinite products, expansions as continuedExpand
Best k-digit rational approximation of irrational numbers: Pre-computer versus computer era
TLDR
The rational approximation of some of the famous irrational numbers such as the π and the exponential function e is focused on by the ingenuity of super-mathematicians in several parts of the world during the pre-computer era spanning over centuries. Expand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 58 REFERENCES
Asymptotically Fast Algorithms for the Numerical Multiplication and Division of Polynomials with Complex Coeficients
TLDR
Under reasonable assumptions, polynomial multiplication and discrete Fourier transforms of length n and in l-bit precision are possible in time O(ψ (nl), and division of polynomials in O(n(l+n))). Expand
A SPIGOT ALGORITHM FOR THE DIGITS OF PI
It is remarkable that the algorithm illustrated in Table 1, which uses no floating-point arithmetic, produces the digits of π. The algorithm starts with some 2s, in columns headed by the fractionsExpand
Experimental Evaluation of Euler Sums
TLDR
Numerical computations using an algorithm that can determine, with high confidence, whether or not a particular numerical value can be expressed as a rational linear combination of several given constants are presented. Expand
On the Evaluation of Euler Sums
TLDR
Various series expansions of ζ(r, s) for real numbers r and s are established, which generally involve infinitely many zeta values. Expand
Ramanujan, modular equations, and approximations to Pi or how to compute one billion digits of Pi
The year 1987 was the centenary of Ramanujan’s birth. He died in 1920 Had he not died so young, his presence in modern mathematics might be more immediately felt. Had he lived to have access toExpand
Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity
Complete Elliptic Integrals and the Arithmetic-Geometric Mean Iteration. Theta Functions and the Arithmetic-Geometric Mean Iteration. Jacobi's Triple-Product and Some Number Theoretic Applications.Expand
On the complexity of familiar functions and numbers
This paper examines low-complexity approximations to familiar functions and numbers. The intent is to suggest that it is possible to base a taxonomy of such functions and numbers on theirExpand
The Design and Analysis of Computer Algorithms
TLDR
This text introduces the basic data structures and programming techniques often used in efficient algorithms, and covers use of lists, push-down stacks, queues, trees, and graphs. Expand
Algorithms and Complexity
  • H. Wilf
  • Mathematics, Computer Science
  • Lecture Notes in Computer Science
  • 1997
TLDR
A computation that is guaranteed to take at most cn3 time for input of size n will be thought of as an ‘easy’ computation, and one that needs at most n10 time is also easy. Expand
The Parallel Evaluation of General Arithmetic Expressions
  • R. Brent
  • Mathematics, Computer Science
  • JACM
  • 1974
TLDR
It is shown that arithmetic expressions with n ≥ 1 variables and constants; operations of addition, multiplication, and division; and any depth of parenthesis nesting can be evaluated in time 4 log 2 + 10(n - 1) using processors which can independently perform arithmetic operations in unit time. Expand
...
1
2
3
4
5
...