On the rapid computation of various polylogarithmic constants

  title={On the rapid computation of various polylogarithmic constants},
  author={David H. Bailey and Peter B. Borwein and Simon Plouffe},
  journal={Math. Comput.},
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

The computation is based on the recently discovered Bailey-Borwein-Plouffe algorithm and the use of a new algorithm that simply splits an ordinary fraction into its components.

Computation of the nth decimal digit of π with low memory

An algorithm that computes directly the n-th decimal digit of π in sub-quadratic time and very low memory is presented, leading to intermediate time complexity between linear and quadratic.

N-th digit computation

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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 has

On the Complexity of Algebraic Numbers, and the Bit-Complexity of Straight-Line Programs

Improved uniform TC 0 circuits for division, matrix powering, and related problems are presented, where the improvement is in terms of “majority depth” (initially studied by Maciel and Th´erien).

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 exist

Memory Efficient Arithmetic

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Finding new mathematical identities via numerical computations

  • D. Bailey
  • Mathematics, Computer Science
  • 1998
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.

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 and

History of the formulas and algorithms for pi

A continual battle has been waged just to break the records computing digits of this number; records have been set using rapidly converging series, ultra fast algorithms and really surprising ones, calculating isolated digits.



Asymptotically Fast Algorithms for the Numerical Multiplication and Division of Polynomials with Complex Coeficients

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))).


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 fractions

Experimental Evaluation of Euler Sums

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.

On the Evaluation of Euler Sums

Various series expansions of ζ(r, s) for real numbers r and s are established, which generally involve infinitely many zeta values.

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 to

Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity

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On the complexity of familiar functions and numbers

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The Design and Analysis of Computer Algorithms

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.

Algorithms and Complexity

  • H. Wilf
  • Computer Science
    Lecture Notes in Computer Science
  • 1997
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.

The Parallel Evaluation of General Arithmetic Expressions

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.