The Borwein brothers, Pi and the AGM.

  title={The Borwein brothers, Pi and the AGM.},
  author={Richard P. Brent and Richard P. Brent},
  journal={arXiv: Number Theory},
  • R. Brent, R. Brent
  • Published 21 February 2018
  • Computer Science, Mathematics
  • arXiv: Number Theory
We consider some of Jonathan and Peter Borweins' contributions to the high-precision computation of $\pi$ and the elementary functions, with particular reference to their book "Pi and the AGM" (Wiley, 1987). Here "AGM" is the arithmetic-geometric mean of Gauss and Legendre. Because the AGM converges quadratically, it can be combined with fast multiplication algorithms to give fast algorithms for the {$n$-bit} computation of $\pi$, and more generally the elementary functions. These algorithms… 
1 Citations

The Art of Modern Homo Habilis Mathematicus, or: What Would Jon Borwein Do?



Easy Proofs of Some Borwein Algorithms for π

The gamma function is a model for complex function analysis used in number theory, geometry, and computer science, and in particular in the area of discrete-time analysis.

Even faster integer multiplication

New proofs of Borwein-type algorithms for Pi

ABSTRACT We use a method of translation to recover Borweins' quadratic and quartic iterations. Then, by using the WZ-method, we obtain some initial values which lead to the limit . We use neither the

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

The Life of π: From Archimedes to ENIAC and Beyond

The desire to understand pi, the challenge, and originally the need, to calculate ever more accurate values of pi, the ratio of the circumference of a circle to its diameter, has challenged

The Computation of π to 29,360,000 Decimal Digits Using Borweins’ Quartically Convergent Algorithm

Paper 7: David H. Bailey, “The computation of pi to 29,360,000 decimal digits using Borweins’ quartically convergent algorithm,” Mathematics of Computation, vol. 50 (1988), p. 283–296. Reprinted by

Fast Multiple-Precision Evaluation of Elementary Functions

It is shown that ƒ(x) can be evaluated, with relative error, in the Schönhage-Strassen bound on M(n), the number of single-precision operations required to multiply n-bit integers.

Old and new algorithms for pi

This is a letter to the editor concerning Semjon Adlaj's article "An eloquent formula for the perimeter of an ellipse", AMS Notices 59, 8 (2012), 1094-1099.

The Arithmetic-Geometric Mean and Fast Computation of Elementary Functions

We produce a self contained account of the relationship between the Gaussian arithmetic-geometric mean iteration and the fast computation of elementary functions. A particularly pleasant algorithm

Bessel Functions. (Scientific Books: A Treatise on the Theory of Bessel Functions)

1. Bessel functions before 1826 2. The Bessel coefficients 3. Bessel functions 4. Differential equations 5. Miscellaneous properties of Bessel functions 6. Integral representations of Bessel