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Computer Science - Theoretical Computer Science | Pi - Unleashed

Pi - Unleashed

Arndt, Jörg, Haenel, Christoph

Translated by Lischka, C., Lischka, D.

2001, XII, 270 p.

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  • New and sensational discoveries in Pi computation
  • Everything the reader always wanted to know about Pi
  • With many links to the Internet
  • Instructions for efficiently testing hardware with programs for Pi computation
Never in the 4000-year history of research into pi have results been so prolific as at present. In their book Joerg Arndt and Christoph Haenel describe in easy-to-understand language the latest and most fascinating findings of mathematicians and computer scientists in the field of pi.
Attention is focussed on new methods of computation whose speed outstrips that of predecessor methods by orders of magnitude.

Content Level » Professional/practitioner

Keywords » Internet - Monte Carlo Method - algorithms - computer - continued fraction

Related subjects » Number Theory and Discrete Mathematics - Theoretical Computer Science

Table of contents 

1. The State of Pi Art.- 2. How Random is ??.- 2.1Probabilities.- 2.2 Is ? normal?.- 2.3 So is ? not normal?.- 2.4 The 163 phenomenon.- 2.5 Other statistical results.- 2.6 The Intuitionists and ?.- 2.7 Representation of continued fractions.- 3. Shortcuts to ?.- 3.1Obscurer approaches to ?.- 3.2 Small is beautiful.- 3.3 Squeezing ? through a sieve.- 3.4 ? and chance (Monte Carlo methods).- 3.5 Memorabilia.- 3.6 Bit for bit.- 3.7 Refinements.- 3.8 The ? room in Paris.- 4. Approximations for ?and Continued Fractions.- 4.1Rational approximations.- 4.2 Other approximations.- 4.3 Youthful approximations.- 4.4 On continued fractions.- 5. Arcus Tangens.- 5.1 John Machin's arctan formula.- 5.2 Other arctan formulae.- 6. Spigot Algorithms.- 6.1 The spigot algorithm in detail.- 6.2 Sequence of operations.- 6.3 A faster variant.- 6.4 Spigot algorithm for e.- 7.Gauss and ?.- 7.1 The ? AGM formula.- 7.2 The Gauss AGM algorithm.- 7.3 Schönhage variant.- 7.4 History of a formula.- 8. Ramanujan and ?.- 8.1 Ramanujan's series.- 8.2 Ramanujan's unusual biography.- 8.3 Impulses.- 9. The Borweins and ?.- 10. The BBP Algorithm.- 10.1Binary modulo exponentiation.- 10.2 A C program on the BBP series.- 10.3 Refinements.- 11. Arithmetic.- 11.1Multiplication.- 11.2 Karatsuba multiplication.- 11.3 FFT multiplication.- 11.4 Division.- 11.5 Square root.- 11.6 nth root.- 11.7 Series calculation.- 12. Miscellaneous.- 12.1 A ? quiz.- 12.2 Let numbers speak.- 12.3 A proof that ? = 2.- 12.4 The big change.- 12.5 Almost but not quite.- 12.6 Why always more?.- 12.7 ? and hyperspheres.- 12.8 Viète × Wallis = Osler.- 12.9 Squaring the circle with holes.- 12.10 An (in)finite funnel.- 13.The History of ?.- 13.1 Antiquity.- 13.2 Polygons.- 13.3 Infinite expressions.- 13.4 High-performance algorithms.- 13.5 The hunt for single ? digits.- Table: History of ? in the pre-computer era.- Table: History of ? in the computer era.- Table: History of digit extraction records.- 14. Historical Notes.- 14.1 The earliest squaring the circle in history?.- 14.2 A ? law.- 14.3 The Bieberbach story.- 15.The Future: ?Calculations on the Internet.- 15.1 The binsplit algorithm.- 15.2 The ? project on the Internet.- 16. ?Formula Collection.- 17. Tables.- 17.1 Selected constants to 100 places (base 10).- 17.2 Digits 0 to 2,500 of ? (base 10).- 17.3 Digits 2,501 to 5,000 of ? (base 10).- 17.4 Digits 0 to 2,500 of ? (base 16).- 17.5 Digits 2,501 to 5,000 of ? (base 16).- 17.6 Continued fraction elements 0 to 1,000 of ?.- 17.7 Continued fraction elements 1,001 to 2,000 of ?.- A. Documentation for the hfloat Library.- A.1 What hfloat is (good for).- A.2 Compiling the library.- A.3 Functions of the hfloat library.- A.4 Using hfloats in your own code.- A.5 Computations with extreme precision.- A.6 Precision and radix.- A.7 Compiling & running the ?-example-code.- A.8 Structure of hfloat.- A.9 Organisation of the files.- A. 10 Distribution policy & no warranty.

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