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Method for calculating the value of pi
Borwein's algorithm was devised by Jonathan and Peter Borwein to calculate the value of 1 / π {\displaystyle 1/\pi } . This and other algorithms can be
Borwein's_algorithm
Formula for computing the nth base-16 digit of π
{1}{8k+5}}-{\frac {1}{8k+6}}\right)\right]} The BBP formula gives rise to a spigot algorithm for computing the nth base-16 (hexadecimal) digit of π (and therefore
Bailey–Borwein–Plouffe formula
Bailey–Borwein–Plouffe_formula
Fast method for calculating the digits of π
is called binary splitting. Mathematics portal Bailey–Borwein–Plouffe formula Borwein's algorithm Approximations of π Chudnovsky, David; Chudnovsky, Gregory
Chudnovsky_algorithm
Scottish mathematician (1951–2016)
period 1981–1999. Borwein integral Borwein's algorithm List of University of Waterloo people "Life Summary - Jonathan Michael (Jon) Borwein - People Australia"
Jonathan_Borwein
Quickly converging computation of π
first 200 billion decimal digits of π, with results checked using Borwein's algorithm. Initial value setting: a 0 = 1 b 0 = 1 2 p 0 = 1 t 0 = 1 4 . {\displaystyle
Gauss–Legendre_algorithm
Bailey–Borwein–Plouffe formula: (BBP formula) a spigot algorithm for the computation of the nth binary digit of π Borwein's algorithm: an algorithm to calculate
List_of_algorithms
Mathematical procedure
H. Bailey and J.M. Borwein: "PSLQ: An Algorithm to Discover Integer Relations" (May 14, 2020) Weisstein, Eric W. "PSLQ Algorithm". MathWorld. A Polynomial
Integer_relation_algorithm
Surname list
Peter Borwein (1953–2020), Scotland-born Canadian mathematician and a professor Borwein's algorithm, algorithm devised by Jonathan and Peter Borwein to calculate
Borwein
Varying methods used to calculate pi
typically computed with the Gauss–Legendre algorithm and Borwein's algorithm; the Salamin–Brent algorithm, which was invented in 1976, has also been used
Approximations_of_pi
Canadian mathematician (1953–2020)
the paper which presented the Bailey–Borwein–Plouffe algorithm (discovered by Simon Plouffe) for computing π. Borwein was born into a Jewish family. He became
Peter_Borwein
Algorithm for computing the value of a transcendental number
A spigot algorithm is an algorithm for computing the value of a transcendental number (such as π or e) that generates the digits of the number sequentially
Spigot_algorithm
Algorithm in computational number theory
Lenstra–Lenstra–Lovász (LLL) lattice basis reduction algorithm is a polynomial time lattice reduction algorithm invented by Arjen Lenstra, Hendrik Lenstra and
Lenstra–Lenstra–Lovász lattice basis reduction algorithm
Lenstra–Lenstra–Lovász_lattice_basis_reduction_algorithm
Number, approximately 3.14
Brent–Salamin algorithm doubles the number of digits in each iteration. In 1984, brothers John and Peter Borwein produced an iterative algorithm that quadruples
Pi
Optimization algorithm
unconstrained mathematical optimization. It is a first-order iterative algorithm for minimizing a differentiable multivariate function. The idea is to
Gradient_descent
Sum of the reciprocal of the Mersenne numbers
calculated efficiently. The Erdős–Borwein constant comes up in the average case analysis of the heapsort algorithm, where it controls the constant factor
Erdős–Borwein_constant
Algorithms for calculating square roots
Square root algorithms compute the non-negative square root S {\displaystyle {\sqrt {S}}} of a positive real number S {\displaystyle S} . Since all square
Square_root_algorithms
Mathematical optimization method
of Any EM Algorithm. Scandinavian Journal of Statistics, 35(2), 335-353. Y. H. Dai, M. Al-Baali, and X. Yang, “A positive Barzilai-Borwein-like stepsize
Barzilai–Borwein_method
Algorithmic runtime requirements for common math procedures
ISBN 978-3-642-14518-6. S2CID 7632655. Borwein, P. (1985). "On the complexity of calculating factorials". Journal of Algorithms. 6 (3): 376–380. doi:10.1016/0196-6774(85)90006-9
Computational complexity of mathematical operations
Computational_complexity_of_mathematical_operations
Mathematical formula
Rather than having to compute all of the digits twice by two separate algorithms to ensure that a computation is correct, the final digits of a very long
Bellard's_formula
quartically to 1/π, and other algorithms Chudnovsky algorithm — fast algorithm that calculates a hypergeometric series Bailey–Borwein–Plouffe formula — can be
List of numerical analysis topics
List_of_numerical_analysis_topics
Series related to Ramanujan's pi formulas
which is a consequence of Stirling's approximation. Chudnovsky algorithm Borwein's algorithm Chan, Heng Huat; Chan, Song Heng; Liu, Zhiguo (2004). "Domb's
Ramanujan–Sato_series
Algorithmic technique
techniques such as Toom–Cook multiplication and the Schönhage–Strassen algorithm must be used; with ordinary O(n2) multiplication, binary splitting may
Binary_splitting
1016/0041-5553(67)90113-9. Bauschke, H.H.; Borwein, J.M. (1993). "On the convergence of von Neumann's alternating projection algorithm for two sets". Set-Valued Analysis
Projections_onto_convex_sets
Approximations of π Arithmetic–geometric mean Bailey–Borwein–Plouffe formula Basel problem Borwein's algorithm Buffon's needle Cadaeic Cadenza Chronology of
List_of_topics_related_to_π
Function in analytic number theory
Tidsskrift. B, 71–73. http://www.jstor.org/stable/24529536 Borwein, Peter (2000). "An efficient algorithm for the Riemann zeta function". In Théra, Michel A.
Dirichlet_eta_function
Uses of the constant
(Archimedes' algorithm, see also harmonic mean and geometric mean) For more iterative algorithms, see the Gauss–Legendre algorithm and Borwein's algorithm. ( 2
List_of_formulae_involving_π
Italian mathematician and historian of science (1872–1953)
efficiency of these formulas is significantly worse than of the modern Borwein's algorithm – they converge by only about half a decimal point with each iteration
Giovanni Vacca (mathematician)
Giovanni_Vacca_(mathematician)
American mathematician (born 1948)
digits of pi beginning at an arbitrary position, by means of a simple algorithm. Subsequently, Bailey and Richard Crandall showed that the existence of
David H. Bailey (mathematician)
David_H._Bailey_(mathematician)
Canadian mathematician
1956) is a Canadian mathematician who discovered the Bailey–Borwein–Plouffe formula (BBP algorithm) which permits the computation of the nth binary digit of
Simon_Plouffe
Conjecture on zeros of the zeta function
neighbors (PDF) This unpublished book describes the implementation of the algorithm and discusses the results in detail. Odlyzko, A. M. (1998), The 1021st
Riemann_hypothesis
Approach to mathematics using computation
rediscovered by Enrico Au-Yeung, a student of Jonathan Borwein using computer search and PSLQ algorithm in 1993: ∑ k = 1 ∞ H k 2 k 2 = ∑ k = 1 ∞ 1 k 2 ( 1
Experimental_mathematics
Number divisible only by 1 and itself
tests whether n is a multiple of any integer between 2 and n. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small
Prime_number
Function representing the number of primes less than or equal to a given number
1014 ≤ x ≤ 1.6×1018 Staple, Douglas (19 August 2015). The combinatorial algorithm for computing π(x) (Thesis). Dalhousie University. Retrieved 2015-09-01
Prime-counting_function
Numerical utility
(Burnaby, Canada). A user will input a number and the Calculator will use an algorithm to search for and calculate closed-form expressions or suitable functions
Inverse_Symbolic_Calculator
Subfield of mathematical optimization
Dimitri P. (2015). Convex Optimization Algorithms. Belmont, MA.: Athena Scientific. ISBN 978-1-886529-28-1. Borwein, Jonathan; Lewis, Adrian (2000). Convex
Convex_optimization
Iterative optimization algorithm
The Bregman method is an iterative algorithm to solve certain convex optimization problems involving regularization. The original version is due to Lev
Bregman_method
Card trick and probabilistic concept
theory Geometric distribution Overlapping instructions Pollard's kangaroo algorithm Random walk Self-synchronizing code According to Diaconis & Graham (2012)
Kruskal_count
Mathematical function of two positive real arguments
sequence of geometric means. The arithmetic–geometric mean is used in fast algorithms for exponential, trigonometric functions, and other special functions
Arithmetic–geometric_mean
Testing a predictive model on historical data
quantitative finance, backtesting is an important step before deploying algorithmic strategies in live markets. In economics and finance, backtesting seeks
Backtesting
Canadian computer scientist (born 1980)
Percival began working on a more efficient delta compression algorithm. This new algorithm, called bsdiff, became the new focus of his doctoral research
Colin_Percival
Product of numbers from 1 to n
1145/251634.251638. S2CID 17347501. Borwein, Peter B. (1985). "On the complexity of calculating factorials". Journal of Algorithms. 6 (3): 376–380. doi:10
Factorial
Signed odd unit fractions sum to π/4
technique that can be applied to the Leibniz series. In 1992, Jonathan Borwein and Mark Limber used the first thousand Euler numbers to calculate π to
Leibniz_formula_for_π
Numbers obtained by adding the two previous ones
Fibonacci Quarterly. Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure
Fibonacci_sequence
Fast summation method in mathematics
two n {\displaystyle n} -digit integers. The algorithms based on the method FEE include the algorithms for fast calculation of any elementary transcendental
FEE_method
Logarithm to the base of the mathematical constant e
Development Center. p. 2-5. Goodrich, Michael T.; Tamassia, Roberto (2002). Algorithm Design: Foundations, Analysis, and Internet Examples. John Wiley & Sons
Natural_logarithm
Base-16 numeric representation
implementation of the above algorithm for converting any number to a hex in String representation. Its purpose is to illustrate the above algorithm. To work with data
Hexadecimal
Sum of a number's digits
checking calculations. Digit sums are also a common ingredient in checksum algorithms to check the arithmetic operations of early computers. Earlier, in an
Digit_sum
Existence of a line through two points
(not all on one line) has at least a linear number of ordinary lines. An algorithm can find an ordinary line in a set of n {\displaystyle n} points in time
Sylvester–Gallai_theorem
Inverse of the gamma function
on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC). International Symposium on Symbolic and Numeric Algorithms for Scientific Computing
Inverse_gamma_function
Extension of the factorial function
Vol.46, No. 2, pp. 246–247 (1991). E.A. Karatsuba "Fast Algorithms and the FEE Method". Borwein, J. M.; Zucker, I. J. (1992). "Fast evaluation of the gamma
Gamma_function
History of pi Approximations of pi David H. Bailey; Jonathan M. Borwein; Peter B. Borwein; Simon Plouffe (1997). "The quest for pi" (PDF). Mathematical
Chronology of computation of pi
Chronology_of_computation_of_pi
Subdivision of the plane by lines
number of triangular cells in a Euclidean arrangement, respectively. Algorithms in computational geometry are known for constructing the features of an
Arrangement_of_lines
Georgia Institute of Technology Jesus 2013 United States Founder of the Algorithmic Justice League Julian Gewirtz Harvard College St Edmund 2013 United States
List_of_Rhodes_Scholars
develops Kuṭṭaka, an algorithm very similar to the Extended Euclidean algorithm. 499: Aryabhata describes a numerical algorithm for finding cube roots
Timeline of scientific discoveries
Timeline_of_scientific_discoveries
algebra. Some of the most important methods and algorithms of the 20th century are: the simplex algorithm, the fast Fourier transform, error-correcting
History_of_mathematics
24–26. Berggren, Borwein & Borwein (2004), 26. Berggren, Borwein & Borwein (2004), 20. Gupta (1975), B45–B48 Berggren, Borwein, & Borwein (2004), 24. Sivin
List_of_Chinese_discoveries
Analytic function in mathematics
{O}}\left(k^{-3/4+\varepsilon }\right)\qquad (\forall \varepsilon >0)} Peter Borwein developed an algorithm that applies Chebyshev polynomials to the Dirichlet eta function
Riemann_zeta_function
common multiple Euclidean algorithm Coprime Euclid's lemma Bézout's identity, Bézout's lemma Extended Euclidean algorithm Table of divisors Prime number
List_of_number_theory_topics
Fourth letter in the Greek alphabet
Emiris, Ioannis Z. (2005). Solving polynomial equations: foundations, algorithms, and applications. Springer. Example 2.5.6, p. 120. ISBN 978-3-540-24326-7
Delta_(letter)
Formula that visually represents itself when graphed
his 2001 SIGGRAPH paper on reliable two-dimensional computer graphing algorithms. The paper examines methods for graphing mathematical expressions involving
Tupper's self-referential formula
Tupper's_self-referential_formula
French computer programmer (born 1972)
using a large language model. He updated it in March 2024, making the algorithm considerably faster as well as hardware-independent. In April 2024, Bellard
Fabrice_Bellard
Springer. p. 29. ISBN 978-3-540-36363-7. Richard E. Crandall (2012). Unified algorithms for polylogarithm, L-series, and zeta variants (PDF). perfscipress.com
List of mathematical constants
List_of_mathematical_constants
Shor formulates Shor's algorithm, a quantum algorithm for integer factorization. 1995 – Simon Plouffe discovers Bailey–Borwein–Plouffe formula capable
Timeline_of_mathematics
Mathematical functions
}}}}}},\quad a_{n}=n^{2}(4n^{2}-1),\,b_{n}=3(2n-1)^{2}} A fast algorithm, returning approximations to sl x {\displaystyle \operatorname {sl}
Lemniscate_elliptic_functions
Special mathematical function defined as sin(x)/x
integral Whittaker–Shannon interpolation formula – Signal (re-)construction algorithm Winkel tripel projection – Pseudoazimuthal compromise map projection (cartography)
Sinc_function
Mathematical expression
with the Euclidean algorithm, a procedure for finding the greatest common divisor of two natural numbers m and n. That algorithm introduced the idea
Continued_fraction
Difference between logarithm and harmonic series
of Mersenne primes. An estimation of the efficiency of the euclidean algorithm. Sums involving the Möbius and von Mangolt function. Estimate of the divisor
Euler's_constant
memory capacity; better software in terms of increasing sophistication of algorithms; more advanced visualization facilities; the mixing of numerical and symbolic
Future_of_mathematics
Special function defined by an integral
"Numerical Computation of Real or Complex Elliptic Integrals". Numerical Algorithms. 10 (1): 13–26. arXiv:math/9409227. Bibcode:1995NuAlg..10...13C. doi:10
Elliptic_integral
Disproved mathematical conjecture
conjecture false using the Lenstra–Lenstra–Lovász lattice basis reduction algorithm: lim inf m ( n ) < − 1.009 {\displaystyle \liminf m(n)<-1.009} and
Mertens_conjecture
Prime number of the form 2^n – 1
cases for the special number field sieve algorithm, so often the largest number factorized with this algorithm has been a Mersenne number. As of June 2019[update]
Mersenne_prime
Difficulties arising when analyzing data with many aspects ("dimensions")
correlation between specific genetic mutations and creating a classification algorithm such as a decision tree to determine whether an individual has cancer
Curse_of_dimensionality
et algorithmes, Éditions Tec & Doc, Paris, 2008. xxx+711 pp. . ). Shapiro, Jeremy F. (1979). Mathematical programming: Structures and algorithms. New
Duality_gap
Fixed number that has received a name
number to be proven transcendental. In the computer science subfield of algorithmic information theory, Chaitin's constant is the real number representing
Mathematical_constant
Problem of constructing equal-area shapes
"Adam Adamandy Kochański's approximations of π: reconstruction of the algorithm". The Mathematical Intelligencer. 34 (4): 40–45. arXiv:1111.1739. doi:10
Squaring_the_circle
Online database of integer sequences
search function called SuperSeeker which runs a large number of different algorithms to identify sequences related to the input. Neil Sloane started collecting
On-Line Encyclopedia of Integer Sequences
On-Line_Encyclopedia_of_Integer_Sequences
Branch of mathematical statistics
Sturmfels introduced Gröbner bases for constructing Markov chain Monte Carlo algorithms for conditional sampling from discrete exponential families. Pistone and
Algebraic_statistics
128-bit computer number format
between the significand of the high-order and low-order numbers. Certain algorithms that rely on having a fixed number of bits in the significand can fail
Quadruple-precision floating-point format
Quadruple-precision_floating-point_format
commentary on The Nine Chapters on the Mathematical Art, providing a pi algorithm with a value of 3.14159, while Liu Song and Southern Qi–era mathematician
Science_in_the_ancient_world
as their inventor. Although not the first to do so, al-Kashi gave an algorithm for calculating nth roots which is a special case of the methods given
Timeline of numerals and arithmetic
Timeline_of_numerals_and_arithmetic
Mathematical formula involving a given set of operations
A simple example is the equation x5−x−1=0. Galois theory provides an algorithmic method for deciding whether a particular polynomial equation can be solved
Closed-form_expression
Real root of the polynomial x^5+x+a
2 = 0. {\displaystyle x^{5}-10Cx^{3}+45C^{2}x-C^{2}=0.} The iteration algorithm proceeds as follows: Set Z = 1 − 1728 C {\displaystyle Z=1-1728C} Compute
Bring_radical
Point not between two other points
Dictionary of algorithms and data structures. US National institute of standards and technology. Retrieved 2011-03-24. Borowski, Ephraim J.; Borwein, Jonathan
Extreme_point
the 1930s. Bellman–Ford algorithm for computing the shortest-length path, proposed by Alfonso Shimbel, who presented the algorithm in 1954, but named after
List of examples of Stigler's law
List_of_examples_of_Stigler's_law
Number with all digits equally frequent
Since the LZ compression algorithm compresses asymptotically as well as any ILFSC, this means that the LZ compression algorithm can compress any non-normal
Normal_number
Number, approximately 0.916
Broadhurst, for the first formula, and Ramanujan, for the second formula. The algorithms for fast evaluation of the Catalan constant were constructed by E. Karatsuba
Catalan's_constant
Set of matrices
coefficients". Algorithms Seminar: 169. CiteSeerX 10.1.1.47.9327. {{cite journal}}: Cite uses deprecated parameter |citeseerx= (help) Borwein, Peter B.; Jörgenson
Bohemian_matrices
Principle in mathematical optimization
Théorie et algorithmes, Éditions Tec & Doc, Paris, 2008. xxx+711 pp. ). Shapiro, Jeremy F. (1979). Mathematical programming: Structures and algorithms. New
Duality_(optimization)
Infinite product converging to 2/π
formula as marking the beginning of mathematical analysis and Jonathan Borwein calls its appearance "the dawn of modern mathematics". Using his formula
Viète's_formula
Chinese scientist and statesman (78–139)
Berggren, Borwein & Borwein 2004, p. 27 Arndt & Haenel 2001, p. 177 Wilson 2001, p. 16 Needham 1986, pp. 100–101 Berggren, Borwein & Borwein 2004, pp. 20
Zhang_Heng
Special mathematical function
computations of various mathematical constants by means of the BBP algorithm (Bailey, Borwein & Plouffe 1997). The polylogarithm has two branch points; one
Polylogarithm
Dutch mathematician
purpose is to promote research and exchange of ideas in the field of algorithmic and computational number theory. The first Beeger Lecture was delivered
N._G._W._H._Beeger
Measure of polynomial height
Donald E. (1997). "4.6.2 Factorization of Polynomials". Seminumerical Algorithms. The Art of Computer Programming. Vol. 2 (3rd ed.). Addison-Wesley. pp
Mahler_measure
(1921–2002) - Contributed a number of theories, such as Papoulis–Gerchberg algorithm, A eloquent proof, among others. Themistocles M. Rassias (born 1951) -
List_of_Greek_mathematicians
Ancient mathematical text
11 (2): 112–124. David H. Bailey, Jonathan Borwein (2011). "A Quartically Convergent Square Root Algorithm: An Exercise in Forensic Paleo-Mathematics"
Bakhshali_manuscript
Ratio of the perimeter of Bernoulli's lemniscate to its diameter
257–290. Cox 1984, p. 277. "A113847 - Oeis". Cremona, J. E. (1997). Algorithms for Modular Elliptic Curves (2nd ed.). Cambridge University Press. ISBN 0521598206
Lemniscate_constant
Unsolved problem in number theory
integers as the sum of k {\displaystyle k} terms". Random Structures Algorithms. 1 (3): 245–261. doi:10.1002/rsa.3240010302. Vu, Van (2000). "On a refinement
Erdős–Turán conjecture on additive bases
Erdős–Turán_conjecture_on_additive_bases
number, a representation as a sum of unit fractions found by a greedy algorithm. Sylvester's rank inequality rank(A) + rank(B) − n ≤ rank(AB) on the rank
List of things named after James Joseph Sylvester
List_of_things_named_after_James_Joseph_Sylvester
Number, sum of distinct powers of 4
Sequences, OEIS Foundation Arndt, Jörg (2011), Matters Computational: Ideas, Algorithms, Source Code (PDF), Springer, pp. 59, 750. Golomb, Solomon W. (1966),
Moser–de_Bruijn_sequence
On when a space equals the closed convex hull of its extreme points
Dictionary of algorithms and data structures. US National institute of standards and technology. Retrieved 2011-03-24. Borowski, Ephraim J.; Borwein, Jonathan
Krein–Milman_theorem
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BORWEINS ALGORITHM
BORWEINS ALGORITHM
BORWEINS ALGORITHM
BORWEINS ALGORITHM
BORWEINS ALGORITHM
BORWEINS ALGORITHM
BORWEINS ALGORITHM
BORWEINS ALGORITHM
BORWEINS ALGORITHM
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