Approximations and complex multiplication according to Ramanujan

Chudnovsky, D. V.; Chudnovsky, G. V.

doi:10.1007/978-1-4757-2736-4_63

Cited by 64 publications

(79 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The first half of this paper was well studied by Watson and others in the 1920s and 1930s, while the second half, which presents marvelous series for pi, was decoded and applied only more than 50 years later. (See [61], [62], [63].) Other highlights include: Watson's engaging and readable account ofthe early development of elliptic functions, [30]; several very influential papers by Kurt Mahler; Fields Medalist Alan Baker's 1964 paper on "algebraic independence of logarithms," [40]; and two papers on the irrationality of ~(3) ( [48], [49]) which was established only in 1976. The computational selections include a report on the early computer calculation of pi-to 2037 places on ENIAC in 1949 by Reitwiesner, Metropolis and Von Neumann [34] and the 1961 computation of pi to 100,000 places by Shanks and Wrench [38], both by arctangent methods.…”

Section: ])mentioning

confidence: 99%

Pi: A Source Book

Berggren¹,

Borwein²,

Borwein³

2004

View full text Add to dashboard Cite

Section: ])mentioning

confidence: 99%

Pi: A Source Book

Berggren¹,

Borwein²,

Borwein³

2004

View full text Add to dashboard Cite

“…The method of binary splitting is an algorithm evaluating D-finite series at complex points with rational coordinates with a bit-complexity that is optimal (up to logarithmic factors) with respect to the required precision. This algorithm is extended in [4,18] to a low-cost evaluation algorithm (•) with arbitrary precision (hence, with guaranteed results) for D-finite functions at any point of their Riemann surface. From there, it is possible to produce graphics and tables of numerical values.…”

Section: Survey Of Algorithmsmentioning

confidence: 99%

Esf

Meunier

Salvy

2003

Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation

View full text Add to dashboard Cite

We present our on-going work on the automatic generation of an encyclopedia of special functions on the web, called The Encyclopedia of Special Functions (ESF) 1 . All mathematical formulae in the ESF are computed, typeset and displayed without any human intervention. This is achieved by exploiting a collection of computer algebra algorithms in a systematic way, on top of a specially designed data structure for a class of special functions.

show abstract

“…This includes multiplication, but also evaluation at rational points by binary splitting [4]. A typical application is the numerical evaluation of π in computer algebra systems; we give another one in these proceedings [3].…”

mentioning

confidence: 99%

D-finiteness

Salvy

2005

Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation

View full text Add to dashboard Cite

Differentially finite series are solutions of linear differential equations with polynomial coefficients. P-recursive sequences are solutions of linear recurrences with polynomial coefficients. Corresponding notions are obtained by replacing classical differentiation or difference operators by their qanalogues. All these objects share numerous properties that are described in the framework of "D-finiteness". Our aim in this area is to enable computer algebra systems to deal in an algorithmic way with a large number of special functions and sequences. Indeed, it can be estimated that approximately 60% of the functions described in Abramowitz & Stegun's handbook [1] fall into this category, as well as 25% of the sequences in Sloane's encyclopedia [20,21].In a way, D-finite sequences or series are non-commutative analogues of algebraic numbers: the role of the minimal polynomial is played by a linear operator. Ore [14] described a non-commutative version of Euclidean division and extended Euclid algorithm for these linear operators (known as Ore polynomials). In the same way as in the commutative case, these algorithms make several closure properties effective (see [22]). It follows that identities between these functions or sequences can be proved or computed automatically. Part of the success of the gfun package [17] comes from an implementation of these operations. Another part comes from the possibility of discovering such identities empirically, with Padé-Hermite approximants on power series [2] taking the place of the LLL algorithm on floating-point numbers.The discovery that a series is D-finite is also important from the complexity point of view: several operations can be performed on D-finite series at a lower cost than on arbitrary power series. This includes multiplication, but also evaluation at rational points by binary splitting [4]. A typical application is the numerical evaluation of π in computer algebra systems; we give another one in these proceedings [3].Also, the local behaviour of solutions of linear differential equations in the neighbourhood of their singularities is well understood [9] and implementations of algorithms comput-Copyright is held by author/owner ISSAC'05, July 24-27, 2005, Beijing, China. ACM 1-59593-095-7/05/0007.ing the corresponding expansions are available [24,13]. This gives access to the asymptotics of numerous sequences or to analytic proofs that sequences or functions cannot satisfy such equations [10]. Results of a more algebraic nature are obtained by differential Galois theory [18,19], which naturally shares many subroutines with algorithms for D-finite series.The truly spectacular applications of D-finiteness come from the multivariate case: instead of series or sequences, one works with multivariate series or sequences, or with sequences of series or polynomials,. . . . They obey systems of linear operators that may be of differential, difference, qdifference or mixed types, with the extra constraint that a finite number of initial conditions are sufficient to spec...

show abstract

Approximations and complex multiplication according to Ramanujan

Cited by 64 publications

References 20 publications

Pi: A Source Book

Pi: A Source Book

Esf

D-finiteness

Contact Info

Product

Resources

About