On the Educational Legacies of Jonathan M. Borwein

Borwein, Naomi Simone; Osborn, Judy-anne

doi:10.1007/978-3-030-36568-4_7

Cited by 7 publications

(5 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…It is useful to define auxiliary variables c n+1 := a n − a n+1 = (a n − b n )/2. Using the quotient for a n and the addition formula (11), we see that…”

Section: Preliminaries: Means Elliptic Integrals and Theta Functionsmentioning

confidence: 99%

“…When using (40), we may need to apply a rotation to q, say by a multiple of π/3, in order to ensure that the starting values (θ 2 2 (q 4 ), θ 2 3 (q 4 )) for the AGM lie inH . 11 For z ∈ C\{0}, log(z) = log(|z|) + i arg(z), provided we use the principal values of the logarithms. Thus, if x ∈ R, we can use the complex AGM to compute arctan(x) = ℑ(log(1 + ix)).…”

Section: Fast Algorithms For the Elementary Functionsmentioning

confidence: 99%

“…Jonathan Borwein was fascinated by the constant π, and gave many stimulating talks on this topic. The slides for most of these talks may be found on the memorial website [11]. In my talk [22] at the Jonathan Borwein Commemorative Conference I discussed the reasons for this fascination.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Borwein Brothers, Pi and the AGM

Brent

2020

Springer Proceedings in Mathematics &Amp; Statistics

View full text Add to dashboard Cite

We consider some of Jonathan and Peter Borweins' contributions to the high-precision computation of π 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 π, and more generally the elementary functions. These algorithms run in "almost linear" time O(M(n) log n), where M(n) is the time for n-bit multiplication. We outline some of the results and algorithms given in Pi and the AGM, and present some related (but new) results. In particular, we improve the published error bounds for some quadratically and quartically convergent algorithms for π, such as the Gauss-Legendre algorithm. We show that an iteration of the Borwein-Borwein quartic algorithm for π is equivalent to two iterations of the Gauss-Legendre quadratic algorithm for π, in the sense that they produce exactly the same sequence of approximations to π if performed using exact arithmetic.Keywords: arithmetic-geometric mean, Borwein-Borwein algorithm, Borwein-Borwein quartic algorithm, Brent-Salamin algorithm, Chudnovsky algorithm, computation of π, computational complexity, elliptic integrals, equivalence of algorithms for π, evaluation of elementary functions, Gauss-Legendre algorithm, linear convergence, quadratic convergence, quartic convergence, Ramanujan-Sato algorithms, Sasaki-Kanada algorithm, theta functions.

show abstract

“…It is useful to define auxiliary variables c n+1 := a n − a n+1 = (a n − b n )/2. Using the quotient for a n and the addition formula (11), we see that…”

Section: Preliminaries: Means Elliptic Integrals and Theta Functionsmentioning

confidence: 99%

Section: Fast Algorithms For the Elementary Functionsmentioning

confidence: 99%

See 1 more Smart Citation

The Borwein Brothers, Pi and the AGM

Brent

2020

Springer Proceedings in Mathematics &Amp; Statistics

View full text Add to dashboard Cite

show abstract

“…Consequently, the whole proof effort boils down to proving that constraint ( 14 ) constitutes a convex set which is discussed as follows. There are theorems on convexity that are used in our proof [ 48 – 51 ]:…”

Section: Mathematical Formulationsmentioning

confidence: 99%

An outer approximation method for the road network design problem

Bagloee

Sarvi

2018

PLoS ONE

View full text Add to dashboard Cite

Best investment in the road infrastructure or the network design is perceived as a fundamental and benchmark problem in transportation. Given a set of candidate road projects with associated costs, finding the best subset with respect to a limited budget is known as a bilevel Discrete Network Design Problem (DNDP) of NP-hard computationally complexity. We engage with the complexity with a hybrid exact-heuristic methodology based on a two-stage relaxation as follows: (i) the bilevel feature is relaxed to a single-level problem by taking the network performance function of the upper level into the user equilibrium traffic assignment problem (UE-TAP) in the lower level as a constraint. It results in a mixed-integer nonlinear programming (MINLP) problem which is then solved using the Outer Approximation (OA) algorithm (ii) we further relax the multi-commodity UE-TAP to a single-commodity MILP problem, that is, the multiple OD pairs are aggregated to a single OD pair. This methodology has two main advantages: (i) the method is proven to be highly efficient to solve the DNDP for a large-sized network of Winnipeg, Canada. The results suggest that within a limited number of iterations (as termination criterion), global optimum solutions are quickly reached in most of the cases; otherwise, good solutions (close to global optimum solutions) are found in early iterations. Comparative analysis of the networks of Gao and Sioux-Falls shows that for such a non-exact method the global optimum solutions are found in fewer iterations than those found in some analytically exact algorithms in the literature. (ii) Integration of the objective function among the constraints provides a commensurate capability to tackle the multi-objective (or multi-criteria) DNDP as well.

show abstract

“…Um resultado que associa as condições impostas as aplições ponto-conjunto em todo o texto será anunciado abaixo, muito mais sobre essas equivalências podem ser encontradas em [35] (ii) e pΦppq X B ρ rxs, Φpqqq ď λ}p ´q} para todo p, q P B ρ rxs.…”

Section: Resultados Técnicosunclassified

Uma abordagem quasi-Newton às equações generalizadas com restrições

Marques Carvalho

View full text Add to dashboard Cite

Equações Generalizadas vêm sendo estudadas por muitos matemáticos há várias décadas, mas o estudo do problema com restrições é bem recente. Nesta Tese, faremos uma abordagem Quasi-Newton às Equações Generalizadas com Restrições. Essa abordagem foi feita de duas formas diferentes, primeiro o Método Quasi-Newton com Projeção Inexata, em seguida o Método Quasi-Newton Inexato com Projeção Inexata. De início, provamos os teoremas exigindo apenas uma condição para a atualização B k e depois, para ambos os casos, enunciamos um novo resultado usando a atualização de Broyden. Alguns testes numéricos foram feitos.

show abstract

On the Educational Legacies of Jonathan M. Borwein

Cited by 7 publications

References 25 publications

The Borwein Brothers, Pi and the AGM

The Borwein Brothers, Pi and the AGM

An outer approximation method for the road network design problem

Uma abordagem quasi-Newton às equações generalizadas com restrições

Contact Info

Product

Resources

About