402 research outputs found
The computation of previously inaccessible digits of π<sup>2</sup> and Catalan's constant
We recently concluded a very large mathematical calculation, uncovering objects that until recently were widely considered to be forever inaccessible to computation. Our computations stem from the “BBP” formula for π, which was discovered in 1997 using a computer program implementing the “PSLQ” integer relation algorithm. This formula has the remarkable property that it permits one to directly calculate binary digits of π, beginning at an arbitrary position d, without needing to calculate any of the first d - 1 digits. Since 1997 numerous other BBP-type formulas have been discovered for various mathematical constants, including formulas for π² (both in binary and ternary bases) and for Catalan’s constant. In this article we describe the computation of base-64 digits of π², base-729 digits of π², and base-4096 digits of Catalan’s constant, in each case beginning at the ten trillionth place, computations that involved a total of approximately 1:549 x 1019 floating-point operations. We also discuss connections between BBP-type formulas and the age-old unsolved questions of whether and why constants such as π; π²; log 2, and Catalan’s constant have “random” digits
The Borwein brothers, Pi and the AGM
We consider some of Jonathan and Peter Borweins' contributions to the
high-precision computation of 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 -bit computation of , and more
generally the elementary functions. These algorithms run in almost linear time
, where is the time for -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 ,
such as the Gauss-Legendre algorithm. We show that an iteration of the
Borwein-Borwein quartic algorithm for is equivalent to two iterations of
the Gauss-Legendre quadratic algorithm for , in the sense that they
produce exactly the same sequence of approximations to if performed using
exact arithmetic.Comment: 24 pages, 6 tables. Changed style file and reformatted algorithms in
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Variational Analysis Down Under Open Problem Session
© 2018, Springer Science+Business Media, LLC, part of Springer Nature. We state the problems discussed in the open problem session at Variational Analysis Down Under conference held in honour of Prof. Asen Dontchev on 19–21 February 2018 at Federation University Australia
MM Algorithms for Geometric and Signomial Programming
This paper derives new algorithms for signomial programming, a generalization
of geometric programming. The algorithms are based on a generic principle for
optimization called the MM algorithm. In this setting, one can apply the
geometric-arithmetic mean inequality and a supporting hyperplane inequality to
create a surrogate function with parameters separated. Thus, unconstrained
signomial programming reduces to a sequence of one-dimensional minimization
problems. Simple examples demonstrate that the MM algorithm derived can
converge to a boundary point or to one point of a continuum of minimum points.
Conditions under which the minimum point is unique or occurs in the interior of
parameter space are proved for geometric programming. Convergence to an
interior point occurs at a linear rate. Finally, the MM framework easily
accommodates equality and inequality constraints of signomial type. For the
most important special case, constrained quadratic programming, the MM
algorithm involves very simple updates.Comment: 16 pages, 1 figur
Special Values of Generalized Polylogarithms
We study values of generalized polylogarithms at various points and
relationships among them. Polylogarithms of small weight at the points 1/2 and
-1 are completely investigated. We formulate a conjecture about the structure
of the linear space generated by values of generalized polylogarithms.Comment: 32 page
Local maximum points of explicitly quasiconvex functions
This work concerns generalized convex real-valued functions defined on a nonempty convex subset of a real topological linear space. Its aim is twofold: first, to show that any local maximum point of an explicitly quasiconvex function is a global minimum point whenever it belongs to the intrinsic core of the function’s domain and second, to characterize strictly convex normed spaces by applying this property for a particular class of convex functions
Continuity for s-convex fuzzy processes
In a previous paper we introduced the concept of s-convex fuzzy mapping and
established some properties. In this work we study the continuity for s-convex
fuzzy processes
Multivariate truncated moments problems and maximum entropy
We characterize the existence of the Lebesgue integrable solutions of the
truncated problem of moments in several variables on unbounded supports by the
existence of some maximum entropy -- type representing densities and discuss a
few topics on their approximation in a particular case, of two variables and
4th order moments.Comment: Revised version, to appear in Analysis and Mathematical Physic
Iteratively regularized Newton-type methods for general data misfit functionals and applications to Poisson data
We study Newton type methods for inverse problems described by nonlinear
operator equations in Banach spaces where the Newton equations
are regularized variationally using a general
data misfit functional and a convex regularization term. This generalizes the
well-known iteratively regularized Gauss-Newton method (IRGNM). We prove
convergence and convergence rates as the noise level tends to 0 both for an a
priori stopping rule and for a Lepski{\u\i}-type a posteriori stopping rule.
Our analysis includes previous order optimal convergence rate results for the
IRGNM as special cases. The main focus of this paper is on inverse problems
with Poisson data where the natural data misfit functional is given by the
Kullback-Leibler divergence. Two examples of such problems are discussed in
detail: an inverse obstacle scattering problem with amplitude data of the
far-field pattern and a phase retrieval problem. The performence of the
proposed method for these problems is illustrated in numerical examples
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