اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدThe Borwein brothers, Pi and the AGM
46
0
0.0
(
0
)
تأليف
Richard P. Brent
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We consider some of Jonathan and Peter Borweins contributions to the high-precision computation of $pi$ 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 $pi$, 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 $pi$, such as the Gauss-Legendre algorithm. We show that an iteration of the Borwein-Borwein quartic algorithm for $pi$ is equivalent to two iterations of the Gauss-Legendre quadratic algorithm for $pi$, in the sense that they produce exactly the same sequence of approximations to $pi$ if performed using exact arithmetic.
قيم البحث
اقرأ أيضاً
In this paper, we study properties of the coefficients appearing in the $q$-series expansion of $prod_{nge 1}[(1-q^n)/(1-q^{pn})]^delta$, the infinite Borwein product for an arbitrary prime $p$, raised to an arbitrary positive real power $delta$. We
use the Hardy--Ramanujan--Rademacher circle method to give an asymptotic formula for the coefficients. For $p=3$ we give an estimate of their growth which enables us to partially confirm an earlier conjecture of the first author concerning an observed sign pattern of the coefficients when the exponent $delta$ is within a specified range of positive real numbers. We further establish some vanishing and divisibility properties of the coefficients of the cube of the infinite Borwein product. We conclude with an Appendix presenting several new conjectures on precise sign patterns of infinite products raised to a real power which are similar to the conjecture we made in the $p=3$ case.
We denote by $c_t^{(m)}(n)$ the coefficient of $q^n$ in the series expansion of $(q;q)_infty^m(q^t;q^t)_infty^{-m}$, which is the $m$-th power of the infinite Borwein product. Let $t$ and $m$ be positive integers with $m(t-1)leq 24$. We provide asymp
totic formula for $c_t^{(m)}(n)$, and give characterizations of $n$ for which $c_t^{(m)}(n)$ is positive, negative or zero. We show that $c_t^{(m)}(n)$ is ultimately periodic in sign and conjecture that this is still true for other positive integer values of $t$ and $m$. Furthermore, we confirm this conjecture in the cases $(t,m)=(2,m),(p,1),(p,3)$ for arbitrary positive integer $m$ and prime $p$.
In this short note we prove two elegant generalized continued fraction formulae $$e= 2+cfrac{1}{1+cfrac{1}{2+cfrac{2}{3+cfrac{3}{4+ddots}}}}$$ and $$e= 3+cfrac{-1}{4+cfrac{-2}{5+cfrac{-3}{6+cfrac{-4}{7+ddots}}}}$$ using elementary methods. The first
formula is well-known, and the second one is newly-discovered in arXiv:1907.00205 [cs.LG]. We then explore the possibility of automatic verification of such formulae using computer algebra systems (CASs).
We describe various errors in the mathematical literature, and consider how some of them might have been avoided, or at least detected at an earlier stage, using tools such as Maple or Sage. Our examples are drawn from three broad categories of error
s. First, we consider some significant errors made by highly-regarded mathematicians. In some cases these errors were not detected until many years after their publication. Second, we consider in some detail an error that was recently detected by the author. This error in a refereed journal led to further errors by at least one author who relied on the (incorrect) result. Finally, we mention some instructive errors that have been detected in the authors own published papers.
I argue that European schools of thought on memory and memorization were critical in enabling the growth of the scientific method. After giving a historical overview of the development of the memory arts from ancient Greece through 17th century Europ
e, I describe how the Baconian viewpoint on the scientific method was fundamentally part of a culture and a broader dialogue that conceived of memorization as a foundational methodology for structuring knowledge and for developing symbolic means for representing scientific concepts. The principal figures of this intense and rapidly evolving intellectual milieu included some of the leading thinkers traditionally associated with the scientific revolution; among others, Francis Bacon, Renes Descartes, and Gottfried Leibniz. I close by examining the acceleration of mathematical thought in light of the art of memory and its role in 17th century philosophy, and in particular, Leibniz project to develop a universal calculus.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
