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Charles George Broyden
Charles George Broyden (3 February 1933 – 20 May 2011) was a mathematician who specialized in optimization problems and numerical linear algebra. While a physicist working at English Electric Company from 1961–1965, he adapted the Davidon–Fletcher–Powell formula to solving some nonlinear systems of equations that he was working with, leading to his widely cited 1965 paper, "A class of methods for solving nonlinear simultaneous equations". He was a lecturer at UCW Aberystwyth from 1965–1967. He later became a senior lecturer at University of Essex from 1967–1970, where he independently discovered the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method. The BFGS method has then become a key technique in solving nonlinear optimization problems. Moreover, he was among those who derived the symmetric rank-one updating formula, and his name was also attributed to Broyden's methods and Broyden family of quasi-Newton methods. After leaving the University of Essex, he co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems of sorts arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries. In the more general approach, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. More generally, optimization includes finding "best available" values of some objective function given a def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Linear Algebra
Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to questions in continuous mathematics. It is a subfield of numerical analysis, and a type of linear algebra. Computers use floating-point arithmetic and cannot exactly represent irrational data, so when a computer algorithm is applied to a matrix of data, it can sometimes increase the difference between a number stored in the computer and the true number that it is an approximation of. Numerical linear algebra uses properties of vectors and matrices to develop computer algorithms that minimize the error introduced by the computer, and is also concerned with ensuring that the algorithm is as efficient as possible. Numerical linear algebra aims to solve problems of continuous mathematics using finite precision computers, so its applications to the natural and social sciences ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
English Electric Company
N.º UIC: 9094 110 1449-3 (Takargo Rail) The English Electric Company Limited (EE) was a British industrial manufacturer formed after the Armistice of 11 November 1918, armistice of World War I by amalgamating five businesses which, during the war, had been making munitions, armaments and aeroplanes. It initially specialised in industrial electric motors and transformers, railway locomotives and traction equipment, diesel motors and steam turbines. Its activities were later expanded to include consumer electronics, nuclear reactors, guided missiles, military aircraft and mainframe computers. Two English Electric aircraft designs became landmarks in British aeronautical engineering; the Canberra and the Lightning. In 1960, English Electric Aircraft (40%) merged with Vickers (40%) and Bristol (20%) to form British Aircraft Corporation. In 1968 English Electric's operations were merged with GEC's, the combined business employing more than 250,000 people. Foundation Aiming t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Davidon–Fletcher–Powell Formula
The Davidon–Fletcher–Powell formula (or DFP; named after William C. Davidon, Roger Fletcher, and Michael J. D. Powell) finds the solution to the secant equation that is closest to the current estimate and satisfies the curvature condition. It was the first quasi-Newton method to generalize the secant method to a multidimensional problem. This update maintains the symmetry and positive definiteness of the Hessian matrix. Given a function f(x), its gradient (\nabla f), and positive-definite Hessian matrix B, the Taylor series is :f(x_k+s_k) = f(x_k) + \nabla f(x_k)^T s_k + \frac s^T_k s_k + \dots, and the Taylor series of the gradient itself (secant equation) :\nabla f(x_k+s_k) = \nabla f(x_k) + B s_k + \dots is used to update B. The DFP formula finds a solution that is symmetric, positive-definite and closest to the current approximate value of B_k: :B_= (I - \gamma_k y_k s_k^T) B_k (I - \gamma_k s_k y_k^T) + \gamma_k y_k y_k^T, where :y_k = \nabla f(x_k+s_k) - \na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aberystwyth University
Aberystwyth University ( cy, Prifysgol Aberystwyth) is a Public university, public Research university, research university in Aberystwyth, Wales. Aberystwyth was a founding member institution of the former federal University of Wales. The university has over 8,000 students studying across three academic faculties and 17 departments. Founded in 1872 as University College Wales, Aberystwyth, it became a founder member of the University of Wales in 1894, and changed its name to the ''University College of Wales, Aberystwyth''. In the mid-1990s, the university again changed its name to become the ''University of Wales, Aberystwyth''. On 1 September 2007, the University of Wales ceased to be a federal university and Aberystwyth University became independent again. In 2019, it became the first university to be named "University of the year for teaching quality" by ''The Times/Sunday Times Good University Guide'' for two consecutive years. It is the first university in the world to be aw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Essex
The University of Essex is a public research university in Essex, England. Established by royal charter in 1965, Essex is one of the original plate glass universities. Essex's shield consists of the ancient arms attributed to the Kingdom of Essex, and the motto, "Thought the harder, heart the keener", is adapted from the Anglo-Saxon poem '' The Battle of Maldon''. The university comprises three campuses with its primary campus located within Wivenhoe Park and campuses in Southend-on-Sea and in Loughton. Essex is rated Gold for Teaching Excellence by the TEF since 2017, named University of the Year at the Times Higher Education Awards in 2018, and is ranked an internationally excellent research-intensive university by the REF. Essex's Department of Government received Regius Professorship conferred by Her Majesty, The Queen in 2013 and the university was awarded the Queen's Anniversary Prize on two occasions for advancing Human Rights in 2009 and Social and Economic Researc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Bologna
The University of Bologna ( it, Alma Mater Studiorum – Università di Bologna, UNIBO) is a public research university in Bologna, Italy. Founded in 1088 by an organised guild of students (''studiorum''), it is the oldest university in continuous operation in the world, and the first degree-awarding institution of higher learning. At its foundation, the word ''universitas'' was first coined.Hunt Janin: "The university in medieval life, 1179–1499", McFarland, 2008, , p. 55f.de Ridder-Symoens, Hilde''A History of the University in Europe: Volume 1, Universities in the Middle Ages'' Cambridge University Press, 1992, , pp. 47–55 With over 90,000 students, it is the second largest university in Italy after La Sapienza in Rome. It was the first place of study to use the term ''universitas'' for the corporations of students and masters, which came to define the institution (especially its law school) located in Bologna. The university's emblem carries the motto, ''Alma Mater Studio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Broyden's Method
In numerical analysis, Broyden's method is a quasi-Newton method for finding roots in variables. It was originally described by C. G. Broyden in 1965. Newton's method for solving uses the Jacobian matrix, , at every iteration. However, computing this Jacobian is a difficult and expensive operation. The idea behind Broyden's method is to compute the whole Jacobian only at the first iteration and to do rank-one updates at other iterations. In 1979 Gay proved that when Broyden's method is applied to a linear system of size , it terminates in steps, although like all quasi-Newton methods, it may not converge for nonlinear systems. Description of the method Solving single-variable equation In the secant method, we replace the first derivative at with the finite-difference approximation: :f'(x_n) \simeq \frac, and proceed similar to Newton's method: :x_ = x_n - \frac where is the iteration index. Solving a system of nonlinear equations Consider a system of nonlin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ABS Methods
ABS methods, where the acronym contains the initials of Jozsef Abaffy, Charles G. Broyden and Emilio Spedicato, have been developed since 1981 to generate a large class of algorithms for the following applications: * solution of general linear algebraic systems, determined or underdetermined, * full or deficient rank; * solution of linear Diophantine systems, i.e. equation systems where the coefficient matrix and the right hand side are integer valued and an integer solution is sought; this is a special but important case of Hilbert's tenth problem, the only one in practice soluble; * solution of nonlinear algebraic equations; * solution of continuous unconstrained or constrained optimization. At the beginning of 2007 ABS literature consisted of over 400 papers and reports and two monographs, one due to Abaffy and Spedicato and published in 1989, one due to Xia and Zhang and published, in Chinese, in 1998. Moreover three conferences had been organized in China. Research on AB ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
21st-century British Mathematicians
The 1st century was the century spanning AD 1 (Roman numerals, I) through AD 100 (Roman numerals, C) according to the Julian calendar. It is often written as the or to distinguish it from the 1st century BC (or BCE) which preceded it. The 1st century is considered part of the Classical era, epoch, or History by period, historical period. The 1st century also saw the Christianity in the 1st century, appearance of Christianity. During this period, Europe, North Africa and the Near East fell under increasing domination by the Roman Empire, which continued expanding, most notably conquering Britain under the emperor Claudius (AD 43). The reforms introduced by Augustus during his long reign stabilized the empire after the turmoil of the previous century's civil wars. Later in the century the Julio-Claudian dynasty, which had been founded by Augustus, came to an end with the suicide of Nero in AD 68. There followed the famous Year of Four Emperors, a brief period of civil war and inst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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