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Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics, numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in medicine and biology.
Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals.
In mathematics, extrapolation is a type of estimation, beyond the original observation range, of the value of a variable on the basis of its relationship with another variable. It is similar to interpolation, which produces estimates between known observations, but extrapolation is subject to greater uncertainty and a higher risk of producing meaningless results. Extrapolation may also mean extension of a method, assuming similar methods will be applicable. Extrapolation may also apply to human experience to project, extend, or expand known experience into an area not known or previously experienced so as to arrive at a knowledge of the unknown. The extrapolation method can be applied in the interior reconstruction problem.
Simple rational approximation (SRA) is a subset of interpolating methods using rational functions. Especially, SRA interpolates a given function with a specific rational function whose poles and zeros are simple, which means that there is no multiplicity in poles and zeros. Sometimes, it only implies simple poles.
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.
In mathematics, series acceleration is one of a collection of sequence transformations for improving the rate of convergence of a series. Techniques for series acceleration are often applied in numerical analysis, where they are used to improve the speed of numerical integration. Series acceleration techniques may also be used, for example, to obtain a variety of identities on special functions. Thus, the Euler transform applied to the hypergeometric series gives some of the classic, well-known hypergeometric series identities.
Philippe G. Ciarlet is a French mathematician, known particularly for his work on mathematical analysis of the finite element method. He has contributed also to elasticity, to the theory of plates and shells and differential geometry.
David M. Young Jr. was an American mathematician and computer scientist who was one of the pioneers in the field of modern numerical analysis/scientific computing.
Richard Steven Varga was an American mathematician who specialized in numerical analysis and linear algebra. He was an Emeritus University Professor of Mathematical Sciences at Kent State University and an adjunct Professor at Case Western Reserve University. Varga was known for his contributions to many areas of mathematics, including matrix analysis, complex analysis, approximation theory, and scientific computation. He was the author of the classic textbook Matrix Iterative Analysis. Varga served as the Editor-in-Chief of the journal Electronic Transactions on Numerical Analysis (ETNA).
Yousef Saad is an I.T. Distinguished Professor of Computer Science in the Department of Computer Science and Engineering at the University of Minnesota. He holds the William Norris Chair for Large-Scale Computing since January 2006. He is known for his contributions to the matrix computations, including the iterative methods for solving large sparse linear algebraic systems, eigenvalue problems, and parallel computing. He is listed as an ISI highly cited researcher in mathematics, is the most cited author in the journal Numerical Linear Algebra with Applications, and is the author of the highly cited book Iterative Methods for Sparse Linear Systems. He is a SIAM fellow and a fellow of the AAAS (2011).
Peter Wynn (1931—2017) was an English mathematician. His main achievements concern approximation theory – in particular the theory of Padé approximants – and its application in numerical methods for improving the rate of convergence of sequences of real numbers.
PROSE was the mathematical 4GL virtual machine that established the holistic modeling paradigm known as Synthetic Calculus. A successor to the SLANG/CUE simulation and optimization language developed at TRW Systems, it was introduced in 1974 on Control Data supercomputers. It was the first commercial language to employ automatic differentiation (AD), which was optimized to loop in the instruction-stack of the CDC 6600 CPU.
Vera Faddeeva was a Soviet mathematician. Faddeeva published some of the earliest work in the field of numerical linear algebra. Her 1950 work, Computational methods of linear algebra was widely acclaimed and she won a USSR State Prize for it. Between 1962 and 1975, she wrote many research papers with her husband, Dmitry Konstantinovich Faddeev. She is remembered as an important Russian mathematician, specializing in linear algebra, who worked in the 20th century.
Christine Bernardi was a French mathematician known for her research on numerical analysis of partial differential equations.
Vivette Girault is a French mathematician, whose research expertise lies in numerical analysis, finite element methods and computational fluid dynamics. She has been affiliated with Pierre and Marie Curie University.
Raphaèle Herbin is a French applied mathematician, known for her work on the finite volume method.
Annie A. M. Cuyt is a Belgian computational mathematician known for her work on continued fractions, numerical analysis, Padé approximants, and related topics. She is a professor at the University of Antwerp, and a member of the Royal Flemish Academy of Belgium for Science and the Arts.
Daniel B. Szyld is an Argentinian and American mathematician who is a professor at Temple University in Philadelphia. He has made contributions to numerical and applied linear algebra as well as matrix theory.
Beatrice Meini is an Italian computational mathematician and numerical analyst specializing in numerical linear algebra and its applications to Markov chains, matrix equations, and queueing theory. She is Professor of Numerical Analysis in the Department of Mathematics at the University of Pisa.
References
- 1 2 WNA-Porto-2019: Workshop on Numerical Analysis dedicated to Claude Brezinski and Michela Redivo-Zaglia (PDF), Center of Mathematics of University of Porto, 15 July 2019, retrieved 2022-01-17
- ↑ Michela Redivo-Zaglia at the Mathematics Genealogy Project
- ↑ Curriculum vitae , retrieved 2022-01-17
- 1 2 Publications , retrieved 2022-01-17
- ↑ Reviews of Extrapolation Methods: Alan M. Cohen, MR 1140920; M. Křížek, ZAMM, Bibcode:1993ZaMM...73..236K, doi:10.1002/zamm.19930730912; Jet Wimp, Math. Comp., doi:10.2307/2153136, JSTOR 2153136; P. Wynn, Zbl 0744.65004
- ↑ Review of Extrapolation and Rational Approximation: Manfred Tasche, Zbl 07279818