study of algorithms that use numerical approximation for the problems of mathematical analysis
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Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).
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A - FEdit
- Mathematical techniques to achieve numerical solutions for partial differential equations began to appear about the turn of the century. The first definitive work was carried out by Richardson, who in a paper delivered to the Royal Society in London in 1910 introduced a finite-difference technique for numerical solution of Laplace's equation. Called a "relaxation technique," that approach is still used today to obtain numerical solutions for so-called elliptic partial differential equations (the equations that govern inviscid subsonic flows are such equations). However, modern numerical analysis is usually considered to have begun in 1928, when Courant, Friedrichs, and Lewy published a definitive paper on the numerical solution of so-called hyperbolic partial differential equations (the equations that govern inviscid compressible flow are such equations).
- John David Anderson (1998). A History of Aerodynamics: And Its Impact on Flying Machines. Cambridge University Press. pp. 443. ISBN 978-0-521-66955-9.
- Problems relative to the uniform propagation, or to the varied movements of heat in the interior of solids, are reduced ... to problems of pure analysis, and the progress of this part of physics will depend in consequence upon the advance which may be made in the art of analysis. The differential equations . . . contain the chief results of the theory; they express, in the most general and concise manner, the necessary relations of numerical analysis to a very extensive class of phenomena; and they connect forever with mathematical science one of the most important branches of natural philosophy.
- Joseph Fourier. Theory of Heat, Cambridge, 1878), Chap. 8, p. 151.
G - LEdit
- Numerical analysis is often considered neither beautiful nor, indeed, profound. Pure mathematics is beautiful if your heart goes after the joy of abstraction, applied mathematics is beautiful if you are excited by mathematics as a means to explain the mystery of the world around us. But numerical analysis? Surely, we compute only when everything else fails, when mathematical theory cannot deliver an answer in a comprehensive, pristine form and thus we are compelled to throw a problem onto a number-crunching computer and produce boring numbers by boring calculations. This, I believe, is nonsense.
- A mathematical problem does not cease being mathematical just because we have discretized it. The purpose of discretization is to render mathematical problems, often approximately, in a form accessible to efficient calculation by computers. This, in particular, means rephrasing and approximating analytic statements as a finite sequence of algebraic steps. Algorithms and numerical methods are, by their very design, suitable for computation but it makes them neither simple nor easy as mathematical constructs. Replacing derivatives by finite differences or an infinite-dimensional space by a hierarchy of finite-dimensional spaces does not necessarily lead to a more fuzzy form of reasoning. We can still ask proper mathematical questions with uncompromising rigour and seek answers with the full mathematical etiquette of precise definitions, statements and proofs. The rules of the game do not change at all.
- Arieh Iserles (2009). A First Course in the Numerical Analysis of Differential Equations, p. ix; Cited in Desmond J. Higham. "quotations" at math.fsu.edu, 2006.
M - REdit
- One of the most important theorems in calculus is the Mean Value Theorem (MVT), which is used to prove many theorems of both differential and integral calculus, as well as other subjects such as numerical analysis. MVT is said to be the midwife of calculus - not very important or glamorous by itself, but often helping to deliver other theorems that are of major significance. The proof of the Mean-Value Theorem is based on a special case of it known as Rolle’s Theorem.
- Michel Rolle, in Ulrich L. Rohde, G. C. Jain, Ajay K. Poddar, A. K. Ghosh Introduction to Differential Calculus: Systematic Studies with Engineering Applications for Beginners, John Wiley & Sons, 12-Jan-2012, p. 520
S - ZEdit
- In the 1950s and 1960s, the founding fathers of the field discovered that inexact arithmetic can be a source of danger, causing errors in results that "ought" to be right. The source of such problems is numerical instability: that is, the amplification of rounding errors from microscopic to macroscopic scale by certain modes of computation. These men, including von Neumann, Wilkinson, Forsythe, and Henrici, took great pains to publicize the risks of careless reliance on machine arithmetic. These risks are very real, but the message was communicated all too successfully, leading to the current widespread impression that the main business of numerical analysis is coping with rounding errors. In fact, the main business of numerical analysis is designing algorithms that converge quickly; rounding-error analysis, while often a part of the discussion, is rarely the central issue. If rounding errors vanished, 90% of numerical analysis would remain.
- Lloyd N. Trefethen (2006). Numerical Analysis, Oxford University, at maths.ox.ac.uk. ; Cited in Desmond J. Higham. "quotations" at math.fsu.edu, 2006.
- The subject of numerical analysis has ancient roots, and it has had periods of intense development followed by long periods of consolidation. In many cases, the new developments have coincided with the introduction of new forms of computing machines. For example, many of the basic theorems about computing solutions of ordinary differential equations were proved soon after desktop adding machines became common at the turn of the 20th century. The emergence of the digital computer in the mid-20th century spurred interest in solving partial differential equations and large systems of linear equations, as well as many other topics. The advent of parallel computers similarly stimulated research on new classes of algorithms. However, many fundamental questions remain open, and the subject is an active area of research today.
- L. Ridgway Scott. Numerical Analysis, Princeton University Press, 2011. p. xii-xiii
- A numerical equation is said to be analysed as soon as we discover the several limits, or pairs of numbers, within which all its unequal real roots lie individually, and its equal roots in distinct groups; that is, as soon as these unequal roots, and groups of equal roots, are all separated and severally enclosed, each between two assignable numbers.
- The preliminary step in this complete analysis of an equation would seem to be, first to determine limits as close as practicable, within which all the roots should be comprised, and this was NEWTON’s step. He showed how close superior and inferior limits were to be found without the bounds of which no roots could possibly exist. Bunan cut up NEWTON’s wide interval into a set of partial intervals, and showed that some of these partial intervals might, in like manner, he rejected; and that the real roots were to be found only among the intervals which he retained.