NUMERICAL ANALYSIS

Numerical analysis is that branch of applied mathematics which studies the methods and algorithms to find (approximate) numerical solutions to various mathematical problems, using a finite sequence of arithmetic and logical operations. Most solutions of numerical problems build on the theory of linear algebra. The problems considered include:

Computing values of functions:
- Evaluating polynomials using Horner's rule
- Handling round-off error estimation
Solving linear equations or systems of linear equations, by for example
- Gauss-Jordan elimination
- LU-factorization
Approximating solutions to nonlinear problems, often by linearization especially using Newton's method
Regression
- Least squares-method
Numerically evaluating integrals using numerical integration, also known as quadrature
- Gaussian quadrature
Solving differential equationss and partial differential equations or approximating their solutions
Interpolation using
- Linear interpolation
- Polynomial interpolation, while keeping an eye on Runge's phenomenon
- Splines (linear, quadratic, cubic)
- Wavelets
Extrapolation

When approximating different solutions to numerical problems, three factors about such methods are to be considered:

Stability - the errors in the approximation may not grow in an uncontrolled manner when the computations increase in size and scope.
Accuracy - the numerical approximation should be as accurate as possible and the error should be possible to estimate according to some scale.
Efficiency - the faster the computation, the better is the method. You may however hit tradeoffs between accuracy and efficiency. Therefore, fast convergence of solutions are to be sought at all times.

While numerical analysis employs mathematical axioms, theorems and proofs in theory, it may use empirical results of computation runs to probe new methods and analyze problems. It has thus a unique character when compared to other mathematical sciences.

Computers as Tools for Numerical Analysis

Computers are an essential tool in numerical analysis, but the field predates computers by many centuries, and actually computers were invented to a large extent in order to solve numerical problems, not the other way around. Taylor approximation is a product of the seventeenth and eighteenth centuries that is still very important. The logarithms of the sixteenth century are no longer vital to numerical analysis, but the associated and even prehistoric notion of interpolation continues to solve problems for us.

Problem Taxonomy

A well-conditioned mathematical problem is, roughly speaking, one whose solution changes by only a small amount if the problem data are changed by a small amount. The analogous concept for the numerical algorithm for solving the problem is that of numerical stability: an algorithm for solving a well-conditioned problem is numerically stable if the result of the algorithm changes only a small amount if the data change a little.

An algorithm that solves a well-conditioned problem may or may not be numerically stable. An art of numerical analysis is to find a stable algorithm for solving a mathematical problem.

The study of the generation and propagation of round-off errors in the cause of a computation is an important part of numerical analysis. Subtraction of two nearly equal numbers is an ill-conditioned operation, producing catastrophic loss of significance.

The effect of round-off error is partly quantified in the condition number of an operator.

Finding Zeros

One fundamental problem is the determination of zeros of a given function. Various algorithms have been developed. If the function is differentiable and the derivative is known, then Newton's method is a popular choice.

Partial Differential Equations

Numerical Analysis is also concerned with computing (in an approximate way) the solution of Partial Differential Equations. This is done by first discretizing the equation, bringing it into a finite dimensional subspace, then solving the linear system in this finite dimensional space. The first stage is done by the Finite element method, finite difference methods, or (particularly in engineering) the method of Finite Volumes. The theoretical justification of these methods often involves theorems from functional analysis.

The linear systems that come form discretized Partial Differential Equations can then be solved by a variant of Gauss-Jordan elimination, by some Iterative method such as Conjugate Gradients, or by Multigrid.

Domain	Title
Books	Approximate Methods and Numerical Analysis for Elliptic Complex Equation (Asian Mathematics) (reference) The Algebraic Eigenvalue Problem (Monographs on Numerical Analysis) (reference) Numerical Analysis Via Derive (reference) Iciam/Gamm 95 Hamburg, July 3-7, 1995 Numerical Analysis (reference) Splines in numerical analysis : contributions to the international seminar ISAM-89 held in Weissig (GDR) April 24-28, 1989 (reference) (more book examples)
Source: compiled by the editor from various references; see credits.

Modern Translation: NUMERICAL ANALYSIS

Language		Translations for "NUMERICAL ANALYSIS"; alternative meanings/domain in parentheses.
Danish		numerisk analyse af formgivning af plademetal (numerical analysis of sheet metal forming). (various references)

Dutch		numerieke analyse van metaalfolievervorming (numerical analysis of sheet metal forming), numerieke analyse van bladmetaalvervorming (numerical analysis of sheet metal forming). (various references)

French		analyse numérique. (various references)

German		numerische Analyse der Blechumformung (numerical analysis of sheet metal forming). (various references)

Italian		analisi numerica sulla formatura di lamiere sottili (numerical analysis of sheet metal forming). (various references)

Japanese Kanji		数値解析 . (various references)

Japanese Katakana		すうちかいせき. (various references)

Pig Latin		umericalnay analysisay

Portuguese		análise numérica da formagem de chapa metálica (numerical analysis of sheet metal forming). (various references)
Source: compiled by the editor from various translation references.

Anagrams: NUMERICAL ANALYSIS

Scrabble® Enable2K-Verified Anagrams
	Words within the letters "a-a-a-c-e-i-i-l-l-m-n-n-r-s-s-u-y"
	-5 letters: illuminances, millenarians, misalliances, semiannually, uncinariases, unicamerally.
Source: compiled by the editor from various references; see credits. SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro.

Alternative Orthography: NUMERICAL ANALYSIS

Hexadecimal (or equivalents, 770AD-1900s) (references)

4E 55 4D 45 52 49 43 41 4C 41 4E 41 4C 59 53 49 53

Leonardo da Vinci (1452-1519; backwards) (references)

Binary Code (1918-1938, probably earlier) (references)

01001110 01010101 01001101 01000101 01010010 01001001 01000011 01000001 01001100 00100000 01000001 01001110 01000001 01001100 01011001 01010011 01001001 01010011

HTML Code (1990) (references)

&#78 &#85 &#77 &#69 &#82 &#73 &#67 &#65 &#76 &#32 &#65 &#78 &#65 &#76 &#89 &#83 &#73 &#83

ISO 10646 (1991-1993) (references)

004E 0055 004D 0045 0052 0049 0043 0041 004C 0041 004E 0041 004C 0059 0053 0049 0053

Encryption (beginner's substitution cypher): (references)

Domain	Definition
Economics	Can be described as the technique of using statistical methods to squeeze the maximum information from available data. Source: European Union. (references)
Source: compiled by the editor from various references; see credits.

NUMERICAL ANALYSIS

Specialty Definition: NUMERICAL ANALYSIS

Economics

Specialty Definition: Numerical analysis

Computers as Tools for Numerical Analysis

Problem Taxonomy

Finding Zeros

Partial Differential Equations

See Also

Crosswords: NUMERICAL ANALYSIS

Commercial Usage: NUMERICAL ANALYSIS

Books

Modern Translation: NUMERICAL ANALYSIS

Danish

Dutch

French

German

Italian

Japanese Kanji

Japanese Katakana

Pig Latin

Portuguese

Anagrams: NUMERICAL ANALYSIS

Scrabble® Enable2K-Verified Anagrams

Alternative Orthography: NUMERICAL ANALYSIS

4E 55 4D 45 52 49 43 41 4C 41 4E 41 4C 59 53 49 53

01001110 01010101 01001101 01000101 01010010 01001001 01000011 01000001 01001100 00100000 01000001 01001110 01000001 01001100 01011001 01010011 01001001 01010011

&#78 &#85 &#77 &#69 &#82 &#73 &#67 &#65 &#76 &#32 &#65 &#78 &#65 &#76 &#89 &#83 &#73 &#83

004E 0055 004D 0045 0052 0049 0043 0041 004C 0041 004E 0041 004C 0059 0053 0049 0053

48554739524337354623548354659534353

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