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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.

1 General introduction

A good method possesses the following three characteristics:

Accuracy - the numerical approximation should be as accurate as possible. This requires the algorithm to be numerically stable, as explained in the next section.
Robustness - the algorithm should solve many problems well. This means that it should warn the user, if the result is inaccurate. Hence it should be possible to estimate the error.
Speed - the faster the computation, the better is the method.

Often you will hit tradeoffs between these characteristics. For instance, it usually happens that one method is faster, while the other is more accurate. This means that no algorithm is the best in all cases.

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.

1.1 Conditioning and stability

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. This means that any error committed in the early stages will not grow in an uncontrolled manner.

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.

1.2 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 approximationIn mathematics, the Taylor series of an infinitely often differentiable real (or complex) function f defined on an open interval a − r a + r is the power series : Here, n is the factorial of n and f n a denotes the n''th derivative of f at the point is a product of the seventeenth and eighteenth centuries that is still very important. The logarithmIn mathematics, the logarithm functions are the inverses of the exponential functions. Logarithms are numbers that are substituted in computation for other numbers, to which they bear such a relation that the operations to be performed on the latter are rs of the sixteenth century are no longer vital to numerical analysis, but the associated and even prehistoric notion of interpolationThis article is about interpolation in mathematics. See also interpolation (music . In the mathematical subfield of numerical analysis interpolation is a method of constructing new data points from a discrete set of known data points. According to the Oxf continues to solve problems for us. Floating point number representations are used extensively in modern computers: for many problems, their behavior can be unexpected, unless care is taken using numerical analysis to ensure that they will not misbehave.

1.3 Software

If a computer is to execute some numerical method, this method has to be implemented in some way. The NetlibNetlib , is a repository of software for scientific computing. Netlib comprises a large number of separate programs and libraries. Most of the code is written in Fortran, with some programs in other languages. The legal status of the code is not entirely repository contains various collections of software routines for numerical problems. Commercial products implementing many different numerical algorithms include the IMSL and NAG libraries; a free alternative is the GNU Scientific LibraryThe GNU Scientific Library GSL is a comprehensive programming library for scientific computation. It is released under the GNU General Public License. External link .. A different approach is taken by the Numerical RecipesNumerical Recipes is the generic term for the following books on algorithms and numerical analysis, all by William Press, Saul Teukolsky, William Vetterling and Brian Flannery: Numerical Recipes in C++. The Art of Scientific Computing BooksEnthsiast.com. library, where emphasis is placed on clear understanding of algorithms.

There are a number of computer programs used for performing numerical calculations:

MATLABMATLAB short for "Matrix Laboratory", refers to a both a numerical computing environment and its core programming language. Created by The MathWorks MATLAB allows one to easily manipulate matrices, plot functions and data, implement algorithms, create use is a widely-used program for performing numerical calculations. It comes with its own programming language, in which numerical algorithms can be implemented.
GNU Octave is a free near-clone of Matlab.
R programming language is a widely used system with a focus on data manipulation and statistics. Several hundred freely downloadable specialized packages are available.
Scilab.
IDL programming language.

Many computer algebra systems such as Mathematica or the Maple computer algebra system (free systems include Maxima, calc and Yacas) can also be used for numerical computations.

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