Numerical analysis

Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). This means it deals mainly with real variable or complex variable questions, numerical linear algebra over the real or complex fields, the solution of differential equations, and other related problems arising in the physical sciences and engineering.

General introduction

Some problems in continuous mathematics can be solved exactly by an algorithm. These algorithms are called direct methods. Examples are Gaussian elimination for solving systems of linear equations and the simplex method in linear programming.

Related Topics:
Gaussian elimination - Systems of linear equations - Simplex method - Linear programming

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However, no direct methods exist for most problems. In such cases it is sometimes possible to use an iterative method. Such a method starts from a guess and finds successive approximations that hopefully converge to the solution. Even when a direct method does exist, an iterative method may be preferable because it is more efficient or more stable.

Related Topics:
Iterative method - Converge

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Furthermore, continuous problems must sometimes be replaced by a discrete problem whose solution is known to approximate that of the continuous problem; this process is called discretization. For example, the solution of a differential equation is a function. This function must be represented by a finite amount of data, for instance by its value at a finite number of points at its domain, even though this domain is a continuum.

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The generation and propagation of errors

The study of errors forms an important part of numerical analysis. There are several ways in which error can be introduced in the solution of the problem. Round-off errors arise because it is impossible to represent all real numbers exactly on a finite-state machine (which is what all practical digital computers are). Truncation errors are committed when an iterative method is terminated and the approximate solution differs from the exact solution. Similarly, discretization induces a discretization error because the solution of the discrete problem does not coincide with the solution of the continuous problem.

Related Topics:
Round-off error - Real number - Finite-state machine - Digital computer - Truncation - Terminate - Discretization error

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Once an error is generated, it will generally propagate through the calculation. This leads to the notion of numerical stability: an algorithm is numerically stable if an error, once it is generated, does not grow too much during the calculation. This is only possible if the problem is well-conditioned, meaning that the solution changes by only a small amount if the problem data are changed by a small amount. Indeed, if a problem is ill-conditioned, then any error in the data will grow a lot.

Related Topics:
Numerical stability - Well-conditioned

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However, 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 well-posed mathematical problem. A related art is to find stable algorithms for solving ill-posed problems, which generally requires finding a well-posed problem whose solution is close to that of the ill-posed problem and solving this well-posed problem instead.

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Applications

The algorithms of numerical analysis are routinely applied to solve many problems in science and engineering. Examples are the design of structures like bridges and airplanes (see computational physics and computational fluid dynamics), weather forecasting, climate models, the analysis and design of molecules (computational chemistry), and finding oil reservoirs. In fact, almost all supercomputers are continually running numerical analysis algorithms.

Related Topics:
Computational physics - Computational fluid dynamics - Weather forecasting - Climate model - Computational chemistry - Supercomputer

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As a consequence, efficiency plays an important role and a heuristic method may be preferred above a method with a solid theoretic foundation because it is more efficient. Generally, numerical analysis uses empirical results of computation runs to probe new methods and analyze problems, though it of course also employs mathematical axioms, theorems and proofs.

Related Topics:
Empirical - Axiom - Theorem - Proof

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