Numerical Analysis

The analyzes numerical is a discipline of the Mathématiques. It is interested as well in the theoretical bases as with the practical application of the methods making it possible to solve, by purely numerical calculations, problems of analyzes mathematical.

More formally, the analyzes numerical is the study of the algorithms making it possible to solve the continuous mathematical problems of (distinguished from the discrete Mathématiques). That means that it is mainly occupied to answer numerically questions with variable real or complex like the Linear algebra numerical about the real or complex fields, the search for numerical solution of differential equations and other problems dependant occurring in the Physical sciences and the Ingénierie.

Its implementation practical and its scopes of application are described more completely in the numerical article Calcul.

General introduction

Certain problems of continuous mathematics can be solved in an exact way by an algorithm. These algorithms are called direct methods . Examples are the elimination of Gauss-Jordan for the resolution of a system of linear equations and the Algorithme of the simplex in linear Programming.

However, no direct method is known for certain problems (moreover, for a class of problems known as complete NP S, no calculation algorithm direct in polynomial time is known to date). In such cases, it is sometimes possible to use a iterative Méthode to try to determine an approximation of the solution. Such a method starts since a value guessed or estimated coarsely and finds approximations successive which should converge towards the solution under certain conditions. Even when a direct method exists however, an iterative method can be preferable because it is often more effective and even often more stable (in particular it generally makes it possible to correct minor errors in intermediate calculations).

In addition, certain continuous problems can sometimes be replaced by a discrete problem whose solution is known to approach that of the problem continu ; this process is called discretization . For example the solution of a differential equation is a function. This function can be represented in a way approached by a quantity finished of data, for example by its value in a finished number of points of its field of definition, even if this field is continuous.

The use of the numerical analysis is largely facilitated by the Ordinateurs. The increase in the availability and the power of the computers since second half of the 20th century allowed the application of the numerical analysis in many scientific disciplines, techniques and economic, with often of the revolutionary effects.

The generation and propagation of the errors

The study of the errors trains an important part of the numerical analysis. The errors introduced into the solution of a problem have several origins. The errors rounding occur because it is impossible to represent in practice all the real numbers exactly on a Machine in finished states (what are in the final analysis all the numerical Ordinateur S). The errors of Troncature are made for example when a iterative Méthode is finished and that the approximate solution obtained differs from the exact solution. In a similar way, the Discretization of a problem (also called Quantification in the practical applications of numerical calculation) induced a Error of discretization (Error of quantification in the practical applications) because the solution of the discrete problem does not coincide exactly with the solution of the continuous problem.

Once the error is generated, it will be generally propagated throughout calculation. That led to the numerical concept of Stability: an algorithm is numerically stable if an error, once generated, does not grow too much during the calculation (in a method of calculating iterative, a too large error can in certain cases make diverge the algorithm which will not manage to approach the solution). That is not possible that if the problem is Bien conditioned, which means that the solution changes only one small quantity if the facts of the case are changed of a weak amount. Thus, if a problem is badly conditioned, then the least error in the data will cause a very important error in the found solution.

However, an algorithm which solves a problem well conditioned can be or not be numerically stable. All the art of the numerical analysis consists in finding an algorithm stable to solve a mathematical problem posed well. Art related is to find algorithms stable allowing to solve problems badly posed, which generally requires the search for a problem posed well to which the solution is close to the badly posed problem, then to solve in the place this second well posed problem.

Fields of studies

The field of the numerical analysis is divided into various disciplines according to the type of problem to solve, and each discipline studies various methods of resolution of the corresponding problems.

Among the examples of methods of numerical analyzes, here are some used for to discretize a system of equations: the Finite element method, the Method the finite differences, method of the divided Differences, the Method of finished volumes,…

Calculation of the values of functions

One of the simplest problems is the evaluation of a function at a given point. But even the evaluation of a similar polynomial is not as obvious as it there paraît : the Méthode of Horner is often more effective than the elementary method based on the Coefficient S of the developed polynomial and the simple amount of its terms. Generally, it is important to estimate in advance and to control the rounding errors occurring during the use of arithmetic operations in Floating decimal point.

Interpolation, extrapolation and regression

The interpolation tries to solve or approach the solution with the problem suivant : being given the known value of a certain function in a certain number of points, which value takes this function in another unspecified point located between two points donnés ? A very simple method is to use the linear Interpolation, which supposes that the unknown function evolves/moves linearly between each pair of known successive points. This method can be generalized in polynomial Interpolation, which is sometimes more precise (one can in chiffer the precision if the Dérivée S from the function are known until the order NR for an interpolation with NR points) and requires smaller tables of known values, but it suffers from the Phénomène of Runge.

Other methods of interpolation use functions located such as the Spline S or the Compression by ondelettes.

The Extrapolation is very similar to the interpolation, except that this time one wants to determine the value of a function in a point located out of the interval of the known points. In certain cases (for example for the extrapolation of cyclic, logarithmic curves or exponential values of functions), it is possible to reduce a problem of extrapolation in an even infinite field of definition very wide, with a problem of interpolation in the finished subspace containing the known points.

The regression is also similar, but takes into account the fact that the known data are also vague. Being given certain points, and value of a function at these points measures it (with an estimated maximum error), one wants to determine the unknown function. The Method of least squares is a popular way to proceed.

Solution of equations and systems of equations

Another fundamental problems are the calculation of the solutions of a equation given. Two cases are commonly distinguished, according to whether the equation is linear or not.

Many efforts were devoted to the development of methods of resolution of linear systems of equations. The methods standards include the elimination of Gauss-Jordan, and the Décomposition LU. The iterative methods such as the Méthode of the gradient combined are generally preferred on the broad systems of equations.

The algorithms of search for roots of a function are used to solve the nonlinear equations (they are named thus because the root of a function is an argument for which the function turns over zero). If the function is differentiable and that its derivative is known, then the Méthode of Newton is a popular choice. The Linéarisation is another technique for the solution of nonlinear equations.

Optimization

The problems of optimization seek the point to which a given function is maximum (or minimal). Often, such a point must also satisfy some Contrainte S.

The field of application of optimization itself is cut out under-fields, according to the form of the objective function and that of the constraint. For example, the linear Programming treats case where the objective function and the constraints all are linear. A famous method of linear programming is the Algorithme of the simplex.

The method of the multiplying of Lagrange can be used to reduce the problems of optimization with constraints in problems of optimization without constraints.

Evaluation of the integrals

Numerical, also known integration like numerical squaring, research the value of an Integral definite. The popular methods are based on the Formules of Newton-Dimensions (with for example the Méthode of the median point or the Méthode of the trapezoids) or use the Méthodes of squaring of Gauss. However if the dimension of the field of integration becomes broad, these methods become expensive also prohibitivement. In this situation, one can use a Méthode of Monte Carlo, a Méthode of quasi-Assembles-Carlo or, in modestly broad dimensions, the method of the incomplete grids.

Differential equations

Articles principaux : ordinary differential numerical Equation, numerical Equation with the derivative partial.

The numerical analysis also treats calculation (in an approximate way) of the solutions of differential equations, that they are ordinary differential equations, or partial derivative equations.

The partial differential equations are solved by discretizing the equation initially, by bringing it in a subspace of finished Dimension. This can be carried out by a Finite element method, a Méthode of the finished differences or, particularly in engineering, a Méthode of finished volumes. The theoretical justification of these methods often implies theorems of the analyzes functional. This reduces the problem to the solution of an algebraic equation.

See too

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numerical Calculation

External bonds

numerical Systems design course www.math-linux.com
Examples of numerical analysis numerical www.euro-se.fr
Methods of analysis optional course with DEA of the Dynamic and Statistical Modeling of the Complex Systems, a numerical introduction to the methods of analysis very largely used in physics in order to solve the algebraic or differential equations.
numerical Resolutions of linear system
Numerical analysis DMOZ category
numerical Systems design course
Numerical Master Modeling in Mechanics, University of the La Rochelle

References

Nick Trefethen , '' The definition off numerical analysis '', appeared in SIAM News , November 1992.
/'' Numerische Mathematik '' : complete copies digitized on line of volumes 1-66, covering the years 1959 to 1994, of a well-known newspaper of numerical analysis.

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