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.

Areas of study

The field of numerical analysis is divided in different disciplines according to the problem that is to be solved.

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Computing values of functions

One of the simplest problems is the evaluation of a function at a given point. But even evaluating a polynomial is not straightforward: the Horner scheme is often more efficient than the obvious method. Generally, it is important to estimate and control round-off errors arising from the use of floating point arithmetic.

Related Topics:
Horner scheme - Round-off error - Floating point

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Interpolation, extrapolation and regression

Interpolation solves the following problem: given the value of some unknown function at a number of points, what value does that function have at some other point between the given points? A very simple method is to use linear interpolation, which assumes that the unknown function is linear between every pair of successive points. This can be generalized to polynomial interpolation, which is sometimes more accurate but suffers from Runge's phenomenon. Other interpolation methods use localized functions like splines or wavelets.

Related Topics:
Interpolation - Linear interpolation - Polynomial interpolation - Runge's phenomenon - Spline - Wavelet

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Extrapolation is very similar to interpolation, except that now we want to find the value of the unknown function at a point which is outside the given points.

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Regression is also similar, but it takes into account that the data is imprecise. Given some points, and a measurement of the value of some function at these points (with an error), we want to determine the unknown function. The least squares-method is one popular way to achieve this.

Related Topics:
Regression - Least squares

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Solving equations and systems of equations

Another fundamental problem is computing the solution of some given equation. Two cases are commonly distinguished, depending on whether the equation is linear or not.

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Much effort has been put in the development of methods for solving systems of linear equations. Standard methods are Gauss-Jordan elimination and LU-factorization. Iterative methods such as the conjugate gradient method are usually preferred for large systems.

Related Topics:
Gauss-Jordan elimination - LU-factorization - Iterative method - Conjugate gradient method

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Root-finding algorithms are used to solve nonlinear equations (they are so named since a root of a function is an argument for which the function yields zero). If the function is differentiable and the derivative is known, then Newton's method is a popular choice. Linearization is another technique for solving nonlinear equations.

Related Topics:
Root-finding algorithm - Differentiable - Newton's method - Linearization

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Optimization

Main article: Optimization (mathematics).

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Optimization problems ask for the point at which a given function is maximized (or minimized). Often, the point also has to satisfy some constraints.

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The field of optimization is further split in several subfields, depending on the form of the objective function and the constraint. For instance, linear programming deals with the case that both the objective function and the constraints are linear. A famous method in linear programming is the simplex method.

Related Topics:
Linear programming - Simplex method

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The method of Lagrange multipliers can be used to reduce optimization problems with constraints to unconstrained optimization problems.

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Evaluating integrals

Main article: Numerical integration.

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Numerical integration, also known as numerical quadrature, asks for the value of a definite integral. Popular methods use some Newton-Cotes formula, for instance the midpoint rule or the trapezoid rule, or Gaussian quadrature. However, if the dimension of the integration domain becomes large, these methods become prohibitively expensive. In this situation, one may use a Monte Carlo method, a quasi-Monte Carlo method, or, in modestly large dimensions, the method of sparse grids.

Related Topics:
Quadrature - Integral - Newton-Cotes formula - Gaussian quadrature - Monte Carlo method - Quasi-Monte Carlo method - Sparse grid

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Differential equations

Main articles: Numerical ordinary differential equations, Numerical partial differential equations.

Related Topics:
Numerical ordinary differential equations - Numerical partial differential equations

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Numerical analysis is also concerned with computing (in an approximate way) the solution of differential equations, both ordinary differential equations and partial differential equations.

Related Topics:
Differential equation - Partial differential equation

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Partial differential equations are solved by first discretizing the equation, bringing it into a finite-dimensional subspace. This can be done by a finite element method, a finite difference method, or (particularly in engineering) a finite volume method. The theoretical justification of these methods often involves theorems from functional analysis. This reduces the problem to the solution of an algebraic equation.

Related Topics:
Finite element method - Finite difference - Finite volume method - Functional analysis

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