Much effort has been put in the development of methods for solving systems of linear equations. Standard direct methods, i. Iterative methods such as the Jacobi methodGauss—Seidel methodsuccessive over-relaxation and conjugate gradient method are usually preferred for large systems.
Much effort has been put in the development of methods for solving systems of linear equations.
Standard direct methods, i. Iterative methods such as the Jacobi methodGauss—Seidel methodsuccessive over-relaxation and conjugate gradient method are usually preferred for large systems. General iterative methods can be developed using a matrix splitting.
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 Numerical problems the derivative is known, then Newton's method is a popular choice. Linearization is another technique for solving nonlinear equations.
Solving eigenvalue or singular value problems[ edit ] Several important problems can be phrased in terms of eigenvalue decompositions or singular value decompositions. For instance, the spectral image compression algorithm  is based on the singular value decomposition.
The corresponding tool in statistics is called principal component analysis. Mathematical optimization Optimization problems ask for the point Numerical problems which a given function is maximized or minimized.
Often, the point also has to satisfy some constraints. 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. The method of Lagrange multipliers can be used to reduce optimization problems with constraints to unconstrained optimization problems.
Numerical integration Numerical integration, in some instances also known as numerical quadratureasks for the value of a definite integral. Popular methods use one of the Newton—Cotes formulas like the midpoint rule or Simpson's rule or Gaussian quadrature. These methods rely on a "divide and conquer" strategy, whereby an integral on a relatively large set is broken down into integrals on smaller sets.
In higher dimensions, where these methods become prohibitively expensive in terms of computational effort, one may use Monte Carlo or quasi-Monte Carlo methods see Monte Carlo integrationor, in modestly large dimensions, the method of sparse grids. Numerical ordinary differential equations and Numerical partial differential equations Numerical analysis is also concerned with computing in an approximate way the solution of differential equationsboth ordinary differential equations and partial differential equations.
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 methoda 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.
List of numerical analysis software and Comparison of numerical analysis software Since the late twentieth century, most algorithms are implemented in a variety of programming languages.
The Netlib repository contains various collections of software routines for numerical problems, mostly in Fortran and C. Also, any spreadsheet software can be used to solve simple problems relating to numerical analysis.For unconstrained problems with twice-differentiable functions, some critical points can be found by finding the points where the gradient of the objective function is zero (that is, the stationary points).
More generally, a zero subgradient certifies that a local minimum has been found for minimization problems with convex functions and other locally Lipschitz functions.
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The shooting method works by considering the boundary conditions as a multivariate function of initial conditions at some point, reducing the boundary value problem to finding the initial conditions that give a root. The advantage of the shooting method is that it takes advantage of the speed and adaptivity of methods for initial value problems.
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Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).
Numerical analysis naturally finds applications in all fields of engineering and the physical sciences, but in the 21st century also.