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Several important problems can be phrased in terms of [[eigenvalue decomposition]]s or [[singular value decomposition]]s. For instance, the [[image compression|spectral image compression]] algorithm<ref>[http://online.redwoods.cc.ca.us/instruct/darnold/maw/single.htm The Singular Value Decomposition and Its Applications in Image Compression]</ref> is based on the singular value decomposition. The corresponding tool in statistics is called [[principal component analysis]].
 
===Optimization===
{{Main|Mathematical optimization}}
 
Optimization problems ask for the point at which a given function is maximized (or minimized). Often, the point also has to satisfy some [[Constraint (mathematics)|constraint]]s.
 
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.
 
===Evaluating integrals===
{{Main|Numerical integration}}
 
Numerical integration, in some instances also known as numerical [[quadrature (mathematics)|quadrature]], asks 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 method|Monte Carlo]] or [[quasi-Monte Carlo method]]s (see [[Monte Carlo integration]]), or, in modestly large dimensions, the method of [[sparse grid]]s.
 
===Differential equations===
{{main|Numerical ordinary differential equations|Numerical partial differential equations}}
 
Numerical analysis is also concerned with computing (in an approximate way) the solution of [[differential equation]]s, both ordinary differential equations and [[partial differential equation]]s.
 
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.
 
 
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