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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.
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.
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.
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.Main article: Optimization (mathematics).
Optimization problems ask for the point at 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 an unconstrained optimization problems.
Main article: Numerical integration.
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 expansive. In this situation, one may use a Monte Carlo method.
Main articles: Numerical ordinary differential equations, Numerical partial differential equations .
Numerical analysis is also concerned with computing (in an approximate way) the solution of differential equations, both 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 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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