Numerical analysis
Numerical analysis is that branch of applied mathematics which studies the methods and algorithms to find (approximate) numerical solutions to various mathematical problems, using a finite sequence of arithmetic and logical operations. Most solutions of numerical problems build on the theory of linear algebra.
A good method possesses the following three characteristics:
While numerical analysis employs mathematical axioms, theorems and proofs in theory, it may use empirical results of computation runs to probe new methods and analyze problems. It has thus a unique character when compared to other mathematical sciences.
A well-conditioned mathematical problem is, roughly speaking, one whose solution changes by only a small amount if the problem data are changed by a small amount. The analogous concept for the numerical algorithm for solving the problem is that of numerical stability: an algorithm for solving a well-conditioned problem is numerically stable if the result of the algorithm changes only a small amount if the data change a little. This means that any error committed in the early stages will not grow in an uncontrolled manner.
An algorithm that solves a well-conditioned problem may or may not be numerically stable. An art of numerical analysis is to find a stable algorithm for solving a mathematical problem.
The study of the generation and propagation of round-off errors in the cause of a computation is an important part of numerical analysis. Subtraction of two nearly equal numbers is an ill-conditioned operation, producing
catastrophic loss of significance.
The effect of round-off error is partly quantified in the condition number of an operator.
Computers are an essential tool in numerical analysis, but the field predates computers by many centuries, and actually computers were invented to a large extent in order to solve numerical problems, not the other way around. Taylor approximation is a product of the seventeenth and eighteenth centuries that is still very important. The logarithms of the sixteenth century are no longer vital to numerical analysis, but the associated and even prehistoric notion of interpolation continues to solve problems for us.
Floating point number representations are used extensively in modern computers: for many problems, their behavior can be unexpected, unless care is taken using numerical analysis to ensure that they will not misbehave.
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.
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.
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.
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.
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 Conjugate Gradients are usually preferred for large systems.
The problem of solving nonlinear equations is usually solved by linearization. If the function is differentiable and the derivative is known, then Newton's method is a popular choice.
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.
Numerical integration, also known as numerical quadrature, asks for the value of an 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.
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. General introduction
Often you will hit tradeoffs between these characteristics. For instance, it usually happens that one method is faster, while the other is more accurate. This means that no algorithm is the best in all cases.Conditioning and stability
Computers as tools for numerical analysis
Areas of study
Computing values of functions
Interpolation, extrapolation and regression
Solving equations
Optimization
Main article: Optimization (mathematics).Evaluating integrals
Main article: Numerical integration.Differential equations
Main articles: Numerical ordinary differential equations, Numerical partial differential equations.