Numerical analysis is that branch of applied mathematics which studies the methods and algorithms to find (approximate) numerical solutions to various mathematical problems. The problems considered include
- solving equations or systems of equations
- computing values of functions
- numerically evaluating integrals
- solving differential equations and partial differential equations
Computers are an essential tool in numerical analysis, but the field predates computers by many centuries. 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.
The study of the generation and propagation of round-off errors in the cause of a computation is an important part of numerical analysis. The effect of round-off error is partly quantified in the condition number of an operator. Subtraction of two nearly equal numbers is an ill-conditioned operation, producing catastrophic loss of significance. Using well-conditioned operations helps achieve numerical stability.
One fundamental problem is the determination of zeros of a given function. Various algorithms have been developed. If the function is differentiable and the derivative is known, then Newton's method is a popular choice.
Numerical Analysis is also concerned with computing (in an approximate way) the solution of Partial Differential Equations. This is done by first discretizing the equation, bringing it into a finite dimensional subspace, then solving the linear system in this finite dimensional space. The first stage is done by the Finite element method, finite difference methods, or (particularly in engineering) the method of Finite Volumes. The theoretical justification of these methods often involves theorems from functional analysis.
The linear systems that come form discretized Parial Differential Equations can then be solved by a variant of Gauss-Jordan elimination, by some Iterative method such as Conjugate Gradients, or by Multigrid.
- Mention: Gaussian quadrature, least squares, Gram-Schmidt Process