Truncation errors are committed when an iterative method is terminated and the approximate solution differs from the exact solution. Round-off errors arise because it is impossible to represent all real numbers exactly on a finite-state machine (which is what all practical digital computers are).

There are several ways in which error can be introduced in the solution of the problem. The study of errors forms an important part of numerical analysis. The generation and propagation of errors

This function must be represented by a finite amount of data, for instance by its value at a finite number of points at its domain, even though this domain is a continuum. For example, the solution of a differential equation is a function. Even when a direct method does exist, an iterative method may be preferable because it is more efficient or more stable.įurthermore, continuous problems must sometimes be replaced by a discrete problem whose solution is known to approximate that of the continuous problem this process is called discretization. Such a method starts from a guess and finds successive approximations that hopefully converge to the solution. In such cases it is sometimes possible to use an iterative method. However, no direct methods exist for most problems. Examples are Gaussian elimination for solving systems of linear equations and the simplex method in linear programming. These algorithms are called direct methods. Some problems can be solved exactly by an algorithm. The latter choice describes the field of numerical analysis. In these situations, one has two options left: either one tries to find an approximate solution using asymptotic analysis or one seeks a numerical solution. Examples are finding the integral of exp(− x 2) (see error function) and solving a general polynomial equation of degree five or higher (see Abel-Ruffini theorem).

Many problems in continuous mathematics do not possess a closed-form solution. 2.4 Solving eigenvalue or singular value problems.2.3 Solving equations and systems of equations.2.2 Interpolation, extrapolation and regression.1.3 The generation and propagation of errors.

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