iteration
Quick Summary
In mathematics, "iteration" refers to the process of repeating a set of operations or instructions. This is a fundamental concept in numerical analysis, where algorithms are used to find approximate solutions to complex problems. Iteration is also a key control structure in computer programming, allowing for the repetition of steps to solve problems efficiently.12
A significant area related to iteration is the study of Julia sets and fractals, which explore the behavior of functions when repeatedly applied. In numerical analysis, iteration is often used to refine solutions, such as in Newton's iterative method for finding roots of equations. The concept of iteration also appears in perturbation theory, where a problem is solved by comparing it to a similar, known problem and then iteratively refining the solution.314
Ask Britannica AI about this topic
Britannica doesn't have a full article on this topic, so we've curated a list of our relevant coverage.
foundations of mathematicsFoundational logicā¦2 is the process of iteration; that is, 2 is the function which to every function f assigns its iterate 2(f) = f ā f, where (f ā f)(x) = f(f(x)). There are some type-theoretical difficulties with this construction, but these can be overcome if quantification over types is allowed;ā¦
Read Moreā¦2 is the process of iteration; that is, 2 is the function which to every function f assigns its iterate 2(f) = f ā f, where (f ā f)(x) = f(f(x)). There are some type-theoretical difficulties with this construction, but these can be overcome if quantification over types is allowed;ā¦
Read More
operations researchDeriving solutions from modelsā¦procedures are said to be iterative (repetitive) if they proceed through successively improved solutions until either an optimal solution is reached or further calculation cannot be justified. A rational basis for terminating such a processāknown as āstopping rulesāāinvolves the determination of the point at which the expected improvement of theā¦
Read Moreā¦procedures are said to be iterative (repetitive) if they proceed through successively improved solutions until either an optimal solution is reached or further calculation cannot be justified. A rational basis for terminating such a processāknown as āstopping rulesāāinvolves the determination of the point at which the expected improvement of theā¦
Read More
weather forecastingNumerical weather prediction (NWP) modelsThis process is repeated step after step. In principle, the process could continue indefinitely. In practice, small errors creep into the calculations, and they accumulate. Eventually, the errors become so large by this cumulative process that there is no point in continuing.
Read MoreThis process is repeated step after step. In principle, the process could continue indefinitely. In practice, small errors creep into the calculations, and they accumulate. Eventually, the errors become so large by this cumulative process that there is no point in continuing.
Read More