Iteration means the act of repeating a process usually with the aim of approaching a desired goal or target or result. Each repetition of the process is also called an "iteration," and the results of one iteration are used as the starting point for the next iteration.
In mathematics, an iterated function is a function which is composed with itself, possibly ad infinitum, in a process called iteration. In this process, starting from some initial value, the result of applying a given function is fed again in the function as input, and this process is repeated...
i.e. applying a function repeatedly, using the output from one iteration as the input to the next. Iteration of apparently simple functions can produce complex behaviours and difficult problems - for examples, see the Collatz conjecture
Collatz conjecture
The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture , Kakutani's problem , the Thwaites conjecture , Hasse's algorithm The Collatz conjecture is a...
In computational mathematics, an iterative method is a mathematical procedure that generates a sequence of improving approximate solutions for a class of problems. A specific implementation of an iterative method, including the termination criteria, is an algorithm of the iterative method...
s which are used to produce approximate numerical solutions to certain mathematical problems. Newton's method
Newton's method
In numerical analysis, Newton's method , named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots of a real-valued function. The algorithm is first in the class of Householder's methods, succeeded by Halley's method...
is an example of an iterative method.
Computing
Iteration in computing is the repetition of a process
Process (computing)
In computing, a process is an instance of a computer program that is being executed. It contains the program code and its current activity. Depending on the operating system , a process may be made up of multiple threads of execution that execute instructions concurrently.A computer program is a...
A computer program is a sequence of instructions written to perform a specified task with a computer. A computer requires programs to function, typically executing the program's instructions in a central processor. The program has an executable form that the computer can use directly to execute...
. It can be used both as a general term, synonymous with repetition, and to describe a specific form of repetition with a mutable state.
Recursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other the nested images that occur are a form of infinite recursion. The term has a variety of meanings specific to a variety of disciplines ranging from...
is an example of iteration, but typically using a recursive notation, which is typically not the case for iteration.
However, when used in the second (more restricted) sense, iteration describes the style of programming used in imperative programming languages. This contrasts with recursion, which has a more declarative approach.
Here is an example of iteration relying on destructive assignment, in imperative pseudocode
Pseudocode
In computer science and numerical computation, pseudocode is a compact and informal high-level description of the operating principle of a computer program or other algorithm. It uses the structural conventions of a programming language, but is intended for human reading rather than machine reading...
:
var i, a = 0 // initialize a before iteration for i from 1 to 3 // loop three times
{
a = a + i // increment a by the current value of i
}
print a // the number 6 is printed
In this program fragment, the value of the variable i changes over time, taking the values 1, 2 and 3. This changing value—or mutable state—is characteristic of iteration.
Iteration can be approximated using recursive techniques in functional programming languages. The following example is in Scheme. Note that the following is recursive (a special case of iteration) because the definition of "how to iterate", the iter function, calls itself in order to solve the problem instance. Specifically it uses tail recursion, which is properly supported in languages like Scheme so it does not use large amounts of stack space.
sum
number -> numbe
to sum the first n natural number
(define (sum n)
(if (and (integer? n) (> n 0))
(let iter ([n n] [i 1])
(if (= n 1)
i
(iter (- n 1) (+ n i))))
((assertion-violation
'sum "invalid argument" n))))
In computer programming, an iterator is an object that enables a programmer to traverse a container. Various types of iterators are often provided via a container's interface...
is an object that wraps iteration.
Iteration is also performed using a worksheet, or by using solver or goal seek functions available in Excel. Many implicit equations like the Colebrook equation can be solved in the convenience of a worksheet by designing suitable calculation algorithms.
Many of the engineering problems like solving Colebrook equations reaches 8-digit accuracy in as small as 12 iterations and a maximum of 100 iterations is sufficient to reach a 15-digit accurate result.
.
Project management
Iterations in a project context may refer to the technique of developing and delivering incremental components of business functionality, product development or process design. This is most often associated with agile software development
Agile software development
Agile software development is a group of software development methodologies based on iterative and incremental development, where requirements and solutions evolve through collaboration between self-organizing, cross-functional teams...
, but could potentially be any material. A single iteration results in one or more bite-sized but complete packages of project work that can perform some tangible business function. Multiple iterations recurse to create a fully integrated product. This is often contrasted with the waterfall model
Waterfall model
The waterfall model is a sequential design process, often used in software development processes, in which progress is seen as flowing steadily downwards through the phases of Conception, Initiation, Analysis, Design, Construction, Testing, Production/Implementation and Maintenance.The waterfall...
In computer science a for loop is a programming language statement which allows code to be repeatedly executed. A for loop is classified as an iteration statement....
Iterative and Incremental development is at the liver of a cyclic software development process developed in response to the weaknesses of the waterfall model...
In mathematics, an iterated function is a function which is composed with itself, possibly ad infinitum, in a process called iteration. In this process, starting from some initial value, the result of applying a given function is fed again in the function as input, and this process is repeated...
In computer programming, an iterator is an object that enables a programmer to traverse a container. Various types of iterators are often provided via a container's interface...
In most computer programming languages, a while loop is a control flow statement that allows code to be executed repeatedly based on a given boolean condition. The while loop can be thought of as a repeating if statement....
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Iteration in mathematics may refer to the process of iterating a function