“iteration”的概念、定义、翻译、参考文献-科学参考

iteration

Physics

The process of successive approximations used as a technique for solving a mathematical problem. The technique can be used manually but is widely used by computers.

Mathematics

A method uses iteration if it yields successive approximations to a required value by repetition of a certain procedure. Examples are fixed‐point iteration and Newton’s method for finding a root of an equation f(x) = 0.

Statistics

See iterative algorithm.

Chemical Engineering

A repeating calculation in which the key variables are recycled in the calculation in order eventually to reach a convergence as the solution.

Computer

1. The repetition of a numerical or nonnumerical process where the results from one or more stages are used to form the input to the next. Generally the recycling of the process continues until some preset bound is achieved, or the process result is constantly repeated. This is one of the key ideas used in the design of numerical methods (see also iterative methods).
An iterative process is m-stage if the new value is derived from m previous values; it is m-stage, sequential if the new value depends upon the last m values, i.e.
$x^{k + 1} = G k (x^{k}, x^{k - 1}, \dots, x^{k - m + 1})$
The iteration is stationary if the function Gk is independent of k, i.e. the new value is calculated from the old values using the same formula. For example,
$x^{k + 1} = \frac{1}{2} (x^{k} + a / x^{k})$
is a stationary, one-stage iteration (used for evaluating the square root of a); this is a particular application of Newton’s method. The secant method is a stationary two-stage sequential iteration. False position is an example of a nonsequential iteration.
2. (of a formal language). See Kleene star.

Logic

In propositional dynamic logic, a method of constructing a program $α^{*}$ from a simpler program $α$ . The intended interpretation of $α^{*}$ is an instruction to carry out program $α$ some arbitrary but finite number of times.

单词	iteration
释义	iteration Physics The process of successive approximations used as a technique for solving a mathematical problem. The technique can be used manually but is widely used by computers. Mathematics A method uses iteration if it yields successive approximations to a required value by repetition of a certain procedure. Examples are fixed‐point iteration and Newton’s method for finding a root of an equation f(x) = 0. Statistics See iterative algorithm. Chemical Engineering A repeating calculation in which the key variables are recycled in the calculation in order eventually to reach a convergence as the solution. Computer 1. The repetition of a numerical or nonnumerical process where the results from one or more stages are used to form the input to the next. Generally the recycling of the process continues until some preset bound is achieved, or the process result is constantly repeated. This is one of the key ideas used in the design of numerical methods (see also iterative methods). An iterative process is m-stage if the new value is derived from m previous values; it is m-stage, sequential if the new value depends upon the last m values, i.e. $x^{k + 1} = G k (x^{k}, x^{k - 1}, \dots, x^{k - m + 1})$ The iteration is stationary if the function Gk is independent of k, i.e. the new value is calculated from the old values using the same formula. For example, $x^{k + 1} = \frac{1}{2} (x^{k} + a / x^{k})$ is a stationary, one-stage iteration (used for evaluating the square root of a); this is a particular application of Newton’s method. The secant method is a stationary two-stage sequential iteration. False position is an example of a nonsequential iteration. 2. (of a formal language). See Kleene star. Logic In propositional dynamic logic, a method of constructing a program $α^{}$ from a simpler program $α$ . The intended interpretation of $α^{}$ is an instruction to carry out program $α$ some arbitrary but finite number of times.
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