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In mathematics, an iterated function is a … In mathematics, an iterated function is a function X → X (that is, a function from some set X to itself) which is obtained by composing another function f : X → X with itself a certain number of times. The process of repeatedly applying the same function is called iteration. In this process, starting from some initial object, the result of applying a given function is fed again in the function as input, and this process is repeated. For example on the image on the right: L = ( K ), M = ( K ) = ( K ),with the circle‑shaped symbol of function composition. Iterated functions are objects of study in computer science, fractals, dynamical systems, mathematics and renormalization group physics.ematics and renormalization group physics.
, 在数学中，迭代函数是在碎形和动力系统中深入研究的对象。迭代函数是重复的与自身复合的函数，这个过程叫做迭代。
, Итерация (от лат. iteratio «повторение») — … Итерация (от лат. iteratio «повторение») — результат повторного применения какой-либо математической операции. Так, если есть некоторая функция от (отображающая область определения в себя), то функции называются соответственно второй, третьей,..., n-й итерациями функции . Например, полагая , получают . Индекс n называют показателем итерации, а переход от функции к — итерированием. Итерации появляются при решении различного рода уравнений и систем уравнений итерационными методами (например метод итерации), которые играют важную роль в теории интегральных уравнений. Итеративные методы играют важную роль в теории игр и математическом программировании.рии игр и математическом программировании.
, En mathématiques, une fonction itérée est … En mathématiques, une fonction itérée est une fonction obtenue par composition répétée d’une autre fonction avec elle-même un certain nombre de fois.La procédure consistant à appliquer la même fonction à plusieurs reprises s’appelle itération. Les fonctions itérées apparaissent en informatique, dans les systèmes dynamiques, les groupes de renormalisation et sont à la base des fractales.alisation et sont à la base des fractales.
, Un algoritmo iterativo è una tipologia di … Un algoritmo iterativo è una tipologia di algoritmo costituito da una sequenza di azioni che viene ripetuta, finché è necessaria la ripetizione stessa (un ciclo). Tutte le operazioni che richiedono la ripetizione di una stessa azione più volte, ma in numero finito sono dette procedure iterative. Ad ogni iterazione, l'esecutore svolge un compito. Al termine verifica se tale compito vada ripetuto mediante una condizione di ripetizione.to mediante una condizione di ripetizione.
, 写像あるいは函数の反復（はんぷく、英: iteration）とは、同じ写像あるいは函数を繰り返し適用する操作である。写像の繰り返しや反復合成とも呼ぶ。ある初期値に写像の反復を適用することで得られる点列を軌道という。 年利で増える残高計算、世代ごとに増減する生物の個体数の計算、ニュートン法のような数値計算で方程式の解を求める問題など、反復によって表すことができるさまざまな科学・数学の問題がある。
, En matemáticas, función iterada es una función que es compuesta consigo misma, en forma repetida, en un proceso llamado iteración. Las funciones iteradas son objeto de profundos estudios en el campo de los fractales y sistemas dinámicos.
, Em matemática, função iterada é uma função que é composta consigo mesma, em forma repetida, em um processo chamado iteração.As funções iteradas são objeto de profundos estudos no campo dos fractais e sistemas dinâmicos.
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在数学中，迭代函数是在碎形和动力系统中深入研究的对象。迭代函数是重复的与自身复合的函数，这个过程叫做迭代。
, Em matemática, função iterada é uma função que é composta consigo mesma, em forma repetida, em um processo chamado iteração.As funções iteradas são objeto de profundos estudos no campo dos fractais e sistemas dinâmicos.
, In mathematics, an iterated function is a … In mathematics, an iterated function is a function X → X (that is, a function from some set X to itself) which is obtained by composing another function f : X → X with itself a certain number of times. The process of repeatedly applying the same function is called iteration. In this process, starting from some initial object, the result of applying a given function is fed again in the function as input, and this process is repeated. For example on the image on the right: L = ( K ), M = ( K ) = ( K ),with the circle‑shaped symbol of function composition.cle‑shaped symbol of function composition.
, En mathématiques, une fonction itérée est … En mathématiques, une fonction itérée est une fonction obtenue par composition répétée d’une autre fonction avec elle-même un certain nombre de fois.La procédure consistant à appliquer la même fonction à plusieurs reprises s’appelle itération. Les fonctions itérées apparaissent en informatique, dans les systèmes dynamiques, les groupes de renormalisation et sont à la base des fractales.alisation et sont à la base des fractales.
, En matemáticas, función iterada es una función que es compuesta consigo misma, en forma repetida, en un proceso llamado iteración. Las funciones iteradas son objeto de profundos estudios en el campo de los fractales y sistemas dinámicos.
, Un algoritmo iterativo è una tipologia di … Un algoritmo iterativo è una tipologia di algoritmo costituito da una sequenza di azioni che viene ripetuta, finché è necessaria la ripetizione stessa (un ciclo). Tutte le operazioni che richiedono la ripetizione di una stessa azione più volte, ma in numero finito sono dette procedure iterative. Ad ogni iterazione, l'esecutore svolge un compito. Al termine verifica se tale compito vada ripetuto mediante una condizione di ripetizione.to mediante una condizione di ripetizione.
, Итерация (от лат. iteratio «повторение») — … Итерация (от лат. iteratio «повторение») — результат повторного применения какой-либо математической операции. Так, если есть некоторая функция от (отображающая область определения в себя), то функции называются соответственно второй, третьей,..., n-й итерациями функции . Например, полагая , получают . Индекс n называют показателем итерации, а переход от функции к — итерированием. Итерации появляются при решении различного рода уравнений и систем уравнений итерационными методами (например метод итерации), которые играют важную роль в теории интегральных уравнений. Итеративные методы играют важную роль в теории игр и математическом программировании.рии игр и математическом программировании.
, 写像あるいは函数の反復（はんぷく、英: iteration）とは、同じ写像あるいは函数を繰り返し適用する操作である。写像の繰り返しや反復合成とも呼ぶ。ある初期値に写像の反復を適用することで得られる点列を軌道という。 年利で増える残高計算、世代ごとに増減する生物の個体数の計算、ニュートン法のような数値計算で方程式の解を求める問題など、反復によって表すことができるさまざまな科学・数学の問題がある。
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, 写像の反復
, Итерация (математика)
, Fonction itérée
, Iterated function
, Función iterada
, Algoritmo iterativo
, 迭代函数
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