Скорость сходимости

Скорость сходимости является основной характеристикой численных методов решения уравнений и оптимизации.

Понятие скорости сходимости

Пусть $\left\{x_{n}\right\}$ — сходящаяся последовательность приближений некоторого алгоритма нахождения корня уравнения или экстремума функции $x^{*}$ , тогда:

Говорят, что метод обладает линейной сходимостью, если $\exists \alpha \in (0,\;1):\quad \exists N\in \mathbb {N} ,\;\forall n\geq N\quad ||x_{n}-x^{*}||<\alpha ||x_{n-1}-x^{*}||$ .

Говорят, что метод обладает сходимостью степени $\beta$ , если $\exists \alpha \in (0,\;1]:\quad \exists N\in \mathbb {N} ,\;\forall n\geq N\quad ||x_{n}-x^{*}||<\alpha ||x_{n-1}-x^{*}||^{\beta }$ .

Отметим, что обычно скорость сходимости методов не превышает квадратичной. В редких случаях метод может обладать кубической скоростью сходимости (метод Чебышёва).

Практическое определение

Пусть $\left\{x_{n}\right\}$ — последовательность приближений рассматриваемого алгоритма нахождения корня $x^{*}$ некоторого уравнения, тогда скорость сходимости $\beta$ определяют из уравнения:

$||x_{n}-x^{*}||<\alpha ||x_{n-1}-x^{*}||^{\beta }$

Для упрощения его переписывают в виде:

$\log ||x_{n}-x^{*}||<\log \alpha +\beta \log ||x_{n-1}-x^{*}||$

Непосредственно скорость сходимости оценивают по тангенсу угла наклона логарифмического графика зависимости $||x_{n}-x^{*}||$ от $||x_{n-1}-x^{*}||$ .

Литература по теме

Амосов А. А., Дубинский Ю. А., Копченова Н. В. Вычислительные методы для инженеров. — М.: Мир, 1998.
Бахвалов Н. С., Жидков Н. П., Кобельков Г. Г. Численные методы. — 8-е изд.. — М.: Лаборатория Базовых Знаний, 2000.
Волков Е. А. Численные методы. — М.: Физматлит, 2003.

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