numpy.gradient

numpy.gradient(f, *varargs, axis=None, edge_order=1) [source]

Возвращает градиент N-мерного массива.

Градиент вычисляется с использованием центральных разностей второго порядка в внутренних точках и либо разностей первого или второго порядка (прямой или обратной) на границах. Полученный градиент имеет ту же форму, что и входной массив.

Параметры

farray_like

N-мерный массив, содержащий образцы скалярной функции.

varargslist of scalar or array, optional

Расстояние между значениями f. По умолчанию единичное расстояние для всех измерений. Расстояние может быть задано с помощью:

единичного скаляра для задания расстояния между образцами для всех измерений.
N скаляров для задания постоянного расстояния между образцами для каждого измерения. Например, dx, dy, dz, …
N массивов для задания координат значений вдоль каждого измерения F. Длина массива должна соответствовать размеру соответствующего измерения
Любое сочетание N скаляров/массивов со значением 2. и 3.

Если axis задан, количество varargs должно быть равно количеству осей. По умолчанию: 1.

edge_order{1, 2}, optional

Градиент вычисляется с использованием разностей N-го порядка на границах. По умолчанию: 1.

Новая в версии 1.9.1.

axisNone or int or tuple of ints, optional

Градиент вычисляется только вдоль заданной оси или осей. По умолчанию (axis = None) вычисляется градиент для всех осей входного массива. axis может быть отрицательным, в этом случае он отсчитывается от последней к первой оси.

Новая в версии 1.11.0.

Возвращает

gradientndarray or list of ndarray: Набор ndarrays (или один ndarray, если есть только одно измерение), соответствующий производным f по каждому измерению. Каждая производная имеет ту же форму, что и f.

Примечания

Предполагая, что $f\in C^{3}$ (т.е., $f$ имеет как минимум 3 непрерывные производные) и пусть

$h_{*}$ является неоднородным шагом, мы минимизируем «ошибку согласованности» $\eta_{i}$ между истинным градиентом и его оценкой из линейной комбинации соседних узлов сетки:

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<img alt="\eta_{i} = f_{i}^{\left(1\right)} - \left[ \alpha f\left(x_{i}\right) + \beta f\left(x_{i} + h_{d}\right) + \gamma f\left(x_{i}-h_{s}\right) \right]" src="data:image/svg+xml;base64,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

Подставив

$f(x_{i} + h_{d})$

и $f(x_{i} - h_{s})$ со своим разложением в ряд Тейлора, это переводится в решение следующей системы линейных уравнений:

<img alt="{\displaystyle \left\{ \begin{array}{r} \alpha+\beta+\gamma=0 \\ \beta h_{d}-\gamma h_{s}=1 \\ \beta h_{d}^{2}+\gamma h_{s}^{2}=0 \end{array} \right.}" 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Полученное приближение для $f_{i}^{(1)}$ равно следующему:

<img alt="{\hat f_{i}^{(1)}} = \frac{ h_{s}^{2}f\left(x_{i} + h_{d}\right) + \left(h_{d}^{2} - h_{s}^{2}\right)f\left(x_{i}\right) - h_{d}^{2}f\left(x_{i}-h_{s}\right)} { h_{s}h_{d}\left(h_{d} + h_{s}\right)} + \mathcal{O}\left(\frac{h_{d}h_{s}^{2} + h_{s}h_{d}^{2}}{h_{d} + h_{s}}\right)" src="data:image/svg+xml;base64,<?xml version='1.0' encoding='UTF-8'?>
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Стоит отметить, что если $h_{s}=h_{d}$ (т.е. данные равномерно распределены), мы получаем стандартное приближение второго порядка:

<img alt="\hat f_{i}^{(1)}= \frac{f\left(x_{i+1}\right) - f\left(x_{i-1}\right)}{2h} + \mathcal{O}\left(h^{2}\right)" src="data:image/svg+xml;base64,<?xml version='1.0' encoding='UTF-8'?>
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С помощью аналогичной процедуры можно получить приближения вперёд/назад, используемые для границ.

Ссылки

1: Квартеррони А., Сакко Р., Салери Ф. (2007) Численные методы (Тексты по прикладной математике). Нью-Йорк: Springer.
2: Дурран Д. Р. (1999) Численные методы для волновых уравнений в геофизической гидродинамике. Нью-Йорк: Springer.
3: Форнберг Б. (1988) Генерация формул конечных разностей на произвольно расположенных сетках, Математика вычислений 51, № 184: 699-706. PDF.

Примеры

>>> f = np.array([1, 2, 4, 7, 11, 16], dtype=float)
>>> np.gradient(f)
array([1. , 1.5, 2.5, 3.5, 4.5, 5. ])
>>> np.gradient(f, 2)
array([0.5 ,  0.75,  1.25,  1.75,  2.25,  2.5 ])

Шаг также можно указать с помощью массива, представляющего координаты значений F по измерениям. Например, равномерный шаг:

>>> x = np.arange(f.size)
>>> np.gradient(f, x)
array([1. ,  1.5,  2.5,  3.5,  4.5,  5. ])

Или неравномерный:

>>> x = np.array([0., 1., 1.5, 3.5, 4., 6.], dtype=float)
>>> np.gradient(f, x)
array([1. ,  3. ,  3.5,  6.7,  6.9,  2.5])

Для двумерных массивов возвращаемые значения будут двумя массивами, упорядоченными по оси. В этом примере первый массив представляет градиент по строкам, а второй — по столбцам:

>>> np.gradient(np.array([[1, 2, 6], [3, 4, 5]], dtype=float))
[array([[ 2.,  2., -1.],
       [ 2.,  2., -1.]]), array([[1. , 2.5, 4. ],
       [1. , 1. , 1. ]])]

В этом примере шаг также указан: равномерный для оси=0 и неравномерный для оси=1

>>> dx = 2.
>>> y = [1., 1.5, 3.5]
>>> np.gradient(np.array([[1, 2, 6], [3, 4, 5]], dtype=float), dx, y)
[array([[ 1. ,  1. , -0.5],
       [ 1. ,  1. , -0.5]]), array([[2. , 2. , 2. ],
       [2. , 1.7, 0.5]])]

Можно указать, как обрабатываются границы, используя edge_order

>>> x = np.array([0, 1, 2, 3, 4])
>>> f = x**2
>>> np.gradient(f, edge_order=1)
array([1.,  2.,  4.,  6.,  7.])
>>> np.gradient(f, edge_order=2)
array([0., 2., 4., 6., 8.])

Ключевое слово axis можно использовать для указания подмножества осей, для которых вычисляется градиент

>>> np.gradient(np.array([[1, 2, 6], [3, 4, 5]], dtype=float), axis=0)
array([[ 2.,  2., -1.],
       [ 2.,  2., -1.]])

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