Scipy 简明教程

SciPy - Special Package

特殊包中提供的函数是通用函数，遵循广播和自动数组循环。

The functions available in the special package are universal functions, which follow broadcasting and automatic array looping.

让我们看看一些最常用的特殊函数 −

Let us look at some of the most frequently used special functions −

  1. Cubic Root Function

  2. Exponential Function

  3. Relative Error Exponential Function

  4. Log Sum Exponential Function

  5. Lambert Function

  6. Permutations and Combinations Function

  7. Gamma Function

我们现在简要了解一下这些函数中的每一个函数。

Let us now understand each of these functions in brief.

Cubic Root Function

此三次方根函数的语法为 – scipy.special.cbrt(x)。这将获取 x 的按元素立方根。

The syntax of this cubic root function is – scipy.special.cbrt(x). This will fetch the element-wise cube root of x.

让我们考虑以下示例。

Let us consider the following example.

from scipy.special import cbrt
res = cbrt([10, 9, 0.1254, 234])
print res

上述程序将生成以下输出。

The above program will generate the following output.

[ 2.15443469 2.08008382 0.50053277 6.16224015]

Exponential Function

指数函数的语法为 – scipy.special.exp10(x)。这将按元素计算 10**x。

The syntax of the exponential function is – scipy.special.exp10(x). This will compute 10**x element wise.

让我们考虑以下示例。

Let us consider the following example.

from scipy.special import exp10
res = exp10([2, 9])
print res

上述程序将生成以下输出。

The above program will generate the following output.

[1.00000000e+02  1.00000000e+09]

Relative Error Exponential Function

此函数的语法为 – scipy.special.exprel(x)。它生成相对误差指数，(exp(x) - 1)/x。

The syntax for this function is – scipy.special.exprel(x). It generates the relative error exponential, (exp(x) - 1)/x.

当 x 接近零时，exp(x) 接近 1，因此 exp(x) - 1 的数值计算会遭受精度灾难性损失。然后实现 exprel(x) 以避免当 x 接近零时发生的精度损失。

When x is near zero, exp(x) is near 1, so the numerical calculation of exp(x) - 1 can suffer from catastrophic loss of precision. Then exprel(x) is implemented to avoid the loss of precision, which occurs when x is near zero.

让我们考虑以下示例。

Let us consider the following example.

from scipy.special import exprel
res = exprel([-0.25, -0.1, 0, 0.1, 0.25])
print res

上述程序将生成以下输出。

The above program will generate the following output.

[0.88479687 0.95162582 1.   1.05170918 1.13610167]

Log Sum Exponential Function

此函数的语法为 – scipy.special.logsumexp(x)。它有助于计算输入元素指数之和的对数。

The syntax for this function is – scipy.special.logsumexp(x). It helps to compute the log of the sum of exponentials of input elements.

让我们考虑以下示例。

Let us consider the following example.

from scipy.special import logsumexp
import numpy as np
a = np.arange(10)
res = logsumexp(a)
print res

上述程序将生成以下输出。

The above program will generate the following output.

9.45862974443

Lambert Function

此函数的语法为 – scipy.special.lambertw(x)。它也被称为 Lambert W 函数。Lambert W 函数 W(z) 定义为 w * exp(w) 的反函数。换句话说，对于任何复数 z，W(z) 的值满足 z = W(z) * exp(W(z))。

The syntax for this function is – scipy.special.lambertw(x). It is also called as the Lambert W function. The Lambert W function W(z) is defined as the inverse function of w * exp(w). In other words, the value of W(z) is such that z = W(z) * exp(W(z)) for any complex number z.

Lambert W 函数是一个多分支函数，具有无限多个分支。每个分支给出方程 z = w exp(w) 的一个解。此处，分支由整数 k 索引。

The Lambert W function is a multivalued function with infinitely many branches. Each branch gives a separate solution of the equation z = w exp(w). Here, the branches are indexed by the integer k.

我们考虑以下示例。这里，Lambert W 函数是 w exp(w) 的反函数。

Let us consider the following example. Here, the Lambert W function is the inverse of w exp(w).

from scipy.special import lambertw
w = lambertw(1)
print w
print w * np.exp(w)

上述程序将生成以下输出。

The above program will generate the following output.

(0.56714329041+0j)
(1+0j)

Permutations & Combinations

让我们先讨论排列和组合，以便能清楚地理解它们。

Let us discuss permutations and combinations separately for understanding them clearly.

Combinations − 组合函数的语法是 - scipy.special.comb(N,k)。让我们考虑以下示例 -

Combinations − The syntax for combinations function is – scipy.special.comb(N,k). Let us consider the following example −

from scipy.special import comb
res = comb(10, 3, exact = False,repetition=True)
print res

上述程序将生成以下输出。

The above program will generate the following output.

220.0

Note − 仅在 exact = False 的情况下接受数组参数。如果 k > N、N < 0 或 k < 0，则返回 0。

Note − Array arguments are accepted only for exact = False case. If k > N, N < 0, or k < 0, then a 0 is returned.

Permutations − 排列函数的语法是 - scipy.special.perm(N,k)。一次对 N 个对象进行 k 个排列，即 N 的 k 个排列。这也称为“部分排列”。

Permutations − The syntax for combinations function is – scipy.special.perm(N,k). Permutations of N things taken k at a time, i.e., k-permutations of N. This is also known as “partial permutations”.

让我们考虑以下示例。

Let us consider the following example.

from scipy.special import perm
res = perm(10, 3, exact = True)
print res

上述程序将生成以下输出。

The above program will generate the following output.

720

Gamma Function

伽马函数通常被称为广义阶乘，因为 z*gamma(z) = gamma(z+1) 并且 gamma(n+1) = n!，其中“n”是自然数。

The gamma function is often referred to as the generalized factorial since z*gamma(z) = gamma(z+1) and gamma(n+1) = n!, for a natural number ‘n’.

组合函数的语法是 - scipy.special.gamma(x)。一次对 N 个对象进行 k 个排列，即 N 的 k 个排列。这也称为“部分排列”。

The syntax for combinations function is – scipy.special.gamma(x). Permutations of N things taken k at a time, i.e., k-permutations of N. This is also known as “partial permutations”.

组合函数的语法是 - scipy.special.gamma(x)。一次对 N 个对象进行 k 个排列，即 N 的 k 个排列。这也称为“部分排列”。

The syntax for combinations function is – scipy.special.gamma(x). Permutations of N things taken k at a time, i.e., k-permutations of N. This is also known as “partial permutations”.

from scipy.special import gamma
res = gamma([0, 0.5, 1, 5])
print res

上述程序将生成以下输出。

The above program will generate the following output.

[inf  1.77245385  1.  24.]