SciPy Tutorial - A Powerful Tool for Python Scientific Computing

SciPy Tutorial

SciPy is a Python scientific computing library that provides many useful algorithms and tools for numerical computing, optimization, statistical analysis, signal processing, image processing, linear algebra, and other fields. This tutorial will introduce some common functions of SciPy and give corresponding sample codes.

Install

Install SciPy using pip:

pip install scipy

Numeral Calculations

array operation

SciPy's numpymodules provide a set of functions for array manipulation, such as creating arrays, indexing, slicing, reshaping, transposing, and more. Here are some common examples:

import numpy as np

# 创建数组
a = np.array([1, 2, 3])
b = np.zeros((2, 3))
c = np.ones((3, 4))
d = np.random.rand(2, 2)

# 索引和切片
print(a[0], b[1, 2], c[:, 1], d[0, :])

# 重塑和转置
e = np.reshape(a, (3, 1))
f = np.transpose(b)

Integral and Differential Equations

SciPy's integratemodules provide some integration functions, such as solving definite integrals, numerical integration, differential equations, etc. Here are some common examples:

from scipy import integrate

# 求解定积分
f = lambda x: np.sin(x)
result, error = integrate.quad(f, 0, np.pi)
print(result)

# 数值积分
g = lambda x: np.exp(-x**2)
result = integrate.quad(g, -np.inf, np.inf)
print(result)

# 求解微分方程
h = lambda t, y: -y + np.sin(t)
result = integrate.solve_ivp(h, [0, 5], [0])
print(result.y)

optimization

SciPy's optimizemodule provides some optimization functions, such as minimization function, maximization function, constrained optimization, etc. Here are some common examples:

from scipy import optimize

# 最小化函数
f = lambda x: (x[0] - 1)**2 + (x[1] - 2.5)**2
result = optimize.minimize(f, [0, 0])
print(result.x)

# 最大化函数
g = lambda x: -(x[0]**2 + x[1]**2)
result = optimize.minimize(g, [0, 0])
print(result.x)

# 约束优化
h = lambda x: x[0]**2 + x[1]**2
constraints = [{
    
    'type': 'eq', 'fun': lambda x: x[0] + x[1] - 1}]
result = optimize.minimize(h, [0, 0], constraints=constraints)
print(result.x)

Statistical Analysis

The SciPy statsmodule provides some probability distribution and statistical functions, such as normal distribution, t distribution, F distribution, chi-square distribution, etc. Here are some common examples:

from scipy import stats

# 正态分布
x = np.linspace(-5, 5, 100)
pdf = stats.norm.pdf(x)
cdf = stats.norm.cdf(x)

# t分布
x = np.linspace(-5, 5, 100)
pdf = stats.t.pdf(x, df=10)
cdf = stats.t.cdf(x, df=10)

# F分布
x = np.linspace(0, 5, 100)
pdf = stats.f.pdf(x, dfn=10, dfd=20)
cdf = stats.f.cdf(x, dfn=10, dfd=20)

# 卡方分布
x = np.linspace(0, 10, 100)
pdf = stats.chi2.pdf(x, df=5)
cdf = stats.chi2.cdf(x, df=5)

signal processing

SciPy's signalmodules provide some signal processing functions, such as Fourier transform, filter, convolution, etc. Here are some common examples:

from scipy import signal

# 傅里叶变换
t = np.linspace(0, 1, 1000)
x = np.sin(2 * np.pi * 5 * t) + np.sin(2 * np.pi * 10 * t)
y = np.fft.fft(x)
freq = np.fft.fftfreq(len(x), t[1] - t[0])

# 滤波器
b, a = signal.butter(4, 0.2)
x = np.random.rand(1000)
y = signal.filtfilt(b, a, x)

# 卷积
x = np.array([1, 2, 3])
y = np.array([4, 5, 6])
z = np.convolve(x, y)

Linear Algebra

The SciPy linalgmodule provides some linear algebra functions, such as matrix inversion, matrix eigenvalue, singular value decomposition, etc. Here are some common examples:

from scipy import linalg

# 矩阵求逆
a = np.array([[1, 2], [3, 4]])
b = linalg.inv(a)

# 矩阵求特征值和特征向量
a = np.array([[1, 2], [3, 4]])
w, v = linalg.eig(a)
python
# 奇异值分解
a = np.array([[1, 2], [3, 4], [5, 6]])
u, s, vh = linalg.svd(a)

This tutorial introduces some commonly used functions of SciPy, including numerical computation, optimization, statistical analysis, signal processing, linear algebra, etc. These functions can help us solve many practical problems, such as numerical integration, function optimization, signal filtering, data fitting, etc. By studying this tutorial, we can better understand the functions and usage of SciPy, and further improve the ability of Python scientific computing.

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