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Book3_Ch13_Python_Codes/Bk3_Ch13_01.ipynb

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+###############
+# Authored by Weisheng Jiang
+# Book 3  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2025
+###############
+
+import math  # 导入数学模块
+import numpy as np  # 导入 numpy 模块，用于数值计算
+import matplotlib.pyplot as plt  # 导入 matplotlib 模块，用于绘图
+from matplotlib import cm  # 导入 matplotlib 的 colormap，用于颜色映射
+import streamlit as st  # 导入 streamlit 模块，用于创建交互式界面
+from sympy import symbols, lambdify  # 导入 sympy 模块，用于符号运算
+import plotly.graph_objects as go  # 导入 plotly 的 graph_objects，用于交互式绘图
+
+## 定义生成网格的函数
+def mesh_square(x1_0, x2_0, r, num):
+    rr = np.linspace(-r, r, num)  # 在指定范围生成均匀分布的点
+    xx1, xx2 = np.meshgrid(rr, rr)  # 创建二维网格
+    xx1 = xx1 + x1_0  # 平移网格中心到 x1_0
+    xx2 = xx2 + x2_0  # 平移网格中心到 x2_0
+    return xx1, xx2, rr  # 返回生成的网格和坐标数组
+
+## 定义绘制 3D 曲面的函数
+def plot_surf(xx1, xx2, ff):
+    norm_plt = plt.Normalize(ff.min(), ff.max())  # 归一化颜色映射范围
+    colors = cm.coolwarm(norm_plt(ff))  # 使用 coolwarm 颜色映射
+    fig = plt.figure()  # 创建图形
+    ax = fig.add_subplot(projection='3d')  # 添加 3D 坐标轴
+    surf = ax.plot_surface(xx1, xx2, ff, facecolors=colors, shade=False)  # 绘制曲面
+    surf.set_facecolor((0, 0, 0, 0))  # 设置背景为透明
+    ax.set_xlabel('$\it{x_1}$')  # 设置 x 轴标签
+    ax.set_ylabel('$\it{x_2}$')  # 设置 y 轴标签
+    ax.set_zlabel('$\it{f}$($\it{x_1}$,$\it{x_2}$)')  # 设置 z 轴标签
+    return fig  # 返回绘制的图形
+
+## 定义绘制等高线的函数
+def plot_contourf(xx1, xx2, ff):
+    fig, ax = plt.subplots()  # 创建图形和坐标轴
+    cntr2 = ax.contourf(xx1, xx2, ff, levels=15, cmap="RdBu_r")  # 绘制填充等高线
+    fig.colorbar(cntr2, ax=ax)  # 添加颜色条
+    ax.set_xlabel('$\it{x_1}$')  # 设置 x 轴标签
+    ax.set_ylabel('$\it{x_2}$')  # 设置 y 轴标签
+    ax.grid(linestyle='--', linewidth=0.25, color=[0.5, 0.5, 0.5])  # 设置网格样式
+    return fig  # 返回绘制的图形
+
+## 设置 Streamlit 侧边栏交互控件
+with st.sidebar:
+    st.latex(r'f(x_1,x_2) = ax_1^2 + bx_1x_2 + cx_2^2 + dx_1 + ex_2 + f')  # 显示数学公式
+    a = st.slider('a', min_value=-2.0, max_value=2.0, step=0.1)  # 滑动条选择参数 a
+    b = st.slider('b', min_value=-2.0, max_value=2.0, step=0.1)  # 滑动条选择参数 b
+    c = st.slider('c', min_value=-2.0, max_value=2.0, step=0.1)  # 滑动条选择参数 c
+    d = st.slider('d', min_value=-2.0, max_value=2.0, step=0.1)  # 滑动条选择参数 d
+    e = st.slider('e', min_value=-2.0, max_value=2.0, step=0.1)  # 滑动条选择参数 e
+    f = st.slider('f', min_value=-2.0, max_value=2.0, step=0.1)  # 滑动条选择参数 f
+
+## 初始化网格和符号表达式
+x1, x2 = symbols('x1 x2')  # 定义符号变量
+x1_0, x2_0 = 0, 0  # 设置网格中心
+r, num = 2, 30  # 设置网格半径和点数
+xx1, xx2, x1_array = mesh_square(x1_0, x2_0, r, num)  # 生成网格
+x2_array = x1_array  # y 轴坐标与 x 轴一致
+
+## 使用 matplotlib 可视化
+f_sym = a * x1**2 + b * x1 * x2 + c * x2**2 + d * x1 + e * x2 + f  # 定义目标函数
+f_fcn = lambdify([x1, x2], f_sym)  # 将符号函数转换为数值函数
+ff = f_fcn(xx1, xx2)  # 计算目标函数值
+fig_1 = plot_surf(xx1, xx2, ff)  # 绘制 3D 曲面
+fig_2 = plot_contourf(xx1, xx2, ff)  # 绘制等高线
+
+## 使用 Plotly 可视化
+fig_surface = go.Figure(go.Surface(
+    x=x1_array, y=x2_array, z=ff, showscale=False, colorscale='RdYlBu_r'))  # 绘制交互式 3D 曲面
+fig_surface.update_layout(autosize=True, width=800, height=600)  # 设置布局大小
+st.plotly_chart(fig_surface)  # 在 Streamlit 中显示 3D 曲面图
+
+fig_contour = go.Figure(data=go.Contour(
+    z=ff, x=x1_array, y=x2_array, colorscale='RdYlBu_r'))  # 绘制交互式等高线
+fig_contour.update_layout(autosize=True, width=600, height=600)  # 设置布局大小
+st.plotly_chart(fig_contour)  # 在 Streamlit 中显示等高线图

Book3_Ch13_Python_Codes/Bk3_Ch13_02.ipynb

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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "73bd968b-d970-4a05-94ef-4e7abf990827",
+   "metadata": {},
+   "source": [
+    "Chapter 14\n",
+    "\n",
+    "# 斐波那契数列\n",
+    "Book_3《数学要素》 | 鸢尾花书：从加减乘除到机器学习 (第二版)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9518a62e-6d72-4d77-b6c7-f210be3b5aff",
+   "metadata": {},
+   "source": [
+    "这段代码的核心功能是生成并显示斐波那契数列的前 $n$ 项。斐波那契数列的定义是：\n",
+    "\n",
+    "$$\n",
+    "F(0) = 0, \\quad F(1) = 1\n",
+    "$$\n",
+    "\n",
+    "对于 $n \\geq 2$ 的情况，数列的递推关系为：\n",
+    "\n",
+    "$$\n",
+    "F(n) = F(n-1) + F(n-2)\n",
+    "$$\n",
+    "\n",
+    "每一项等于前两项之和，从而形成一系列逐步递增的值。代码中的函数 `fib(n)` 是递归地定义的，用于计算第 $n$ 项的值。递归关系表示，如果 $n \\leq 1$，函数直接返回 $n$；否则，它会调用自身来计算前两项 $F(n-1)$ 和 $F(n-2)$，并返回它们的和。这种递归调用直到到达基准条件，即 $F(0)$ 或 $F(1)$，从而逐步回溯并计算最终结果。\n",
+    "\n",
+    "在主程序中，定义了变量 `n = 10`，表示我们希望显示数列的前 $10$ 项。然后，代码通过 `for` 循环依次计算每一项 $F(i)$，从 $i = 0$ 到 $i = 9$，并打印输出。这段代码的结果是输出斐波那契数列的前 $10$ 项，通过逐项调用 `fib(i)` 函数并显示计算结果。"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "85c1aae2-7f61-4f09-b193-8567ae2946ce",
+   "metadata": {},
+   "source": [
+    "## 定义斐波那契数列生成函数"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "d15d8d04-e0ee-48d9-85f0-d910263dd962",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def fib(n):\n",
+    "    # 计算斐波那契数列第n项\n",
+    "    if n <= 1:\n",
+    "        return (n)  # 如果n小于等于1，返回n\n",
+    "    else:\n",
+    "        return (fib(n-1) + fib(n-2))  # 否则递归计算前两项之和"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "cda79a3c-5f7a-4fe8-9c7a-85b488586c85",
+   "metadata": {},
+   "source": [
+    "## 输出斐波那契数列的前n项"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "13119ff3-f4ae-4846-94b1-674f97bbeaba",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "0\n",
+      "1\n",
+      "1\n",
+      "2\n",
+      "3\n",
+      "5\n",
+      "8\n",
+      "13\n",
+      "21\n",
+      "34\n"
+     ]
+    }
+   ],
+   "source": [
+    "n = 10  # 设定要显示的项数\n",
+    "for i in range(n):\n",
+    "    print(fib(i))  # 打印斐波那契数列的第i项"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "85a80909-2aac-49ed-bb7a-f8cc6b80ee7d",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ecd322f4-f919-4be2-adc3-69d28ef25e69",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.12.7"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}