### A Pluto.jl notebook ###
# v0.19.32

#> [frontmatter]
#> title = "使用Julia进行科学计算"
#> date = "2023-12-09"
#> description = "介绍用Julia进行科学计算的基础知识和案例，包括矩阵运算、优化问题、求解微分方程等"
#> 
#>     [[frontmatter.author]]
#>     name = "陈伟凯"
#>     url = "https://weikaichen.gitlab.io/"

using Markdown
using InteractiveUtils

# ╔═╡ 12b8cfbb-54f6-4f21-b32f-d6907d25b61f
using LinearAlgebra

# ╔═╡ f83777c2-ca3d-4f16-84a9-dfeb3a1a1687
using JuMP

# ╔═╡ bd3576f5-a74c-4a7c-b048-c27556b8130b
using Optimization, OptimizationNLopt, ForwardDiff

# ╔═╡ db17bfd6-af45-4221-b1d5-89fa78c09540
using DifferentialEquations

# ╔═╡ 989a5785-d79c-4d85-aad1-6324665e3bea
using ModelingToolkit

# ╔═╡ fdb1f332-07aa-4151-b19b-ef7ac9320392
begin
	using PlutoUI
	TableOfContents(title="📚目录", depth = 4)
end

# ╔═╡ ce899fc3-f69e-41e5-b24d-3d8a3517c9cb
lastname = "陈"

# ╔═╡ 5982fabc-b8c4-4bd6-94a9-dd489524d11f
firstname = "某某"

# ╔═╡ e164cfde-b6e9-48bb-9d44-94a0815db1f9
fullname = lastname * firstname

# ╔═╡ 0ec80455-532d-4774-a5fd-e052fad4329b
id = "20201209"

# ╔═╡ 1fef0dc0-92aa-11ee-09c2-135124aaf0ae
md"""
# 使用Julia进行科学计算

[**陈伟凯**](https://weikaichen.gitlab.io/)·中国人民大学经济学院·2023/12/9


本节介绍使用Julia进行科学计算的基础知识，包括矩阵运算，最优化问题以及微分方程求解。

!!! warning "注意"
	请在Pluto Notebook中完成练习，我们将根据同学们最后提交的Notebook给成绩，请修改后面的代码，确保你的姓名和学号在下表中正确显示


| 姓名 | 学号|
|:---|----:|
| $fullname | $id    |
"""


# ╔═╡ d36fb63b-acbc-411a-a1a4-f98780f6ba92
md"""
----------------
"""

# ╔═╡ 97d8d58a-7212-4d98-a530-12ce2e7f38aa
md"""
## 一、 矩阵运算

### 1. `LinearAlgebra` 模块简介

Julia 的 `LinearAlgebra` 模块是为了处理线性代数运算而设计的，提供了许多用于矩阵和向量操作的功能。该模块包含了一系列用于解线性方程组、计算特征值和特征向量、矩阵分解等任务的函数。Julia 的 `LinearAlgebra` 采用了高效的实现，旨在提供快速且可扩展的线性代数操作。

以下是 `LinearAlgebra` 模块的一些关键功能：

1. **矩阵和向量运算：** 提供了常用的线性代数运算，如矩阵乘法、向量点积、矩阵转置等。

2. **特征值和特征向量：** 可以通过 `eigen` 函数计算矩阵的特征值和特征向量，这在许多科学和工程应用中是关键的操作。

3. **奇异值分解（SVD）：** 通过 `svd` 函数进行奇异值分解，这在数据降维和矩阵逆的计算中经常用到。

4. **LU 分解和 QR 分解：** 通过 `lu` 和 `qr` 函数，可以进行LU分解和QR分解，用于解线性方程组和求解最小二乘问题。

5. **逆和行列式：** 通过 `inv` 函数计算矩阵的逆，以及 `det` 函数计算矩阵的行列式。

6. **Cholesky 分解：** 提供了 `cholesky` 函数用于进行 Cholesky 分解，特别适用于正定对称矩阵。

7. **伪逆：** 通过 `pinv` 函数可以计算矩阵的伪逆。

`LinearAlgebra` 模块中的这些功能使得 Julia 成为进行科学计算和工程仿真的强大工具，尤其是在需要高性能数值计算的场景下。Julia 通过其创新性的设计和即时编译技术，为用户提供了高性能和易用性的线性代数工具。

下面我们通过例子介绍一些常见的函数，更多内容可参考[文档](https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/) 和以下书籍：

- [Introduction to Applied Linear Algebra – Vectors, Matrices, and Least Squares](https://web.stanford.edu/~boyd/vmls/)
- [Numerical Linear Algebra with Julia | SIAM Publications Library](https://epubs.siam.org/doi/10.1137/1.9781611976557)
"""

# ╔═╡ b57b29ab-bfa8-41e6-9dbe-3b999617f0ff
md"""
### 2. 向量和矩阵及其运算
"""

# ╔═╡ 2eb13948-5502-4c22-a314-f3a62dbb8e0a
md"""
#### （1） 定义
"""

# ╔═╡ 318ba4e0-1d1c-419a-a9eb-56a363b766cd
md"""
在 Julia 中向量可以用数组表示， 下面是向量和矩阵的定义方式。
"""

# ╔═╡ 80eb8afd-c95b-406f-a94d-dcf8f83f36d0
vector = [1, 2] # column vector

# ╔═╡ fc760eb4-65f2-45f9-a06b-96924b5dc98d
typeof(vector)

# ╔═╡ 6b72bf4a-d00a-479a-9345-3bd4ba0bc8ca
md"""
注意不要忘了中间的逗号（也可以用分号），否则就会变成一个``1\times 2`` 的矩阵：
"""

# ╔═╡ d4f58ea9-9317-41fb-9233-0613043b3308
[1 2] 

# ╔═╡ e11680af-a905-4de0-8b77-3f6c25ab8b9e
matrix = [1 2; 3 4]

# ╔═╡ 95b85062-bab0-4f8b-96c8-e49d921d8560
size(matrix) # dimensions

# ╔═╡ 367e5e30-dd8e-450a-8b30-4f840f448592
md"""
几种特殊的矩阵：
- 零矩阵： ```zeros(size)```
- 全1矩阵： ``` ones(size)```
- 单位矩阵: ```I```
- 随机矩阵: ``` rand(size)``` 或 ``` randn(size)```
"""

# ╔═╡ 24f798b1-0af2-4361-b1a9-507ab354fd44
zeros(2,3)

# ╔═╡ 306525fb-d178-4b3c-9ade-d49fd6115fdd
zeros(3,3)+I

# ╔═╡ b69dd3b2-b61c-4940-9271-6ad7a7bc5006
ones(2,3)

# ╔═╡ a03bd7a2-e658-465c-beed-dd60cf0d7dc7
randn(2,3) # standard normally-distributed

# ╔═╡ 02f8cc08-b799-450b-b154-9c5a418726a8
rand(2,3) # uinform distribution over [0,1]

# ╔═╡ 11bf188b-70ab-48cf-834a-8ce8fcb7528d
md"""
#### （2） 基本运算和常用函数
"""

# ╔═╡ fae318d3-bd9b-4031-9f43-39b9059ffebe
md"""
矩阵的基本运算和常用函数包括：

| 描述 | 运算或函数 |
| :--- | :---: |
| 相加和相减 | `+`、`-` |
| 内积 | `dot()` |
| 乘法 | `*` |
| 转置 | `transpose()`、`'`|
| 求逆 | `inv()` |
| 行列式 | `det()` |
| 特征值和特征向量 | `eigen()` |
| 秩 | `rank()` |
| 范数 | `norm()` |
| 迹 | `tr()` |
| 幂次方 | `^` |
| 对角线元素 | `diag()` |
| 行数和列数 | `size()` |
| 维度 | `ndims()` |
"""

# ╔═╡ 28ed3b6a-99dc-44dc-a08c-ee81376e5768
matrix * vector

# ╔═╡ 73ed64b9-de73-4849-87d4-bc3abfd67498
matrix'

# ╔═╡ d968e1c1-3cd7-445e-92a8-d26761bac138
transpose(matrix)

# ╔═╡ fe83bad7-f2d7-4086-852f-854a731c3657
inv(matrix) * matrix

# ╔═╡ 629c7d4c-1307-4489-8b4f-8d91d9859015
det(matrix) 

# ╔═╡ 39a56c5b-069f-4351-ad17-c5cbe2ce89b4
matrix[:,2] # the second column

# ╔═╡ b63ff98e-c1f1-486b-9bad-4f3862d8dee4
λ, v = eigen(matrix)

# ╔═╡ 810e06a7-a577-480c-a8af-6b01219d147f
matrix * v[:,1]

# ╔═╡ 3bbf5e7d-8d42-461c-84ca-58874e095efd
λ[1] * v[:,1]

# ╔═╡ 749e1bba-7a0c-4315-b89c-bae5ca0a297b
rank(matrix) == rank(matrix')

# ╔═╡ 2ee40583-e641-4d28-8b6e-d2d3410cb6f5
norm(matrix)

# ╔═╡ 7afa29c6-c0bc-4d93-b97b-76a792bfe116
tr(matrix) == sum(diag(matrix))

# ╔═╡ 46273ffa-b284-4fcc-beb7-de631ddba4dd
size(ones(2,3))

# ╔═╡ c5ad3fa6-bc21-40bb-996d-716d5065b37d
ndims(zeros(2,3))

# ╔═╡ 7bef2ef3-e82f-4f3c-b689-6dd1dd566ead
md"""
!!! info "练习"
	**现值**。 用向量 ``c`` 表示现金流， ``r`` 代表利率，各期的贴现因子可以表示为 ``(1+r)^{-t}``,则现值可以表示为什么？
"""

# ╔═╡ 874b3bad-77b2-4a2a-b902-8b327ab50289
c = repeat([100],8) #cahs flow

# ╔═╡ f06b5708-e545-489e-a791-93affbb382c9
r = 0.05 #5% per-period interest rate

# ╔═╡ 740cd4b6-d813-4321-b01f-813d771044c1
function PV(c,r)
	# your code below
	return 
end

# ╔═╡ a41a186a-7d82-4a51-a995-5239c5ef05b2
md"""
#### （4） 解线性方程组

求解方程 `` Ax = b``。 当``A``可逆时，我们知道 ``x= A^{-1}b``. Julia可以用```A\b```来求解，或者用 ```inv(A)*b```，下面比较两种计算方法的效率。
"""

# ╔═╡ e542b73c-fb20-4ebc-b91d-3734cdcccb89
dim = 100

# ╔═╡ 36557a21-64c6-4e4c-8d36-8c07f59fcf0d
randnmatrix = randn(dim,dim)

# ╔═╡ 9702e6cf-b2e2-4361-b40b-f170b6329806
randnvector = randn(dim)

# ╔═╡ 90cf8142-c01d-4ef3-a921-b6d161f1e9c8
@time x1 = randnmatrix\randnvector

# ╔═╡ 50554ed5-62bb-4831-83fc-04906234e38b
@time x2 = inv(randnmatrix) * randnvector

# ╔═╡ 0dcb016d-9911-47e6-99f9-3692a6276284
norm(randnvector - randnmatrix * x1)

# ╔═╡ 73692f8e-b04b-4e93-a33f-11c02d9d7739
norm(randnvector - randnmatrix * x2)

# ╔═╡ 24e57996-0384-4bf5-a774-c4661bd89d66
md"""
可以看出用 ```A\b``` 更好，这是因为它会自动根据矩阵的特性选择合适的算法（如LU分解）。这在矩阵 ``A`` 是特殊矩阵（如下三角、稀疏矩阵等）时表现更为突出。因此，采用推荐使用 ```A\b```。
"""

# ╔═╡ 48a339b5-bf1d-4032-8e53-e0ed79880d9f
md"""
!!! info "练习"
	给定简单线性生产体系``(A,\ell)``，写一个函数返回劳动价值向量。
"""

# ╔═╡ a5a3bf09-f996-495d-acfb-206489c3850a
A = [0.3 0.4; 0.2 0.5]

# ╔═╡ 04113621-00bb-481c-a950-45b88e99ec78
ℓ = [0.2, 0.3] # note that this is a column vector

# ╔═╡ b93e1c46-223f-42e8-8728-c11ac8359a10
function labor_value(A,ℓ)
	# revise the code below
	v = zero(length(ℓ))  #a column vector of labor value
	return v 
end

# ╔═╡ bab094b7-80fb-409a-b815-d705c0ad3be5
md"""
## 二、 优化问题

### 1. 简介

Julia目前有比较多成熟的Package处理优化问题，比较流行的方案有`JuMP.jl` 和 `Optimization.jl`，它们的设计理念和用途有一些不同，使用那一个库要根据具体的问题。以下是它们的主要区别和特点的比较：

- `JuMP.jl`:
   1. **建模语言：** `JuMP` 是一个建模语言，其主要设计目的是提供一个直观而高级的接口，使用户能够以数学方式自然地表示优化问题。
   2. **自动化：** `JuMP` 处理了很多与优化相关的底层细节，如求解器接口、约束管理等，使用户能够更专注于问题的建模而非求解细节。
   3. **易用性：** 由于其建模语言的特性，`JuMP` 对于初学者来说相对容易上手，尤其适用于需要频繁更改和调整问题的场景。

- `Optimization.jl`:
   1. **底层控制：** `Optimization.jl` 更注重对底层优化算法的直接控制。它提供了一组优化算法和工具，使用户能够更精确地控制算法的参数和行为。
   2. **连续优化：** `Optimization.jl` 主要专注于连续优化问题，适用于需要最小化或最大化目标函数的情况，同时支持约束和无约束问题。
   3. **扩展性：** 由于 `Optimization.jl` 是更底层的库，用户可以比较容易地实现自定义的优化算法或进行更高级的定制。 

"""

# ╔═╡ ab2abce0-8a20-40c6-87f1-1dff4bd123e0
md"""
### 2. `JuMP.jl` 优化示例 

我们首先简单地介绍两个使用 `JuMP.jl` 的例子。
"""

# ╔═╡ c5b5bd50-7c1a-49b5-86d5-108c65345144
md"""
#### （1） 背包问题

背包问题是一个经典的优化问题：给定一组物品和一个具有固定容量的容器，选择一个具有最大组合值的物品子集，既能装入容器，又不超出容量。

它类似于旅行打包，行李重量限制就是容量，目标是打包最有利可图的物品组合。我们可以把背包问题表述为整数线性规划：
```math
\begin{aligned}
\max \; & \sum_{i=1}^n c_i x_i      \\
s.t. \; & \sum_{i=1}^n w_i x_i \le C, \\
        & x_i \in \{0,1\},\quad \forall i=1,\ldots,n,
\end{aligned}
```
其中 ``C`` 是容量，要在 ``n`` 个物品中选择，每个物品的重量是 ``w_i``， 价值为 ``c_i``， ``x_i=1``代表物品 ``i`` 被选上。
"""

# ╔═╡ a290cff5-e133-4bd6-b107-f98b99a45768
md"""
假定一共有5个物品
"""

# ╔═╡ bc08814b-06a3-4cce-b61c-341e236a07a6
n = 5

# ╔═╡ f7b7abfe-ec6b-4607-8ff3-10b5922821e3
md"""
假定容量为10， 价值和重量如下
"""

# ╔═╡ 2849b64a-4ded-4dac-a79e-083a5f4fce99
begin
	capacity = 10.0;
	profit = [5.0, 3.0, 2.0, 7.0, 4.0];
	weight = [2.0, 8.0, 4.0, 2.0, 5.0];
end

# ╔═╡ 6b64adff-9cee-436d-9295-dce838ee4577
md"""
接着我们在一个 `JuMP` 模型表示以上背包问题。首先要调用 `JuMP`库，并采用 `HiGHS`求解器。
"""

# ╔═╡ 2f2a0b69-402b-4973-9aa7-32ccd38077b4
import HiGHS

# ╔═╡ a11e7736-fa65-4051-b086-c46f6f3dbb90
md"""
首先，我们将创建一个模型对象，用于在构建每个部件时保存模型元素。我们还将设置求解器，一旦模型构建完成，最终将调用该求解器来求解模型。
"""

# ╔═╡ 231ccf62-98b3-4c7d-ad11-414e478ba5a9
md"""
``` julia
 model = Model(HiGHS.Optimizer)
```
"""

# ╔═╡ fc45d891-69ca-4a5a-bdfa-8d24617fa6c9
md"""
接下来，我们需要代表选择哪些项目的决策变量：
"""

# ╔═╡ dd8fae4c-02d9-4918-84fb-a6710a2034d1
md"""
``` julia
@variable(model, x[1:n], Bin) #Bin 代表这是一个Binary变量，取值为0或1
```
"""

# ╔═╡ 2c258ed5-6193-4a15-be04-83b2c38249c3
md"""
我们现在要对这些变量进行限制，使它们的总权重小于或等于给定的容量：
"""

# ╔═╡ 40233233-b7ce-48ae-b380-b2b7f8e1ddb4
md"""
```julia
@constraint(model, sum(weight[i] * x[i] for i in 1:n) <= capacity)
```
"""

# ╔═╡ bf8b1c35-69cc-4513-a6b1-da3fcbf92259
md"""
最后，我们的目标是使所选物品的价值最大化：
"""

# ╔═╡ be0404e4-3394-45e4-933b-fc99a1ed39bf
md"""
```julia
@objective(model, Max, sum(profit[i] * x[i] for i in 1:n))
```
"""

# ╔═╡ dc90bb53-24be-437b-a2a6-0f28fb68e5ae
md"""
现在，我们可以解决优化问题并检查结果。
"""

# ╔═╡ 99133210-655e-418e-b7ab-252d2f68d5eb
# ╠═╡ show_logs = false
md"""
```julia
JuMP.optimize!(model)
```
"""

# ╔═╡ b9a7ffd0-9443-471d-820c-20990ad44e1f
md"""
我们可以把背包问题的解写成下面的函数 `solve_knapsack_problem`
"""

# ╔═╡ b38fe88b-1b65-447c-b444-6e7bf65d8755
function solve_knapsack_problem(;
    profit::Vector{Float64},
    weight::Vector{Float64},
    capacity::Float64,
)
    n = length(weight)
    # The profit and weight vectors must be of equal length.
    @assert length(profit) == n
    model = Model(HiGHS.Optimizer)
    set_silent(model)
    @variable(model, x[1:n], Bin)
    @objective(model, Max, profit' * x)
    @constraint(model, weight' * x <= capacity)
    JuMP.optimize!(model)
    @assert termination_status(model) == OPTIMAL
    @assert primal_status(model) == FEASIBLE_POINT
    println("Objective is: ", objective_value(model))
    println("Solution is:")
    for i in 1:n
        print("x[$i] = ", round(Int, value(x[i])))
        println(", c[$i] / w[$i] = ", profit[i] / weight[i])
    end
    chosen_items = [i for i in 1:n if value(x[i]) > 0.5]
    return return chosen_items
end

# ╔═╡ ff9f41a7-c12d-4ca2-892f-38ede15d5880
solve_knapsack_problem(; profit = profit, weight = weight, capacity = capacity)

# ╔═╡ 137c51e9-b43d-4632-ba7c-ae24fbe97fae
md"""
可以看出，在这个例子中，被选上的物品为（1、4 和 5），其价值与重量比最高。
"""

# ╔═╡ 8675748f-2f97-430b-bfe1-961aaf9b9a42
md"""
!!! info "练习"
	1. 如果容量 ```C=20``` 结果如何？
	2. 如果我们获准携带的每件物品超过一件，会发生什么情况？试修改新的函数 `solve_knapsack_problem_new` 
"""

# ╔═╡ ba5bfece-de13-4515-a854-2c53b4030e83
function solve_knapsack_problem_new(;
    profit::Vector{Float64},
    weight::Vector{Float64},
    capacity::Float64,
)
    n = length(weight)
    @assert length(profit) == n
    model = Model(HiGHS.Optimizer)
    set_silent(model)
	#------------------------
	# Revise the code below
    @variable(model, x[1:n], Bin)
    @objective(model, Max, profit' * x)
    @constraint(model, weight' * x <= capacity)
	#------------------------
    JuMP.optimize!(model)
    @assert termination_status(model) == OPTIMAL
    @assert primal_status(model) == FEASIBLE_POINT
    println("Objective is: ", objective_value(model))
    println("Solution is:")
    for i in 1:n
        print("x[$i] = ", round(Int, value(x[i])))
        println(", c[$i] / w[$i] = ", profit[i] / weight[i])
    end
    chosen_items = [i for i in 1:n if value(x[i]) > 0.5]
    return return chosen_items
end

# ╔═╡ 0e04bb72-bcb9-414b-ac05-0b6b5bb892b4
solve_knapsack_problem_new(;profit,weight,capacity)

# ╔═╡ f6a5e3b8-7bb6-4ad2-8c8f-086bc727dab9
md"""
#### （2） 最大似然估计（MLE）

本例使用非线性优化计算正态分布参数(均值和方差)的最大似然估计值 (MLE)。
"""

# ╔═╡ 4219234b-1d9f-46ba-af54-57805f7052aa
md"""
假定 ``X_i\sim \mathcal{N}(\mu,\sigma)``, 已知数据 ``x_i`` 后， 似然函数为：
```math
\mathcal{L}(\mu,\sigma^2|x) = (2\pi\sigma^2)^{-n/2} \exp\left\{-\frac{1}{2\sigma^2} \sum_{i=1}^n (x_i-\mu)^2\right\}
```
下面我们选择 ``\mu`` 和 `` \sigma^2`` 以最大化 ``\ln(\mathcal{L}(\mu,\sigma^2|x))``
"""

# ╔═╡ 66f17199-a2f1-4a6e-a915-6039e0de5361
begin
	import Ipopt
	import Random
	import Statistics
	import Test
end

# ╔═╡ 38ae453c-fada-4aee-90ec-f5fa3e2eb0f1
function example_mle()
    n = 1_000
    Random.seed!(1234)
    data = randn(n)
    model = Model(Ipopt.Optimizer)
    set_silent(model)
	#---------------------------------------------
    @variable(model, μ, start = 0.0)
    @variable(model, σ >= 0.0, start = 1.0)
    @objective(
        model,
        Max,
        n / 2 * log(1 / (2 * π * σ^2)) -
        sum((data[i] - μ)^2 for i in 1:n) / (2 * σ^2)
    )
	#----------------------------------------------
    JuMP.optimize!(model)
    println("μ             = ", value(μ))
    println("mean(data)    = ", Statistics.mean(data))
    println("σ^2           = ", value(σ)^2)
    println("var(data)     = ", Statistics.var(data))
    println("MLE objective = ", objective_value(model))
    Test.@test value(μ) ≈ Statistics.mean(data) atol = 1e-3
    Test.@test value(σ)^2 ≈ Statistics.var(data) atol = 1e-2
    # ---------------------------------------------
	# You can even do constrained MLE! Revise the code below
    # add constrain here using @constraint(model, ?) 
	# and uncomment the folloing lines
	#-----------------------------------------------
 #    optimize!(model)
 #    Test.@test value(μ) ≈ value(σ)^2
 #    println()
 #    println("With constraint μ == σ^2:")
 #    println("μ                         = ", value(μ))
 #    println("σ^2                       = ", value(σ)^2)
 #    println("Constrained MLE objective = ", objective_value(model))
    return
end

# ╔═╡ 1539d457-be39-4621-ad5e-1fe27c113103
example_mle()

# ╔═╡ cbe97b3f-4f4b-42c8-b6b9-8e07c817a64c
md"""
!!! info "练习"
	修改上述函数，增加约束 ``\mu = \sigma^2``
"""

# ╔═╡ 1655a1c4-fe05-4049-b9c7-4e54d8f2c958
md"""
### 3. 使用 `Optimization.jl`

下面我们以最小化 Rosenbrock 方程为例：
```math
L(u,p) = (p_1 - u_1)^2 + p_2 (u_2 - u_1)^2
```
 其中变量 ``u_i \in [-1,1]``， 已知参数 ``p_1 = 1.0``，  ``p_2 = 100.0``。 
"""

# ╔═╡ b005647b-7de3-4ab7-ba14-d74bac0f9c5c
function rosenbrock()
	# Define the problem to optimize
	L(u, p) = (p[1] - u[1])^2 + p[2] * (u[2] - u[1]^2)^2
	u0 = zeros(2)
	p = [1.0, 100.0]
	optfun = OptimizationFunction(L, Optimization.AutoForwardDiff())
	prob = OptimizationProblem(optfun, u0, p, lb = [-1.0, -1.0], ub = [1.0, 1.0])

	# Solve the optimization problem
	sol = solve(prob, NLopt.LD_LBFGS())

	# Analyze the solution
	@show sol.u, L(sol.u, p)
	return prob, sol
end

# ╔═╡ 4ae70bb1-53ab-4a9f-82c3-73d6717237fb
rosenbrock_prob, rosenbrock_sol = rosenbrock()

# ╔═╡ 0a9781e2-9567-44fc-9eda-ee131a2e783a
md"""
即在 ``u=(1,1)`` 处， 函数 ``L`` 取得最小值 0.
"""

# ╔═╡ 40ef4ab5-ae93-446a-9a24-9fc119e5b437
md"""
## 三、 微分方程

### 1. 简介

`DifferentialEquations.jl` 提供了丰富的工具和算法，可用于处理各种不同类型的微分方程，包括常微分方程 (ODEs)、偏微分方程 (PDEs)、随机微分方程 (SDEs) 等，是 Julia 生态系统中广泛使用的微分方程（和差分方程）求解工具之一。以下是 `DifferentialEquations.jl` 的主要特点和功能：

1. **广泛的微分方程支持：** `DifferentialEquations.jl` 支持求解常微分方程 (ODEs)、偏微分方程 (PDEs)、随机微分方程 (SDEs)、延迟微分方程 (DDEs) 等多种类型的微分方程。

2. **高性能求解器：** 提供了一系列高性能的数值求解器，如 `ode45`、`ode23`、`Tsit5` 等，以便用户选择适合其问题的最优求解器。

3. **自适应步长控制：** 支持自适应步长控制，确保在求解过程中对解的精度进行有效的控制。

4. **并行计算：** 充分利用 Julia 的并行计算能力，使得在多核处理器上进行并行求解变得更加高效。

5. **事件处理：** 提供了处理事件的机制，允许在方程求解的特定事件发生时触发用户定义的动作。

6. **敏感性分析：** 支持对微分方程解的敏感性分析，以便研究参数变化对解的影响。

`DifferentialEquations.jl` 是 `SciML`生态的核心，后者是Julia中将高性能科学计算与机器学习相结合的生态，致力于开发科学计算工具，这些工具可与下一代机器学习工作流无缝结合。这包括

- 用于标准科学计算建模和仿真的高性能、高精度工具
- 与可微分编程和自动微分兼容
- 建立复杂多尺度模型的工具
- 处理逆问题、模型校准、控制和贝叶斯分析的方法
- 为数值方程求解器生成高效代码的符号建模工具
- 自动发现（生物）物理方程的方法
更多内容可以查看[`SciML`的官方文档](https://docs.sciml.ai/Overview/stable/)。
"""

# ╔═╡ 8e6779f7-987b-423a-93f6-7f96a96492ae
md"""
### 2. 示例

下面我们将通过两个例子介绍微分方程的数值求解。

#### （1） 一阶常微分方程

```math
	\frac{du}{dt}=f(u,p,t) = \alpha u
```
"""

# ╔═╡ 11f0528b-8f9f-464f-9980-6972bb6fa1ac
begin
	f(u, p, t) = 1.01 * u
	u0 = 1 / 2
	tspan = (0.0, 1.0)
	prob = ODEProblem(f, u0, tspan)
	sol = solve(prob, Tsit5(), reltol = 1e-8, abstol = 1e-8)
end

# ╔═╡ f8de904a-15d3-46fb-83f2-a484ad599752
begin
	using Plots
	plot(sol.t, t -> 0.5 * exp(1.01t), lw = 6, alpha = 0.6, label = "The Exact Solution")
	plot!(sol, linewidth = 3, ls = :dash, title = "Solution to the Linear OED",
    xaxis = "Time (t)", yaxis = "u(t)", label = "Numerial Solution") 
end

# ╔═╡ d9592891-dad9-4d9b-81a0-5122b08f3d87
md"""
#### （2） 捕食者-猎物动态

```math
\begin{align}
\frac{dx}{dt} &= \alpha x - \beta x y\\
\frac{dy}{dt} &= -\gamma y + \delta x y
\end{align}
```
其中 ``x(t)`` 和 ``y(t)`` 分别代表🐰 和 🐺 的数量。 我们采用 `ModelingToolkit` 的框架来定义该模型。
"""

# ╔═╡ 5a71f058-8892-4604-8a8d-2e9a7a467b97
md"""
假如我们知道参数的具体值和变量的初始值,我们可以模拟出该系统的动态。
"""

# ╔═╡ a5a8f609-183b-4d0d-a1d1-937fa59096bb
function simulation_predator()
	@parameters α=1.5 β=1.0 γ=3.0 δ=1.0
	ModelingToolkit.@variables t 🐰(t)=10.0 🐺(t)=2.0
	D = Differential(t)
	eqs = [D(🐰) ~ α * 🐰 - β * 🐰 * 🐺,
    D(🐺) ~ -γ * 🐺 + δ * 🐰 * 🐺]

	@named sys = ODESystem(eqs, t)
	simpsys = structural_simplify(sys)
	prob = ODEProblem(simpsys, [], (0.0, 10))
	return prob, solve(prob,saveat = 0.05)
end

# ╔═╡ 1b6f3705-f8b4-49d9-9b07-4f5fa4679a4f
rabit_prob, rabit_sol = simulation_predator();

# ╔═╡ 9929a0cf-0c25-4eff-89b3-59d649af1b12
rabit_sol

# ╔═╡ ca2bef82-f389-437a-9077-3b332e132fe4
plot(rabit_sol, lw = 2, alpha = 0.6, label=["Rabit" "Wolf"])

# ╔═╡ 940f474a-3615-41fd-8f44-fd76fb3583b9
plot(rabit_sol, idxs = [(1,2)], xaxis = "Rabit", yaxis = "Wolf", 
	label = "(Rabit(t),Wolf(t))")

# ╔═╡ aac54db7-de1c-44c6-a69c-1de66db33357
md"""
## 四、 随堂作业

### 1. 随机技术矩阵

考虑生产价值体系：`` p = (1+r)pM``, 假定非负矩阵 ``M`` 有生产性，即其最大特征值小于1。

1. 写一个函数：``M \to (r,p)``.
2. 写一个函数生产随机的矩阵 ``M``，注意 ``M``非负且有生产性
"""

# ╔═╡ a0baebbd-1cff-4909-9c11-43ff513609d0
# your answer here; you could add cell by `Ctrl + Enter`

# ╔═╡ a51b4603-aac9-44e6-8397-e9792d3abf9a
md"""
### 2. 拟合微分方程

假如我们现在观察到了捕食者-猎物动态中 🐰 和 🐺 的时间序列数据， 如何找到对应的参数呢？ 请阅读以下代码，大致说明拟合的思路。
"""

# ╔═╡ dbd80f15-7452-4144-baf4-1d0b52f3af3b
md"""
```julia
using DifferentialEquations, Optimization, OptimizationPolyalgorithms, SciMLSensitivity
using ForwardDiff, Plots

##------------生成数据----------------------
function lotka_volterra!(du, u, p, t)
    x, y = u
    α, β, δ, γ = p
    du[1] = dx = α * x - β * x * y
    du[2] = dy = -δ * y + γ * x * y
end

# Initial condition
u0 = [1.0, 1.0]

# Simulation interval
tspan = (0.0, 10.0)

# LV equation parameter. p = [α, β, δ, γ]
p = [1.5, 1.0, 3.0, 1.0]

# Setup the ODE problem, then solve
prob = ODEProblem(lotka_volterra!, u0, tspan, p)
datasol = solve(prob, saveat = 1)
data = Array(datasol)
##---------------------------------------------
## Now do the optimization process
## 重点看下面的拟合过程
## 
function loss(newp)
    newprob = remake(prob, p = newp)
    sol = solve(newprob, saveat = 1)
    loss = sum(abs2, sol .- data)
    return loss, sol
end

adtype = Optimization.AutoForwardDiff()
pguess = [1.0, 1.2, 2.5, 1.2]
optf = Optimization.OptimizationFunction((x, p) -> loss(x), adtype)
optprob = Optimization.OptimizationProblem(optf, pguess)

result_ode = Optimization.solve(optprob, PolyOpt(),maxiters = 200)
```
"""

# ╔═╡ ddaee183-331a-4e54-bfc3-53868cd27e5d
your_answer = "在这里写你的答案"

# ╔═╡ 487c95c8-9aca-4e6b-a22c-70d03d918bad
md"""
拟合的思路是：$your_answer
"""

# ╔═╡ b84c7275-5d35-419d-bfe7-2e455b1878ca
keep_working(text=md"The answer is not quite right.") = Markdown.MD(Markdown.Admonition("danger", "Keep working on it!", [text]));

# ╔═╡ 61c0dbe6-000c-47ce-b122-8c235776ec7e
correct(text=md"Great! You got the right answer! Let's move on to the next section.") = Markdown.MD(Markdown.Admonition("correct", "Got it!", [text]));

# ╔═╡ e60551cd-79e2-4dcf-96d6-38e4eaaeb162
if PV(100*(1.05).^(0:2),0.05)!=300
	keep_working()
else
	correct(md"""**Great!**  You figured it out.  Keep going.""")
end

# ╔═╡ 2bd64dbf-5bc9-4b1e-8920-4d0e2950fa81
if labor_value([0.2 0.4;0.5 0.3],[0.3,0.3]) != [1,1]
	keep_working()
else
	correct(md"""**Great!**  You figured it out.  Keep going.""")
end

# ╔═╡ 12287354-2693-43fa-9f98-cf6000c4405b
hint(text) = Markdown.MD(Markdown.Admonition("hint", "Hint", [text]));

# ╔═╡ b970978d-95e4-4b9f-9f9c-dee1e1bea696
hint(md"""
1. `eigen(A).values` 可以得到特征值；
2. 特征值可能是复数， 用 `abs` 可以计算其绝对值
3. 如果矩阵 ``A`` 的最大特征值是 ``\lambda(A)``， 那么 ``\lambda(kA) = k\lambda(A)``.
""")

# ╔═╡ d1a3aa89-8d17-44fb-9515-5eb458eb72e4
almost(text) = Markdown.MD(Markdown.Admonition("warning", "Almost there!", [text]));

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deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "8fdda4c692503d44d04a0603d9ac0982054635f9"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.1+0"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "b4bfde5d5b652e22b9c790ad00af08b6d042b97d"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.15.0+0"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "730eeca102434283c50ccf7d1ecdadf521a765a4"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.2+0"

[[deps.Xorg_xcb_util_cursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_jll", "Xorg_xcb_util_renderutil_jll"]
git-tree-sha1 = "04341cb870f29dcd5e39055f895c39d016e18ccd"
uuid = "e920d4aa-a673-5f3a-b3d7-f755a4d47c43"
version = "0.1.4+0"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "330f955bc41bb8f5270a369c473fc4a5a4e4d3cb"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.6+0"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "691634e5453ad362044e2ad653e79f3ee3bb98c3"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.39.0+0"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "e92a1a012a10506618f10b7047e478403a046c77"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.5.0+0"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.13+0"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "49ce682769cd5de6c72dcf1b94ed7790cd08974c"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.5+0"

[[deps.eudev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "gperf_jll"]
git-tree-sha1 = "431b678a28ebb559d224c0b6b6d01afce87c51ba"
uuid = "35ca27e7-8b34-5b7f-bca9-bdc33f59eb06"
version = "3.2.9+0"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "a68c9655fbe6dfcab3d972808f1aafec151ce3f8"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.43.0+0"

[[deps.gperf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3516a5630f741c9eecb3720b1ec9d8edc3ecc033"
uuid = "1a1c6b14-54f6-533d-8383-74cd7377aa70"
version = "3.1.1+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.4.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.8.0+0"

[[deps.libevdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "141fe65dc3efabb0b1d5ba74e91f6ad26f84cc22"
uuid = "2db6ffa8-e38f-5e21-84af-90c45d0032cc"
version = "1.11.0+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libinput_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "eudev_jll", "libevdev_jll", "mtdev_jll"]
git-tree-sha1 = "ad50e5b90f222cfe78aa3d5183a20a12de1322ce"
uuid = "36db933b-70db-51c0-b978-0f229ee0e533"
version = "1.18.0+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.mtdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "814e154bdb7be91d78b6802843f76b6ece642f11"
uuid = "009596ad-96f7-51b1-9f1b-5ce2d5e8a71e"
version = "1.1.6+0"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.52.0+1"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "9c304562909ab2bab0262639bd4f444d7bc2be37"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+1"
"""

# ╔═╡ Cell order:
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