442 - Matrix Chain Multiplication***

//题目刚做是思路出现了问题,栈中元素只需要row 和column 即可。
//字典可以包含一个char,但更好的做法是将char型转换成下标
//新函数append()
#include <iostream>
#include <string>
#include <cstdio>
#include <stack>
using namespace std;
struct Type
{
	int row;
	int column;
	Type():row(0),column(0){}
	Type(int a,int b):row(a),column(b){}
};
bool operator!=(const Type &a,const Type &b)//要定义成const型,否则编译出错
{
	return (a.column!=b.column || a.row!=b.row);//这里曾经出错,!=
}
int main()
{
	//freopen("data.in","r",stdin);
	//freopen("data.out","w",stdout);
	Type p[30];
	int n;
	cin>>n;
	for(int i=0;i<n;i++)
	{
		char ch;
		int a,b;
		cin>>ch>>a>>b;
		p[ch-'A'].row=a;
		p[ch-'A'].column=b;
	}
	string line;
	while(cin>>line)
	{		
		int num=0;
		stack<Type> s;
		line.insert(0,"(");//这里插入元素为字符串型
		line.append(")");
		int flag=1;
		for(int i=0;i<line.size() && flag==1;i++)
		{
			if(line[i]=='(')
			{
				s.push(Type(-1,0));
			}
			else if(line[i]==')')
			{
				Type t1,t2;
				if(!s.empty() && s.top()!=Type(-1,0))
				{
					t2=s.top();
					s.pop();
				}
				while(!s.empty() && s.top()!=Type(-1,0))
				{
					t1=s.top();
					if(t1.column!=t2.row)
					{
						printf("error\n");
						flag=0;
						break;
					}
					else
					{
						s.pop();
						num+=t1.row*t1.column*t2.column;
						t2.row=t1.row;
					}
				}
				s.pop();
				s.push(t2);
			}
			else 
			{
				s.push(p[line[i]-'A']);
			}
		}
		if(flag==1)
			printf("%d\n",num);
	}
	return 0;
}


内容概要:本文详细介绍了900W或1Kw,20V-90V 10A双管正激可调电源充电机的研发过程和技术细节。首先阐述了项目背景,强调了充电机在电动汽车和可再生能源领域的重要地位。接着深入探讨了硬件设计方面,包括PCB设计、磁性器件的选择及其对高功率因数的影响。随后介绍了软件实现,特别是程序代码中关键的保护功能如过流保护的具体实现方法。此外,文中还提到了充电机所具备的各种保护机制,如短路保护、欠压保护、电池反接保护、过流保护和过温度保护,确保设备的安全性和可靠性。通讯功能方面,支持RS232隔离通讯,采用自定义协议实现远程监控和控制。最后讨论了散热设计的重要性,以及为满足量产需求所做的准备工作,包括提供详细的PCB图、程序代码、BOM清单、磁性器件和散热片规格书等源文件。 适合人群:从事电力电子产品研发的技术人员,尤其是关注电动汽车充电解决方案的专业人士。 使用场景及目标:适用于需要高效、可靠充电解决方案的企业和个人开发者,旨在帮助他们快速理解和应用双管正激充电机的设计理念和技术要点,从而加速产品开发进程。 其他说明:本文不仅涵盖了理论知识,还包括具体的工程实践案例,对于想要深入了解充电机内部构造和工作原理的人来说是非常有价值的参考资料。
import cvxpy as cp import numpy as np import pandas as pd data = pd.read_excel('附件1.xlsx') X = data.iloc[1:, 1:2] Y = data.iloc[1:, 2:3] W = data.iloc[2:, 3:4] # 收集点垃圾量(i=1..n) xi = X.values.flatten() # 转换为一维数组 (n,) yi = Y.values.flatten() # 转换为一维数组 (n,) wi = W.values.flatten() # 转换为一维数组 (n,) # 定义参数 n = 30 # 收集点数量(i=1..n,0为处理厂) K = 30 # 车辆总数 Q = 5 # 最大载重 # 计算距离矩阵dij(i,j=0..n) dij = np.zeros((n+1, n+1)) for i in range(n+1): for j in range(n+1): dij[i, j] = np.sqrt((xi[i] - xi[j])**2 + (yi[i] - yi[j])**2) x = cp.Variable((n+1, n+1, K), boolean=True) # x_ijv: 车辆v从i到j y = cp.Variable((n, K), boolean=True) # y_iv: 车辆v服务收集点i(i=1..n,索引0..n-1) z = cp.Variable(K, boolean=True) # z_v: 使用车辆v u = cp.Variable((n+1, K), nonneg=True) # u_iv: 车辆v访问i后的累计载重(i=0..n,u[0,v]=0) objective = cp.Minimize(cp.sum(cp.sum(cp.sum(dij @ x, axis=2), axis=1), axis=0)) # 约束条件 constraints = [] # 1. 每个收集点被 exactly 一辆车服务(i=1..n,对应y的索引0..n-1) for i in range(n): constraints.append(cp.sum(y[i, :]) == 1) # 2. 流量守恒(i=1..n,处理厂i=0的约束在6中) for i in range(1, n+1): # 收集点i(1..n,y索引i-1) for v in range(K): constraints.append(cp.sum(x[i, :, v]) == y[i-1, v]) # 离开i的次数等于服务i的次数 constraints.append(cp.sum(x[:, i, v]) == y[i-1, v]) # 进入i的次数等于服务i的次数 # 3. 载重约束 for v in range(K): constraints.append(cp.sum(wi * y[:, v]) <= Q * z[v]) # 4. MTZ约束(子回路消除) for v in range(K): constraints.append(u[0, v] == 0) # 处理厂出发时载重为0 for i in range(1, n+1): # 收集点i(1..n,wi[i-1]) constraints.append(u[i, v] >= wi[i-1]) # 载重下限 constraints.append(u[i, v] <= Q) # 载重上限 for j in range(1, n+1): if i != j: constraints.append(u[i, v] - u[j, v] + Q * x[i, j, v] <= Q - wi[j-1]) # 5. 车辆使用约束:服务收集点则使用车辆 for i in range(n): for v in range(K): constraints.append(y[i, v] <= z[v]) # 6. 出发返回约束(处理厂i=0,出发到收集点1..n,返回从收集点1..n到0) for v in range(K): constraints.append(cp.sum(x[0, 1:, v]) == z[v]) # 出发次数等于车辆使用 constraints.append(cp.sum(x[1:, 0, v]) == z[v]) # 返回次数等于车辆使用 # 建立问题并求解 problem = cp.Problem(objective, constraints) problem.solve() # 输出结果(示例,需根据实际需求处理) print("最优目标值:", problem.value) print("使用的车辆:", z.value)D:\python\python.exe D:\35433\Documents\数模练习\电工杯\问题1.py D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 1 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 2 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 3 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 4 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 5 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 6 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 7 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 8 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 9 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 10 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 11 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 12 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 13 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 14 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 15 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 16 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 17 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 18 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 19 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 20 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 21 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 22 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 23 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 24 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 25 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 26 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 27 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 28 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 29 times so far. D:\python\Lib\site-packages\cvxpy\expressions\expression.py:674: UserWarning: This use of ``*`` has resulted in matrix multiplication. Using ``*`` for matrix multiplication has been deprecated since CVXPY 1.1. Use ``*`` for matrix-scalar and vector-scalar multiplication. Use ``@`` for matrix-matrix and matrix-vector multiplication. Use ``multiply`` for elementwise multiplication. This code path has been hit 30 times so far. D:\python\Lib\site-packages\cvxpy\reductions\solvers\solving_chain_utils.py:41: UserWarning: The problem has an expression with dimension greater than 2. Defaulting to the SCIPY backend for canonicalization.
05-25
### CVXPY 中矩阵乘法的正确实现方式 在 CVXPY 1.1 及以上版本中,`*` 操作符被重新定义为逐元素乘法(element-wise multiplication),而不是传统的矩阵乘法。因此,在涉及矩阵乘法的操作时,直接使用 `*` 将引发警告或错误。为了消除这些警告并提高代码的兼容性和性能,建议采用以下方法: #### 使用 '@' 运算符 CVXPY 支持 Python 的标准矩阵乘法运算符 `@`,该运算符能够高效地完成矩阵间的乘法操作。这是推荐的方式之一,因为它不仅语义清晰,而且与 NumPy 和 PyTorch 等库保持一致[^1]。 ```python import cvxpy as cp # 定义变量和参数 X = cp.Variable((3, 2)) A = cp.Parameter((2, 3), value=np.random.randn(2, 3)) # 正确使用 @ 实现矩阵乘法 expression = A @ X ``` 此代码片段展示了如何通过 `@` 来替代旧版中的 `*`,从而避免不必要的警告。 #### 使用 `cp.matmul` 除了 `@` 外,还可以显式调用 `cvxpy.matmul()` 函数来进行矩阵乘法。这种方式更加灵活,尤其适用于复杂表达式的构建场景[^1]。 ```python # 显式调用 matmul 方法 expression = cp.matmul(A, X) ``` 需要注意的是,无论是 `@` 还是 `cp.matmul`,都支持广播机制,这意味着当两个张量维度匹配时,它们会自动扩展以适应计算需求[^1]。 #### 避免误用 'multiply' 尽管 `np.multiply` 或者其对应的 CVXPY 版本可用于逐元素乘法,但它并不适合处理传统意义上的矩阵乘法问题。例如,下面的例子演示了两者的区别: ```python # 错误示范:试图用 multiply 替代矩阵乘法 C = np.array([[1, 2], [3, 4]]) D = np.array([[5, 6], [7, 8]]) wrong_result = np.multiply(C, D) # 结果是一个逐元素相乘后的数组 correct_result = C @ D # 执行真正的矩阵乘法 ``` 显然,只有后者才是正确的做法[^4]。 最后值得注意的一点在于输入数据结构的选择上——如果确认所有参与运算的数据均为二维形式,则可以直接选用效率更高的专用函数;但如果存在高维情况或者不确定具体形状的话,则应优先考虑那些具备广泛适用性的解决方案,比如前述提到过的两种主流途径[^3]。 ### 性能考量 从性能角度来看,`@` 和 `cp.matmul` 均经过高度优化,能够在大多数情况下提供最佳表现。然而实际应用过程中还需关注以下几个方面: - **稀疏性**:若待求解模型含有大量零值项,则需特别指定相应属性以便充分利用内存空间减少冗余计算; - **批量化处理能力**:现代硬件架构往往偏好批量作业模式,故设计算法阶段也应当考虑到这一点进而合理规划数据流走向。 综上所述,在升级至最新版 CVXPY 后遇到此类提示信息时,请果断切换到现代化接口上来满足规范要求的同时兼顾执行速度上的优势。
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