Solve a Full Matrix System

First define the coefficient matrix $\mathit{A}$作为客户端内存中的变量，A, and then pass this matrix to thedistributedfunction to create a distributed version of the same matrix,ADist。当您使用distributed功能，MATLAB会使用您的默认群集设置自动启动并行池。

n = 1e3; A = randi(100,n,n); ADist = distributed(A);

You can now define the right hand vector $$b$$。In this example, $$b$$is defined as the row sum of $$A$$, which leads to an exact solution to $$Ax=b$$形式 $$x_{\text{exact}}=[1,。。。,1{]}^T$$。

b = sum（a，2）;bdist = sum（Adist，2）;

自从sum作用于分布式阵列，bDistis also distributed and its data is stored in the memory of the workers of your parallel pool. Finally, define the exact solutions for comparison with the solutions obtained using direct numerical methods.

Xex =一个（n，1）;xdistex =一个（n，1，'distributed');

Now that you have defined your system of linear equations, you can usemldivideto solve the system directly. In MATLAB, you can callmldivideusing the special operator\。You do not have to change your code to solve the distributed system asmldivide对分布式阵列有自动支持。万博1manbetx

Once you have calculated the solution, you can check the error between each element of the obtained result $\mathit{x}$和the expected values of $$x_{\text{exact}}$$。

x = A\b; err = abs(xEx-x); xDist = ADist\bDist; errDist = abs(xDistEx-xDist); figure subplot(2,1,1) semilogy(err,'o');标题（'System of Linear Equations with Full Matrices');ylabel(“绝对错误”);xlabel(X中的元素);Ylim（[10E-17,10E-13]）子图（2,1,2）半学（errdist，'o');标题（'System of Linear Equations with Distributed Full Matrices');ylabel(“绝对错误”);xlabel(X中的元素);ylim ([10 e-17, e-13])

For both the distributed arrays and the arrays stored on the client, the absolute error between the calculated results for $\mathit{x}$和确切的结果 $$x_{\text{exact}}$$是小。对于两种阵列类型，解决方案的精度大致相同。

mean(err)

ANS = 1.6031E-13

mean(errDist)

ANS = 1.2426E-13

Solve a Sparse Matrix System

Distributed arrays can also contain sparse data. To create the coefficient matrix $\mathit{A}$， 利用sprand和speye直接生成一个随机数的稀疏矩阵加上稀疏的身份矩阵。添加身份矩阵有助于防止创建 $\mathit{A}$作为一个单数或近乎单位的矩阵，两者都难以分解。

n = 1e3; density = 0.2; A = sprand(n,n,density) + speye(n); ADist = distributed(A);

Choosing the right hand vector $$b$$as the row sum of $$A$$产生与完整矩阵系统解决方案相同形式的精确解。

b = sum（a，2）;bdist = sum（Adist，2）;Xex =一个（n，1）;xdistex =一个（n，1，'distributed');

In the same way as with full matrices, you can now solve this system of linear equations directly usingmldivide和check the error between the obtained result and its expected value.

x = A\b; err = abs(xEx-x); xDist = ADist\bDist; errDist = abs(xDistEx-xDist); figure subplot(2,1,1) semilogy(err,'o');标题（“具有稀疏矩阵的线性方程系统”);ylabel(“绝对错误”);xlabel(X中的元素);Ylim（[10E-17,10E-13]）子图（2,1,2）半学（errdist，'o');标题（“具有分布式稀疏矩阵的线性方程系统”);ylabel(“绝对错误”);xlabel(X中的元素);ylim ([10 e-17, e-13])

与完整的矩阵系统一样，使用离线阵列和分布式阵列求解线性方程的系统也会产生具有可比精度的解决方案。万博 尤文图斯

mean(err)

ANS = 1.6031E-13

mean(errDist)

ANS = 1.2426E-13

After you are done with your computations, you can delete your parallel pool. Thegcpfunction returns the current parallel pool object so you can delete the current pool.

delete(gcp('nocreate'));

Improving Efficiency of the Solution

对于某些类型的大和稀疏系数矩阵 $\mathit{A}$，与解决系统的直接分解相比，有更有效的方法。在这些情况下，迭代方法在解决线性方程系统方面可能更有效。迭代方法生成一系列近似解决方案，这些解决方案会收敛到最终结果。万博 尤文图斯有关如何使用迭代方法求解具有较大稀疏输入矩阵的线性方程的示例，请参见使用分布式阵列用迭代方法求解线性方程的系统。