https://ww2.mathworks.cn/help/releases/R2020a/parallel-computing/examples/Use-Distributed-Arrays-to-Solve-Systems-of-Linear-Equations-with-Direct-Methods.html

1、求解满阵（Solve a Full Matrix System）

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

b = sum(A,2);
bDist = sum(ADist,2);

xEx = ones(n,1);
xDistEx = ones(n,1,'distributed');

x = A\b;
err = abs(xEx-x);

xDist = ADist\bDist;
errDist = abs(xDistEx-xDist);

figure
subplot(2,1,1)
semilogy(err,'o');
title('System of Linear Equations with Full Matrices');
ylabel('Absolute Error');
xlabel('Element in x');
ylim([10e-17,10e-13])
subplot(2,1,2)
semilogy(errDist,'o');
title('System of Linear Equations with Distributed Full Matrices');
ylabel('Absolute Error');
xlabel('Element in x');
ylim([10e-17,10e-13])

mean(err)

ans = 1.6031e-13

mean(errDist)

ans = 1.2426e-13

2、解稀疏矩阵（Solve a Sparse Matrix System）

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

b = sum(A,2);
bDist = sum(ADist,2);
xEx = ones(n,1);
xDistEx = ones(n,1,'distributed');

x = A\b;
err = abs(xEx-x);

xDist = ADist\bDist;
errDist = abs(xDistEx-xDist);

figure
subplot(2,1,1)
semilogy(err,'o');
title('System of Linear Equations with In-Client Sparse Matrices');
ylabel('Absolute Error');
xlabel('Element in x');
ylim([10e-17,10e-13])
subplot(2,1,2)
semilogy(errDist,'o');
title('System of Linear Equations with Distributed Sparse Matrices');
ylabel('Absolute Error');
xlabel('Element in x');
ylim([10e-17,10e-13])

mean(err)

ans = 1.6031e-13

mean(errDist)

ans = 1.2426e-13