SD.rar_SD_下降迭代_线性方程组

N = 100; %作5个100阶对角方阵Di D = zeros(N,N,5); D1 = [100:-1:1]'; D2 = zeros(100,1); D2(1:50) = 100; D2(51:100) = [99:-2:1]'; D3 = ones(100,1); D3(1:50) = [100:-2:2]'; D4 = ones(100,1); D4(31:70) = 50; D4(1:30) = 100; D5 = ones(100,1); D5(1:10) = 100; D5(11:30) = 50; D(:,:,1) = diag(D1); D(:,:,2) = diag(D2); D(:,:,3) = diag(D3); D(:,:,4) = diag(D4); D(:,:,5) = diag(D5); %随机生成10个正交阵Qj Q = zeros(N,N,10); for j=1:10 M = rand(N); [Q0,R0] = qr(M); Q(:,:,j) = Q0; end %设定精确解 xexact = ones(100,1); xexact(1:20) = 5; xexact(70:100) = 10; %定义迭代初值和误差数组 x0 = zeros(100,1); eps = zeros(100,1,10); eps0 = zeros(1,1,10); n = 100; % %不同特征值分布的最速下降法 % for i=1:5 % A = Q(:,:,1)*D(:,:,i)*Q(:,:,1)'; % b = A*xexact; % x = x0; % eps0(1,1,i) = norm(x - xexact); % r = b - A*x; % for k=1:n % alpha = (r'*r)/(r'*(A*r)); % x = x + alpha*r; % eps(k,1,i) = norm(x - xexact); % r = b - A*x; % end % end % % temp = zeros(n,1); % speed = zeros(5,1); % for i=1:5 % temp(1) = eps(1,1,i)/eps0(1,1,i); % speed(i) = speed(i)+temp(1); % for k=2:n % temp(k) = eps(k,1,i)/eps(k-1,1,i); % speed(i) = speed(i)+temp(k); % end % speed(i) = speed(i)/n; % figure; % plot(temp); % end % speed(1:5,1) % eps(n,1,1:5) %不同特征向量的最速下降法 for j=1:10 A = Q(:,:,j)*D(:,:,1)*Q(:,:,j)'; b = A*xexact; x = x0; eps0(1,1,j) = norm(x - xexact); r = b - A*x; for k=1:n alpha = (r'*r)/(r'*(A*r)); x = x + alpha*r; eps(k,1,j) = norm(x - xexact); r = b - A*x; end end temp = zeros(n,1); speed = zeros(10,1); for j=1:10 temp(1) = eps(1,1,j)/eps0(1,1,j); speed(j) = speed(j)+temp(1); for k=2:n temp(k) = eps(k,1,j)/eps(k-1,1,j); speed(j) = speed(j)+temp(k); end speed(j) = speed(j)/n; figure; plot(temp); end speed(1:10,1) eps(n,1,1:10)