以下是使用GMRES GPU算法的MATLAB代码示例： ```matlab %GMRES-GPU algorithm %The matrix A and the vector b are stored in the device (GPU) %The initial guess x0 is also stored in the device (GPU) %The algorithm solves Ax = b for x function [x, flag, relres, iter] = gmres_gpu(A, b, x0, rtol, maxit) %Transfer A and b from the host (CPU) to the device (GPU) A_gpu = gpuArray(A); b_gpu = gpuArray(b); %Initialize variables n = size(A, 1); x = gpuArray(x0); r = b_gpu - A_gpu * x; normb = norm(b_gpu); normr = norm(r); %Check for initial guess x0 if normr/normb <= rtol flag = 0; relres = normr/normb; iter = 0; return; end %Initialize GMRES variables m = 30; %maximum number of iterations Q = zeros(n, m+1, 'gpuArray'); H = zeros(m+1, m, 'gpuArray'); cs = zeros(m, 1, 'gpuArray'); sn = zeros(m, 1, 'gpuArray'); e1 = [1; zeros(m, 1), 'gpuArray']; Q(:,1) = r/normr; beta = normr; j = 0; while normr/normb > rtol && j < maxit j = j + 1; %Arnoldi process for i = 1:m v = A_gpu * Q(:,i); for k = 1:i H(k,i) = Q(:,k)' * v; v = v - H(k,i) * Q(:,k); end H(i+1,i) = norm(v); Q(:,i+1) = v / H(i+1,i); %Apply Givens rotations for k = 1:i-1 temp = cs(k) * H(k,i) + sn(k) * H(k+1,i); H(k+1,i) = -sn(k) * H(k,i) + cs(k) * H(k+1,i); H(k,i) = temp; end %Compute Givens rotation [cs(i), sn(i), H(i,i)] = givens(H(i,i), H(i+1,i)); H(i+1,i) = 0; %Apply Givens rotation to RHS temp = cs(i) * beta; beta = -sn(i) * beta; beta = cs(i) * (H(i,i) - temp) + sn(i) * H(i+1,i); H(i+1,i) = -sn(i) * (H(i,i) - temp) + cs(i) * H(i+1,i); %Check for convergence if abs(beta) <= rtol y = H(1:i,1:i) \ beta * e1(1:i); x = x + Q(:,1:i) * y; relres = norm(A_gpu * x - b_gpu) / normb; flag = 0; iter = j; return; end end %Solve least squares problem y = H(1:m,1:m) \ beta * e1(1:m); x = x + Q(:,1:m) * y; %Compute residual r = b_gpu - A_gpu * x; normr = norm(r); %Check for convergence if normr/normb <= rtol relres = normr/normb; flag = 0; iter = j; return; end end %Check for maximum number of iterations if j >= maxit flag = 1; else flag = -1; end %Compute relative residual relres = normr/normb; %Return number of iterations iter = j; end %Givens rotation function function [c, s, r] = givens(a, b) if b == 0 c = sign(a); s = 0; r = abs(a); elseif abs(b) > abs(a) temp = a / b; s = sign(b) / sqrt(1 + temp^2); c = s * temp; r = b / s; else temp = b / a; c = sign(a) / sqrt(1 + temp^2); s = c * temp; r = a / c; end end ``` 请注意，此代码需要GPU计算能力。您还需要确保您的MATLAB版本支持GPU计算。
