GMRES matlab

GMRES是一种求解线性方程组的迭代方法，它可以用于求解大型稀疏矩阵的线性方程组。GMRES方法的基本思想是通过迭代求解来逐步逼近线性方程组的解，每次迭代都会产生一个新的近似解，并且通过残差向量的正交化来保证迭代的收敛性。在matlab中，可以使用bGMRES函数来实现GMRES方法的求解，该函数的输入参数包括系数矩阵A、右端向量b和初值x0，输出参数包括残差res、残差向量V、上Hessenberg矩阵H和旋转矩阵R。此外，还可以使用givens函数和backward函数来求解旋转矩阵和后向代替矩阵，从而得到线性方程组的解。

相关问题

matlab gmres 程序

我不太了解matlab gmres程序，但是我可以给你一些提示，GMRES(Generalized Minimal Residual algorithm)是一种迭代方法，用于求解线性方程组的最小二乘解。它会迭代地求解最小二乘解，从而避免了求解大矩阵的计算量。

gmres gpu算法在Matlab中的代码

以下是使用GMRES GPU算法的MATLAB代码示例：

%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。