agmres在Matlab中的代码

agmres是一种求解线性方程组的迭代方法，以下是在Matlab中实现agmres的示例代码：

function [x,flag,relres,iter,resvec] = agmres(A,b,restart,tol,maxit,M1,M2,x0,varargin)
% AGMRES   Aggregated GMRES method for Ax=b.
%   X = AGMRES(A,B) solves the linear system A*X=B using AGMRES.
%   AGMRES(A,B,RESTART,TOL,MAXIT,M1,M2,X0) specifies optional
%   parameters for controlling the solution process.
%
%   Parameters:
%     A       - coefficient matrix
%     B       - right-hand side vector/matrix
%     RESTART - number of iterations between restarts (default 30)
%     TOL     - convergence tolerance (default 1e-6)
%     MAXIT   - maximum number of iterations (default 1000)
%     M1      - preconditioner applied to A (default none)
%     M2      - preconditioner applied to the projected system (default M1)
%     X0      - initial guess (default none)
%
%   Returns:
%     X       - solution vector/matrix
%     FLAG    - convergence flag: 0 = converged, 1 = not converged
%     RELRES  - relative residual norm at termination
%     ITER    - number of iterations performed
%     RESVEC  - history of residual norms
%
%   See also GMRES, CGS, BICGSTAB, LSQR.
%
%   References:
%     A. H. Baker, "Aggregated iterative methods for linear systems,"
%       Technical Report, Department of Mathematics, University of
%       Tennessee, Knoxville, TN, 1994.
%     M. A. Saunders, "Large sparse eigenvalue computations," in
%       Handbook of Linear Algebra, L. Hogben (ed.), CRC Press, Boca
%       Raton, FL, 2007, pp. 39-1-39-22.

% Check input arguments and set default values
if ~exist('restart','var') || isempty(restart), restart = 30; end
if ~exist('tol','var') || isempty(tol), tol = 1e-6; end
if ~exist('maxit','var') || isempty(maxit), maxit = 1000; end
if ~exist('M1','var') || isempty(M1), M1 = []; end
if ~exist('M2','var') || isempty(M2), M2 = M1; end
if ~exist('x0','var') || isempty(x0), x0 = []; end

% Initialize variables
flag = 1; iter = 0;
x = x0; r = b - A*x;
if isempty(M1), z = r; else z = M1\r; end
if isempty(M2), w = z; else w = M2\z; end
resvec = norm(r); normb = norm(b);
if resvec/normb < tol, flag = 0; return; end

% Main loop
while iter < maxit
    iter = iter + 1;
    V = zeros(length(r),restart+1);
    H = zeros(restart+1,restart);
    s = zeros(restart,1);
    
    % Arnoldi process
    V(:,1) = w/norm(w);
    for j = 1:restart
        if isempty(M1), z = A*V(:,j); else z = M1\(A*V(:,j)); end
        if isempty(M2), w = z; else w = M2\z; end
        for i = 1:j
            H(i,j) = w'*V(:,i);
            w = w - H(i,j)*V(:,i);
        end
        H(j+1,j) = norm(w);
        if H(j+1,j) == 0, break; end
        V(:,j+1) = w/H(j+1,j);
        
        % Solve least-squares problem using QR factorization
        [Q,~] = qr(H(1:j+1,1:j),0);
        s(1:j) = Q'\normb;
        if j == restart || norm(H(j+1,1:j)*s(1:j) - H(1:j+1,j+1)*normb) <= tol*normb
            x = x + V(:,1:j)*s(1:j);
            r = b - A*x;
            if isempty(M1), z = r; else z = M1\r; end
            if isempty(M2), w = z; else w = M2\z; end
            resvec = [resvec;norm(r)];
            if resvec(end)/normb < tol, flag = 0; return; end
            break;
        end
    end
end
end

注意：这只是一个示例代码，可能不适用于所有情况，使用前请仔细阅读并理解代码。