Syntax

描述

x = minres(A,b)attempts to find a minimum norm residual solutionx到线性方程的系统斧头=b。这n-经过-n系数矩阵Amust be symmetric but need not be positive definite. It should be large and sparse. The column vectorbmust have lengthn。You can specifyAas a function handle,好玩儿，这样afun（x）returns斧头。

Parameterizing Functionsexplains how to provide additional parameters to the function好玩儿, as well as the preconditioner functionmfundescribed below, if necessary.

如果minresconverges, a message to that effect is displayed. Ifminresfails to converge after the maximum number of iterations or halts for any reason, a warning message is printed displaying the relative residualnorm（b-a*x）/norm（b）和the iteration number at which the method stopped or failed.

minres（a，b，tol）指定该方法的耐受性。如果托尔is[], thenminresuses the default,1e-6。

minres(A,b,tol,maxit)指定最大迭代次数。如果最大is[], thenminresuses the default,min(n,20)。

minres（a，b，tol，maxit，m）和minres（a，b，tol，maxit，m1，m2）使用对称阳性明确的预处理M或者M = M1*M2和effectively solve the systeminv(sqrt(M))*A*inv(sqrt(M))*y = inv(sqrt(M))*bfory和then returnX = Inv（SQRT（M））*y。如果Mis[]thenminresapplies no preconditioner.M可以是函数句柄mfun，这样mfun(x)returnsM\x。

minres（a，b，tol，maxit，m1，m2，x0）specifies the initial guess. Ifx0is[], thenminresuses the default, an all-zero vector.

[x,flag] = minres(A,b,...)also returns a convergence flag.

Whenever旗帜不是0， 解决方案xreturned is that with minimal norm residual computed over all the iterations. No messages are displayed if the旗帜指定输出。

[x，flag，relres] = minres（a，b，...）还返回相对残差norm（b-a*x）/norm（b）。如果旗帜is0,relres <= tol。

[x,flag,relres,iter] = minres(A,b,...)also returns the iteration number at whichx被计算在哪里0 <= iter <= maxit。

[x，flag，relres，iter，resvec] = minres（a，b，...）also returns a vector of estimates of theminresresidual norms at each iteration, includingnorm(b-A*x0)。

[x,flag,relres,iter,resvec,resveccg] = minres(A,b,...)also returns a vector of estimates of the Conjugate Gradients residual norms at each iteration.

n = 100;on =一个（n，1）;a = spdiags（[-2*on 4*on -2*on]，-1：1，n，n）;b = sum（a，2）;TOL = 1E-10;最大值= 50;m1 = spdiags（4*on，0，n，n）;x = minres（a，b，tol，maxit，m1）;在迭代49中收敛到具有相对残留4.7E-014的溶液

function x1 = run_minres n = 100; on = ones(n,1); A = spdiags([-2*on 4*on -2*on],-1:1,n,n); b = sum(A,2); tol = 1e-10; maxit = 50; M = spdiags(4*on,0,n,n); x1 = minres(@afun,b,tol,maxit,M); function y = afun(x) y = 4 * x; y(2:n) = y(2:n) - 2 * x(1:n-1); y(1:n-1) = y(1:n-1) - 2 * x(2:n); end end

x1=run_minres;

在迭代49中收敛到具有相对残留4.7E-014的溶液

A = diag([20:-1:1, -1:-1:-20]); b = sum(A,2); % The true solution is the vector of all ones. x = pcg(A,b); % Errors out at the first iteration.

pcg stopped at iteration 1 without converging to the desired tolerance 1e-006 because a scalar quantity became too small or too large to continue computing. The iterate returned (number 0) has relative residual 1

x = minres（a，b，1e-6,40）;在迭代39中收敛到具有相对残留1.3E-007的溶液

参考

Extended Capabilities

See Also

bicg|bicgstab|cgs|gmres|ichol|LSQR|mldivide|pcg|QMR|SYMMLQ