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Minimum residual method


x = minres(A,b)
[x,flag] = minres(A,b,...)
[x,flag,relres] = minres(A,b,...)
[x,flag,relres,iter] = minres(A,b,...)
[x,flag,relres,iter,resvec] = minres(A,b,...)
[x,flag,relres,iter,resvec,resveccg] = minres(A,b,...)


x = minres(A,b) attempts to find a minimum norm residual solution x to the system of linear equations A*x=b. The n-by-n coefficient matrix A must be symmetric but need not be positive definite. It should be large and sparse. The column vector b must have length n. You can specify A as a function handle, afun, such that afun(x) returns A*x.

Parameterizing Functions explains how to provide additional parameters to the function afun, as well as the preconditioner function mfun described below, if necessary.

If minres converges, a message to that effect is displayed. If minres fails to converge after the maximum number of iterations or halts for any reason, a warning message is printed displaying the relative residual norm(b-A*x)/norm(b) and the iteration number at which the method stopped or failed.

minres(A,b,tol) specifies the tolerance of the method. If tol is [], then minres uses the default, 1e-6.

minres(A,b,tol,maxit) specifies the maximum number of iterations. If maxit is [], then minres uses the default, min(n,20).

minres(A,b,tol,maxit,M) and minres(A,b,tol,maxit,M1,M2) use symmetric positive definite preconditioner M or M = M1*M2 and effectively solve the system inv(sqrt(M))*A*inv(sqrt(M))*y = inv(sqrt(M))*b for y and then return x = inv(sqrt(M))*y. If M is [] then minres applies no preconditioner. M can be a function handle mfun, such that mfun(x) returns M\x.

minres(A,b,tol,maxit,M1,M2,x0) specifies the initial guess. If x0 is [], then minres uses the default, an all-zero vector.

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




minres converged to the desired tolerance tol within maxit iterations.


minres iterated maxit times but did not converge.


Preconditioner M was ill-conditioned.


minres stagnated. (Two consecutive iterates were the same.)


One of the scalar quantities calculated during minres became too small or too large to continue computing.

Whenever flag is not 0, the solution x returned is that with minimal norm residual computed over all the iterations. No messages are displayed if the flag output is specified.

[x,flag,relres] = minres(A,b,...) also returns the relative residual norm(b-A*x)/norm(b). If flag is 0, relres <= tol.

[x,flag,relres,iter] = minres(A,b,...) also returns the iteration number at which x was computed, where 0 <= iter <= maxit.

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


Using minres with a Matrix Input

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; 
M1 = spdiags(4*on,0,n,n);

x = minres(A,b,tol,maxit,M1);
minres converged at iteration 49 to a solution with relative 
residual 4.7e-014

Using minres with a Function Handle

This example replaces the matrix A in the previous example with a handle to a matrix-vector product function afun. The example is contained in a file run_minres that

  • Calls minres with the function handle @afun as its first argument.

  • Contains afun as a nested function, so that all variables in run_minres are available to afun.

The following shows the code for run_minres:

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);

When you enter


MATLAB® software displays the message

minres converged at iteration 49 to a solution with relative 
residual 4.7e-014

Using minres instead of pcg

Use a symmetric indefinite matrix that fails with pcg.

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.

displays the following message:

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

However, minres can handle the indefinite matrix A.

x = minres(A,b,1e-6,40);
minres converged at iteration 39 to a solution with relative 
residual 1.3e-007


[1] Barrett, R., M. Berry, T. F. Chan, et al., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia, 1994.

[2] Paige, C. C. and M. A. Saunders, “Solution of Sparse Indefinite Systems of Linear Equations.” SIAM J. Numer. Anal., Vol.12, 1975, pp. 617-629.

Extended Capabilities

Introduced before R2006a