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minres

Minimum residual method

Syntax

x = minres(A,b)
minres(A,b,tol)
minres(A,b,tol,maxit)
minres(A,b,tol,maxit,M)
minres(A,b,tol,maxit,M1,M2)
minres(A,b,tol,maxit,M1,M2,x0)
[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,...)

Description

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.

Flag

Convergence

0

minres converged to the desired tolerance tol within maxit iterations.

1

minres iterated maxit times but did not converge.

2

Preconditioner M was ill-conditioned.

3

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

4

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.

Examples

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);
       end
end

When you enter

x1=run_minres;

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

References

[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.

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