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lsqr - LSQR method

Syntax

x = lsqr(A,b) lsqr(A,b,tol) lsqr(A,b,tol,maxit) lsqr(A,b,tol,maxit,M) lsqr(A,b,tol,maxit,M1,M2) lsqr(A,b,tol,maxit,M1,M2,x0) [x,flag] = lsqr(A,b,tol,maxit,M1,M2,x0) [x,flag,relres] = lsqr(A,b,tol,maxit,M1,M2,x0) [x,flag,relres,iter] = lsqr(A,b,tol,maxit,M1,M2,x0) [x,flag,relres,iter,resvec] = lsqr(A,b,tol,maxit,M1,M2,x0) [x,flag,relres,iter,resvec,lsvec] = lsqr(A,b,tol,maxit,M1,M2,x0)

Description

x = lsqr(A,b) attempts to solve the system of linear equations A*x=b for x if A is consistent, otherwise it attempts to solve the least squares solution x that minimizes norm(b-A*x). The m-by-n coefficient matrix A need not be square but it should be large and sparse. The column vector b must have length m. A can be a function handle afun such that afun(x,'notransp') returns A*x and afun(x,'transp') returns A'*x. See Function Handles in the MATLAB Programming documentation for more information.

Parametrizing Functions, in the MATLAB Mathematics documentation, explains how to provide additional parameters to the function afun, as well as the preconditioner function mfun described below, if necessary.

If lsqr converges, a message to that effect is displayed. If lsqr 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.

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

lsqr(A,b,tol,maxit) specifies the maximum number of iterations.

lsqr(A,b,tol,maxit,M) and lsqr(A,b,tol,maxit,M1,M2) use n-by-n preconditioner M or M = M1*M2 and effectively solve the system A*inv(M)*y = b for y, where y = M*x. If M is [] then lsqr applies no preconditioner. M can be a function mfun such that mfun(x,'notransp') returns M\x and mfun(x,'transp') returns M'\x.

lsqr(A,b,tol,maxit,M1,M2,x0) specifies the n-by-1 initial guess. If x0 is [], then lsqr uses the default, an all zero vector.

[x,flag] = lsqr(A,b,tol,maxit,M1,M2,x0) also returns a convergence flag.

Flag	Convergence
`0`	`lsqr` converged to the desired tolerance `tol` within `maxit` iterations.
`1`	`lsqr` iterated `maxit` times but did not converge.
`2`	Preconditioner `M` was ill-conditioned.
`3`	`lsqr` stagnated. (Two consecutive iterates were the same.)
`4`	One of the scalar quantities calculated during `lsqr` 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 you specify the flag output.

[x,flag,relres] = lsqr(A,b,tol,maxit,M1,M2,x0) also returns an estimate of the relative residual norm(b-A*x)/norm(b). If flag is 0, relres <= tol.

[x,flag,relres,iter] = lsqr(A,b,tol,maxit,M1,M2,x0) also returns the iteration number at which x was computed, where 0 <= iter <= maxit.

[x,flag,relres,iter,resvec] = lsqr(A,b,tol,maxit,M1,M2,x0) also returns a vector of the residual norm estimates at each iteration, including norm(b-A*x0).

[x,flag,relres,iter,resvec,lsvec] = lsqr(A,b,tol,maxit,M1,M2,x0) also returns a vector of estimates of the scaled normal equations residual at each iteration: norm((A*inv(M))'*(B-A*X))/norm(A*inv(M),'fro'). Note that the estimate of norm(A*inv(M),'fro') changes, and hopefully improves, at each iteration.

Examples

Example 1

n = 100; 
on = ones(n,1); 
A = spdiags([-2*on 4*on -on],-1:1,n,n);
b = sum(A,2); 
tol = 1e-8; 
maxit = 15;
M1 = spdiags([on/(-2) on],-1:0,n,n); 
M2 = spdiags([4*on -on],0:1,n,n);

x = lsqr(A,b,tol,maxit,M1,M2);

displays the following message:

lsqr converged at iteration 11 to a solution with relative 
residual 3.5e-009

Example 2

This example replaces the matrix A in Example 1 with a handle to a matrix-vector product function afun. The example is contained in an M-file run_lsqr that

Calls lsqr with the function handle @afun as its first argument.
Contains afun as a nested function, so that all variables in run_lsqr are available to afun.

The following shows the code for run_lsqr:

function x1 = run_lsqr
n = 100; 
on = ones(n,1); 
A = spdiags([-2*on 4*on -on],-1:1,n,n);
b = sum(A,2); 
tol = 1e-8; 
maxit = 15;
M1 = spdiags([on/(-2) on],-1:0,n,n); 
M2 = spdiags([4*on -on],0:1,n,n);
x1 = lsqr(@afun,b,tol,maxit,M1,M2);

    function y = afun(x,transp_flag)
       if strcmp(transp_flag,'transp')      % y = A'*x
          y = 4 * x;
          y(1:n-1) = y(1:n-1) - 2 * x(2:n);
          y(2:n) = y(2:n) - x(1:n-1);
       elseif strcmp(transp_flag,'notransp') % y = A*x
          y = 4 * x;
          y(2:n) = y(2:n) - 2 * x(1:n-1);
          y(1:n-1) = y(1:n-1) - x(2:n);
       end
    end
end

When you enter

x1=run_lsqr;

MATLAB software displays the message

lsqr converged at iteration 11 to a solution with relative 
residual 3.5e-009

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, "LSQR: An Algorithm for Sparse Linear Equations And Sparse Least Squares," ACM Trans. Math. Soft., Vol.8, 1982, pp.43-71.