MATLAB Function Reference |
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,...) [x,flag,relres] = lsqr(A,b,...) [x,flag,relres,iter] = lsqr(A,b,...) [x,flag,relres,iter,resvec] = lsqr(A,b,...) [x,flag,relres,iter,resvec,lsvec] = lsqr(A,b,...)
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.
Parameterizing Functions Called by Function 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. If maxit
is []
, then lsqr
uses the default, min([m,n,20])
.
lsqr(A,b,tol,maxit,M1) 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 x = M*y
. 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.
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
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:
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
lsqr
with the function handle @afun
as its first argument.
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
See Also
bicg
, bicgstab
, cgs
, gmres
, minres
, norm
, pcg
, qmr
, symmlq
, function_handle
(@
)
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.
lsqnonneg | lt |
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