MATLAB Function Reference    

LSQR implementation of Conjugate Gradients on the Normal Equations



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 afun such that afun(x) returns A*x and afun(x,'transp') returns A'*x.

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) 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.

lsqr(afun,b,tol,maxit,m1fun,m2fun,x0,p1,p2,...) passes parameters p1,p2,... to functions afun(x,p1,p2,...) and afun(x,p1,p2,...,'transp') and similarly to the preconditioner functions m1fun and m2fun.

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

lsqr converged to the desired tolerance tol within maxit iterations.
lsqr iterated maxit times but did not converge.
Preconditioner M was ill-conditioned.
lsqr stagnated. (Two consecutive iterates were the same.)

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 the flag output is specified.

[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.


Alternatively, use this matrix-vector product function

as input to lsqr

See Also

bicg, bicgstab, cgs, gmres, minres, norm, pcg, qmr, symmlq

@ (function handle)


[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 lu