- cupyx.scipy.sparse.linalg.lsmr(A, b, x0=None, damp=0.0, atol=1e-06, btol=1e-06, conlim=100000000.0, maxiter=None)[source]#
Iterative solver for least-squares problems.
lsmr solves the system of linear equations
Ax = b. If the system is inconsistent, it solves the least-squares problem
min ||b - Ax||_2. A is a rectangular matrix of dimension m-by-n, where all cases are allowed: m = n, m > n, or m < n. B is a vector of length m. The matrix A may be dense or sparse (usually sparse).
A (ndarray, spmatrix or LinearOperator) – The real or complex matrix of the linear system.
b (cupy.ndarray) – Right hand side of the linear system with shape
x0 (cupy.ndarray) – Starting guess for the solution. If None zeros are used.
damp (float) –
Damping factor for regularized least-squares. lsmr solves the regularized least-squares problem
min ||(b) - ( A )x|| ||(0) (damp*I) ||_2
where damp is a scalar. If damp is None or 0, the system is solved without regularization.
atol (float) – Stopping tolerances. lsmr continues iterations until a certain backward error estimate is smaller than some quantity depending on atol and btol.
btol (float) – Stopping tolerances. lsmr continues iterations until a certain backward error estimate is smaller than some quantity depending on atol and btol.
conlim (float) – lsmr terminates if an estimate of
cond(A)i.e. condition number of matrix exceeds conlim. If conlim is None, the default value is 1e+8.
maxiter (int) – Maximum number of iterations.
x (ndarray): Least-square solution returned.
istop (int): istop gives the reason for stopping:
0 means x=0 is a solution. 1 means x is an approximate solution to A*x = B, according to atol and btol. 2 means x approximately solves the least-squares problem according to atol. 3 means COND(A) seems to be greater than CONLIM. 4 is the same as 1 with atol = btol = eps (machine precision) 5 is the same as 2 with atol = eps. 6 is the same as 3 with CONLIM = 1/eps. 7 means ITN reached maxiter before the other stopping conditions were satisfied.
itn (int): Number of iterations used.
norm(A^T (b - Ax))
conda (float): Condition number of A.
- Return type
D. C.-L. Fong and M. A. Saunders, “LSMR: An iterative algorithm for sparse least-squares problems”, SIAM J. Sci. Comput., vol. 33, pp. 2950-2971, 2011.