# cupyx.scipy.sparse.linalg.lsmr#

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).

Parameters
• A (ndarray, spmatrix or LinearOperator) – The real or complex matrix of the linear system. `A` must be `cupy.ndarray`, `cupyx.scipy.sparse.spmatrix` or `cupyx.scipy.sparse.linalg.LinearOperator`.

• b (cupy.ndarray) – Right hand side of the linear system with shape `(m,)` or `(m, 1)`.

• 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.

Returns

• 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.

• normr (float): `norm(b-Ax)`

• normar (float): `norm(A^T (b - Ax))`

• norma (float): `norm(A)`

• conda (float): Condition number of A.

• normx (float): `norm(x)`

Return type

tuple

References

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.