# `sparse.linalg.isolve.iterative`¶

Iterative methods for solving linear systems

## Module Contents¶

### Functions¶

 `set_docstring`(header,Ainfo,footer=”“) `bicg`(A,b,x0=None,tol=1e-05,maxiter=None,M=None,callback=None) `bicgstab`(A,b,x0=None,tol=1e-05,maxiter=None,M=None,callback=None) `cg`(A,b,x0=None,tol=1e-05,maxiter=None,M=None,callback=None) `cgs`(A,b,x0=None,tol=1e-05,maxiter=None,M=None,callback=None) `gmres`(A,b,x0=None,tol=1e-05,restart=None,maxiter=None,M=None,callback=None,restrt=None) Use Generalized Minimal RESidual iteration to solve `Ax = b`. `qmr`(A,b,x0=None,tol=1e-05,maxiter=None,M1=None,M2=None,callback=None) Use Quasi-Minimal Residual iteration to solve `Ax = b`.
`set_docstring`(header, Ainfo, footer="")
`bicg`(A, b, x0=None, tol=1e-05, maxiter=None, M=None, callback=None)
`bicgstab`(A, b, x0=None, tol=1e-05, maxiter=None, M=None, callback=None)
`cg`(A, b, x0=None, tol=1e-05, maxiter=None, M=None, callback=None)
`cgs`(A, b, x0=None, tol=1e-05, maxiter=None, M=None, callback=None)
`gmres`(A, b, x0=None, tol=1e-05, restart=None, maxiter=None, M=None, callback=None, restrt=None)

Use Generalized Minimal RESidual iteration to solve `Ax = b`.

A : {sparse matrix, dense matrix, LinearOperator}
The real or complex N-by-N matrix of the linear system.
b : {array, matrix}
Right hand side of the linear system. Has shape (N,) or (N,1).
x : {array, matrix}
The converged solution.
info : int
Provides convergence information:
• 0 : successful exit
• >0 : convergence to tolerance not achieved, number of iterations
• <0 : illegal input or breakdown
x0 : {array, matrix}
Starting guess for the solution (a vector of zeros by default).
tol : float
Tolerance to achieve. The algorithm terminates when either the relative or the absolute residual is below tol.
restart : int, optional
Number of iterations between restarts. Larger values increase iteration cost, but may be necessary for convergence. Default is 20.
maxiter : int, optional
Maximum number of iterations (restart cycles). Iteration will stop after maxiter steps even if the specified tolerance has not been achieved.
M : {sparse matrix, dense matrix, LinearOperator}
Inverse of the preconditioner of A. M should approximate the inverse of A and be easy to solve for (see Notes). Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance. By default, no preconditioner is used.
callback : function
User-supplied function to call after each iteration. It is called as callback(rk), where rk is the current residual vector.
restrt : int, optional
DEPRECATED - use restart instead.

LinearOperator

A preconditioner, P, is chosen such that P is close to A but easy to solve for. The preconditioner parameter required by this routine is `M = P^-1`. The inverse should preferably not be calculated explicitly. Rather, use the following template to produce M:

```# Construct a linear operator that computes P^-1 * x.
import scipy.sparse.linalg as spla
M_x = lambda x: spla.spsolve(P, x)
M = spla.LinearOperator((n, n), M_x)
```
```>>> from scipy.sparse import csc_matrix
>>> from scipy.sparse.linalg import gmres
>>> A = csc_matrix([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float)
>>> b = np.array([2, 4, -1], dtype=float)
>>> x, exitCode = gmres(A, b)
>>> print(exitCode)            # 0 indicates successful convergence
0
>>> np.allclose(A.dot(x), b)
True
```
`qmr`(A, b, x0=None, tol=1e-05, maxiter=None, M1=None, M2=None, callback=None)

Use Quasi-Minimal Residual iteration to solve `Ax = b`.

A : {sparse matrix, dense matrix, LinearOperator}
The real-valued N-by-N matrix of the linear system. It is required that the linear operator can produce `Ax` and `A^T x`.
b : {array, matrix}
Right hand side of the linear system. Has shape (N,) or (N,1).
x : {array, matrix}
The converged solution.
info : integer
Provides convergence information:
0 : successful exit >0 : convergence to tolerance not achieved, number of iterations <0 : illegal input or breakdown
x0 : {array, matrix}
Starting guess for the solution.
tol : float
Tolerance to achieve. The algorithm terminates when either the relative or the absolute residual is below tol.
maxiter : integer
Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved.
M1 : {sparse matrix, dense matrix, LinearOperator}
Left preconditioner for A.
M2 : {sparse matrix, dense matrix, LinearOperator}
Right preconditioner for A. Used together with the left preconditioner M1. The matrix M1*A*M2 should have better conditioned than A alone.
callback : function
User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector.

LinearOperator

```>>> from scipy.sparse import csc_matrix
>>> from scipy.sparse.linalg import qmr
>>> A = csc_matrix([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float)
>>> b = np.array([2, 4, -1], dtype=float)
>>> x, exitCode = qmr(A, b)
>>> print(exitCode)            # 0 indicates successful convergence
0
>>> np.allclose(A.dot(x), b)
True
```