scipy.sparse.linalg.

cgs#

scipy.sparse.linalg.cgs(A, b, x0=None, *, rtol=1e-05, atol=0.0, maxiter=None, M=None, callback=None)[source]#

Solve Ax = b with the Conjugate Gradient Squared method.

Parameters:

A{sparse array, ndarray, LinearOperator}: The real-valued N-by-N matrix of the linear system. Alternatively, A can be a linear operator which can produce Ax using, e.g., scipy.sparse.linalg.LinearOperator.
bndarray: Right hand side of the linear system. Has shape (N,) or (N,1).
x0ndarray: Starting guess for the solution.
rtol, atolfloat, optional: Parameters for the convergence test. For convergence, norm(b - A @ x) <= max(rtol*norm(b), atol) should be satisfied. The default is atol=0. and rtol=1e-5.
maxiterint: Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved.
M{sparse array, ndarray, LinearOperator}: Preconditioner for A. It should approximate the inverse of A (see Notes). Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance.
callbackfunction: User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector.

Returns:

xndarray

The converged solution.

infoint

Provides convergence information:: 0 : successful exit >0 : convergence to tolerance not achieved, number of iterations <0 : parameter breakdown

Notes

The preconditioner M should be a matrix such that M @ A has a smaller condition number than A, see [1].

Array API Standard Support

cgs has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	⛔
PyTorch	⛔	⛔
JAX	⛔	⛔
Dask	⛔	n/a

See Support for the array API standard for more information.

References

[1]

“Preconditioner”, Wikipedia, https://en.wikipedia.org/wiki/Preconditioner

[2]

“Conjugate gradient squared”, Wikipedia, https://en.wikipedia.org/wiki/Conjugate_gradient_squared_method

Examples

>>> import numpy as np
>>> from scipy.sparse import csc_array
>>> from scipy.sparse.linalg import cgs
>>> R = np.array([[4, 2, 0, 1],
...               [3, 0, 0, 2],
...               [0, 1, 1, 1],
...               [0, 2, 1, 0]])
>>> A = csc_array(R)
>>> b = np.array([-1, -0.5, -1, 2])
>>> x, exit_code = cgs(A, b)
>>> print(exit_code)  # 0 indicates successful convergence
0
>>> np.allclose(A.dot(x), b)
True