scipy.optimize.

KrylovJacobian#

class scipy.optimize.KrylovJacobian(rdiff=None, method='lgmres', inner_maxiter=20, inner_M=None, outer_k=10, **kw)[source]#

Find a root of a function, using Krylov approximation for inverse Jacobian.

This method is suitable for solving large-scale problems.

Parameters:

%(params_basic)s

rdifffloat, optional

Relative step size to use in numerical differentiation.

methodstr or callable, optional

Krylov method to use to approximate the Jacobian. Can be a string, or a function implementing the same interface as the iterative solvers in scipy.sparse.linalg. If a string, needs to be one of: 'lgmres', 'gmres', 'bicgstab', 'cgs', 'minres', 'tfqmr'.

The default is scipy.sparse.linalg.lgmres.

inner_maxiterint, optional

Parameter to pass to the “inner” Krylov solver: maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved.

inner_MLinearOperator or InverseJacobian

Preconditioner for the inner Krylov iteration. Note that you can use also inverse Jacobians as (adaptive) preconditioners. For example,

>>> from scipy.optimize import BroydenFirst, KrylovJacobian
>>> from scipy.optimize import InverseJacobian
>>> jac = BroydenFirst()
>>> kjac = KrylovJacobian(inner_M=InverseJacobian(jac))

If the preconditioner has a method named ‘update’, it will be called as update(x, f) after each nonlinear step, with x giving the current point, and f the current function value.

outer_kint, optional

Size of the subspace kept across LGMRES nonlinear iterations. See scipy.sparse.linalg.lgmres for details.

inner_kwargskwargs

Keyword parameters for the “inner” Krylov solver (defined with method). Parameter names must start with the inner_ prefix which will be stripped before passing on the inner method. See, e.g., scipy.sparse.linalg.gmres for details.

%(params_extra)s

Methods

aspreconditioner
matvec
setup
solve
update

See also

root: Interface to root finding algorithms for multivariate functions. See method='krylov' in particular.
scipy.sparse.linalg.gmres
scipy.sparse.linalg.lgmres

Notes

This function implements a Newton-Krylov solver. The basic idea is to compute the inverse of the Jacobian with an iterative Krylov method. These methods require only evaluating the Jacobian-vector products, which are conveniently approximated by a finite difference:

\[J v \approx (f(x + \omega*v/|v|) - f(x)) / \omega\]

Due to the use of iterative matrix inverses, these methods can deal with large nonlinear problems.

SciPy’s scipy.sparse.linalg module offers a selection of Krylov solvers to choose from. The default here is lgmres, which is a variant of restarted GMRES iteration that reuses some of the information obtained in the previous Newton steps to invert Jacobians in subsequent steps.

For a review on Newton-Krylov methods, see for example [1], and for the LGMRES sparse inverse method, see [2].

References

[1]

C. T. Kelley, Solving Nonlinear Equations with Newton’s Method, SIAM, pp.57-83, 2003. DOI:10.1137/1.9780898718898.ch3

[2]

D.A. Knoll and D.E. Keyes, J. Comp. Phys. 193, 357 (2004). DOI:10.1016/j.jcp.2003.08.010

[3]

A.H. Baker and E.R. Jessup and T. Manteuffel, SIAM J. Matrix Anal. Appl. 26, 962 (2005). DOI:10.1137/S0895479803422014

Examples

The following functions define a system of nonlinear equations

>>> def fun(x):
...     return [x[0] + 0.5 * x[1] - 1.0,
...             0.5 * (x[1] - x[0]) ** 2]

A solution can be obtained as follows.

>>> from scipy import optimize
>>> sol = optimize.newton_krylov(fun, [0, 0])
>>> sol
array([0.66731771, 0.66536458])