`scipy.optimize._nonlin:KrylovJacobian`

source: /scipy/optimize/_nonlin.py :1366

Signature

   class     KrylovJacobian (    rdiff    =  None   ,    method    =  lgmres   ,    inner_maxiter    =  20   ,    inner_M    =  None   ,    outer_k    =  10   ,   ** kw      )

Members

Summary

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

Extended Summary

This method is suitable for solving large-scale problems.

Parameters

%(params_basic)s

rdiff : float, optional

Relative step size to use in numerical differentiation.

method : str 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_maxiter : int, 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_M : LinearOperator 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_k : int, optional

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

inner_kwargs : kwargs

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

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 + ω * v /∣ v ∣) - f (x)) / ω

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].

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

✗

Aliases

scipy.optimize.KrylovJacobian