pymor.algorithms.newton

Module Contents

Functions

newton

Newton algorithm.

pymor.algorithms.newton.newton(operator, rhs, initial_guess=None, mu=None, range_product=None, source_product=None, least_squares=False, miniter=0, maxiter=100, atol=0.0, rtol=1e-07, relax='armijo', line_search_params=None, stagnation_window=3, stagnation_threshold=np.inf, error_measure='update', return_stages=False, return_residuals=False)

Newton algorithm.

This method solves the nonlinear equation

A(U, mu) = V

for U using the Newton method.

Parameters

operator

The Operator A. A must implement the jacobian interface method.

rhs

VectorArray of length 1 containing the vector V.

initial_guess

If not None, a VectorArray of length 1 containing an initial guess for the solution U.

mu

The parameter values for which to solve the equation.

range_product

The inner product Operator on operator.range with which the norm of the resiudal is computed. If None, the Euclidean inner product is used.

source_product

The inner product Operator on operator.source with which the norm of the solution and update vectors is computed. If None, the Euclidean inner product is used.

least_squares

If True, use a least squares linear solver (e.g. for residual minimization).

miniter

Minimum amount of iterations to perform.

maxiter

Fail if the iteration count reaches this value without converging.

atol

Finish when the error measure is below this threshold.

rtol

Finish when the error measure has been reduced by this factor relative to the norm of the initial residual resp. the norm of the current solution.

relax

If real valued, relaxation factor for Newton updates; otherwise 'armijo' to indicate that the armijo line search algorithm shall be used.

line_search_params

Dictionary of additional parameters passed to the line search method.

stagnation_window

Finish when the error measure has not been reduced by a factor of stagnation_threshold during the last stagnation_window iterations.

stagnation_threshold

See stagnation_window.

error_measure

If 'residual', convergence depends on the norm of the residual. If 'update', convergence depends on the norm of the update vector.

return_stages

If True, return a VectorArray of the intermediate approximations of U after each iteration.

return_residuals

If True, return a VectorArray of all residual vectors which have been computed during the Newton iterations.

Returns

U

VectorArray of length 1 containing the computed solution

data

Dict containing the following fields:

solution_norms

NumPy array of the solution norms after each iteration.

update_norms

NumPy array of the norms of the update vectors for each iteration.

residual_norms

NumPy array of the residual norms after each iteration.

stages

See return_stages.

residuals

See return_residuals.

Raises

NewtonError

Raised if the Newton algorithm failed to converge.