pymor.algorithms.newton

Module Contents

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)[source]

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 residual 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.