Class NewtonFixed¶
Defined in File NewtonFixed.m
Class Documentation¶
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class NewtonFixed : public handle¶
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Class container for a dumped Newton’s method with affine invariant step.
Solution Algorithm:
Given a zeros of a vectorial function problem of the form \( \mathbf{F} (\mathbf{x}) = \mathbf{0} \), where \( \mathbf{F}: \mathbb{R}^n \rightarrow \mathbb{R}^n \), then the Newton’s method is defined as:
\[ \mathbf{JF}(\mathbf{x}_k)\mathbf{h} = -\mathbf{F}(\mathbf{x}_k). \]The dumped step is defined as:
\[ \mathbf{x}_{k+1} = \mathbf{x}_k + \alpha_k \mathbf{h} \]where \( \alpha_k \) is a dumping coefficient that satisfies:
\[ \left\| \mathbf{JF}(\mathbf{x}_k)^{-1} \mathbf{F}(\mathbf{x}_{k+1}) \right\| \leq \left(1 - \dfrac{\alpha_k}{2}\right) \left\| \mathbf{JF}(\mathbf{x}_k)^{-1} \mathbf{F}(\mathbf{x}_k) \right\| = \left(1 - \dfrac{\alpha_k}{2} \right) \left\| \mathbf{h} \right\|. \]Note:
For more details on the Newton’s method with affine invariant step refer to: https://www.zib.de/deuflhard/research/algorithm/ainewton.en.html.
Public Functions
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function NewtonFixed()¶
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Newton’s solver class constructor.
- Returns
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The Newton’s solver object.
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function set_tolerance(t_tolerance)¶
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Set algorithm tolerance.
- Parameters
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t_tolerance – The algorithm tolerance.
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function get_tolerance()¶
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Get algorithm tolerance.
- Returns
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The algorithm tolerance.
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function set_max_iterations(t_max_iterations)¶
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Set maximum allowed algorithm iterations.
- Parameters
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t_max_iterations – The maximum allowed algorithm iterations.
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function get_max_iterations()¶
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Set maximum allowed algorithm iterations.
- Returns
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The maximum allowed algorithm iterations.
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function enable_verbose()¶
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Enable verbose mode.
- Parameters
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t_alpha – The relaxation factor.
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function disable_verbose()¶
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Disable verbose mode.
- Parameters
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t_alpha – The relaxation factor.
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function out_iterations()¶
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Get algorithm iterations.
- Returns
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The algorithm iterations.
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function out_function_evaluations()¶
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Set function evaluations.
- Returns
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The function evaluations.
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function out_jacobian_evaluations()¶
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Set Jacobian evaluations.
- Returns
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The Jacobian evaluations.
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function out_residuals()¶
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Get function evaluations.
- Returns
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The function evaluations.
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function out_converged()¶
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Get convergence boolean value.
- Returns
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The convergence boolean value.
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function solve_handle(t_function_handle, t_jacobian_handle, x_ini)¶
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Solve non-linear system of equations \( \mathbf{F}(\mathbf{x}) = \mathbf{0} \)
- Parameters
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t_function_handle – The function handle.
t_jacobian_handle – The Jacobian handle.
x_ini – The initial guess vector \( \mathbf{x} \).
- Returns
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The solution vector \( \mathbf{x} \).
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function reset()¶
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Reset solver internal counter and variables.
- Parameters
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t_function_handle – The function handle.
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function eval_function(x)¶
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Perform function \( \mathbf{F}(\mathbf{x}) \) evaluation.
- Parameters
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x – The input vector \( \mathbf{x} \).
- Returns
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The function value \( \mathbf{F}(\mathbf{x}) \).
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function eval_jacobian(x)¶
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Perform function \( \mathbf{JF}(\mathbf{x}) \) evaluation.
- Parameters
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x – The input vector \( \mathbf{x} \).
- Returns
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The Jacobian value \( \mathbf{JF}(\mathbf{x}) \).
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function solve(x_ini)¶
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Solve non-linear system of equations \( \mathbf{F} (\mathbf{x}) = \mathbf{0} \).
- Parameters
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x_ini – The initial guess for the vector \( \mathbf{x} \).
- Returns
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The solution \( \mathbf{x} \) and the output flag: \( 0 \) = success, \( 1 \) = failed because of bad initial point, \( 2 \) = failed because of bad dumping (step got too short).
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function NewtonFixed()¶
