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# Questions tagged [newton-method]

Method for finding successively better approximations to the roots (or zeroes) of a real-valued function

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### On Newton-Raphson Method for Single Degree of Freedom Systems

I am trying to understand the geometric interpretation of the Newton-Raphson method as used in nonlinear structural mechanics. The fundamental governing equation of nonlinear structural mechanics is ...
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### Newton method not converging with square root equations

I am applying Newton's method to solve the following nonlinear equation systems $F = 0$: F = \left[\begin{array}{l} \sqrt{(x_0 - x_4)^2 + (x_1 - x_5)^2} - 6 \\ \sqrt{(x_0 - x_2)^2 +...
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### PETSc non-linear solvers (SNES): specifying single Eval & Jacobian function

The PETSc documentation example of a non-linear solver call has the user provide separate functions for the Jacobian and function evaluations: ...
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### Algorithm to solve system of nonlinear equations

I would like some tips in figuring out a good algorithm to find the solution of the following system. Let $\theta$ be a constant in $(0,1)$, let $i,l=1,...,N$, let $a_{l}$ and $b_{i,l}$ be some ...
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I'm trying to improve the speed of the following iteration to calculate $s_k$: $$B_k^{-1} = \Bigg( I + \frac{s_{k}s_{k-1}^T}{||s_{k-1}||^2}\Bigg)...\Bigg(I+ \frac{s_1s_0^T}{||s_0||^2}\Bigg) B_0^{-1}\\... • 11 0 votes 1 answer 129 views ### Debugging Newton-method used in a CG-approach I am currently proof-checking my program, which is intended to use Newton's method for solving nonlinear equations, using a continuous galerkin approach. Thus, as first step I checked it using a time-... • 563 7 votes 1 answer 733 views ### Lack of quadratic convergence in Newton's method It is well-known that Newton's method can converge quadratically, if initial guess is close enough and if the arising linear systems are solved accurately. I am applying Newton's method to highly ill-... 2 votes 0 answers 118 views ### Nonlinear system with diagonal nonlinearity Consider a nonlinear system of the form \boldsymbol{f}(\boldsymbol{x}) = \boldsymbol{0}_{\mathbb{R}^n} for \boldsymbol{x} \in \mathbb{R}^n, where the function \boldsymbol{f} is given by \begin{... • 186 1 vote 2 answers 762 views ### Implementation of the Jacobian-free Newton method In my calculation (of a simple heat equation, for testing) using the Newton method, I tried to replace the full Jacobian matrix with an approximation vector, i.e. replacing J in$$J(u)\delta u=-F(u)...
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The condition number of function evaluation $$\mathrm{cond}(f,x) := \left| \frac{x f'(x)}{f(x)} \right|$$ is infinite at a root of $f$. Hence it is useless for rescaling a tolerance which defines an ...
I have a trouble of defining a Jacobian matrix for my problem. Basically, I have 4 differential equations to be solved.  \begin{aligned} \dot x_1(t)&=x_2(t)\\ \dot x_2(t)&=p_2(t)−\sqrt 2 ...