Questions tagged [nonlinear-equations]
Solution of nonlinear systems of equations. The equations might be algebraic or differential equations.
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Mathematica Package for validating effective string theory solution
I am asking for Mathematica package that given an input of:
symmetric matrix $G_{\mu\nu}$, antisymmetric matrix $B_{\mu\nu}$ and a scalar function $\Phi$
will check whether it is a solution to the one-...
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An alternative to Levenberg–Marquardt algorithm
When trying to solve for a (over)determined non-linear least square method:
$$\underset{x}{\min}||f(x)||^2_2, f: \mathbb{R}^n \rightarrow \mathbb{R}^m, x\in \mathbb{R}^n, m\geq n$$
we use the Gauss-...
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Analysis of convergence of Newton method
I often used the Newton-Raphson method in material calculation, where I had to solve a small set of nonlinear equations (size=1..5). In most cases, it worked. However, convergence failure is often ...
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How to ensure solution evolves forward in Modified Riks Method by Crisfield?
I am writing a Matlab based script for solving nonlinear FEA problems using Modified Riks method by Crisfield. As a starting point, I am solving $y = x^3 - 2 x^2 - x + 2$. Around the function minima ...
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Solving coupled 2nd-order differential equation
I would appreciate it if you could help me solve the following coupled equation numerically
$$
[-\frac{1}{2} \partial_r^2 + v_0(r) -E]\psi_{\ell} + v_1(r) \psi_{1-\ell}(r) = 0,
$$
where $\ell = 0 , 1$ ...
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Cheap way to keep parameter matrices orthogonal during optimization?
TLDR; I can keep matrix variables approximately orthogonal by taking a single gradient step in the direction of "effective rank" of matrix at each step of iterative solver, is there a more ...
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Solving systems of the form $y_i=UW x_i$ for $U,W$
I'm looking for pointers/examples of solving system of equations $y_i=f_W(f_U(x_i))$ for $W,U$ where
$f_M(x) \approx M x$
$U,W$ are updated simultaneously
$i\in (0, 10^{12})$
Simplest example is ...
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How to impose boundary conditions when solving a nonlinear dynamical system given by the FEM solver
I am solving a nonlinear dynamical system given by a nonlinear elastic problem which takes the following form:
$$ \boldsymbol{M} \ddot{u} + \boldsymbol{K}_{\textrm{NL}}u = 0 ,$$
here $u \in \mathbb{R}...
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Non-dimensionalizing the Ideal MHD System
Non-dimensionalization is a really frustrating topic for me, and I imagine many others, because in school it was glossed over while being really important to implementing a simulation.
I'm writing a ...
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Solving a system of non-linear equations to find relationship between arguments
I have a program that implements a multivariate function, call it $f = \mathcal{Q}(Z,v)$ that I can compute given $Z,v$. The $v$ variable is related to the $f$ variable by another relation, call it $v ...
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How to embed linear elasticity/constrain solver in non-linear soft-body dynamics
I often do simulation of dynamics of mass points connected by strings (e.g. molecular dynamics, soft body dynamics etc.). Typically I do it simply by integration of equations of motion by e.g. verlet ...
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Visualizing a low-dimensional torus in a high-dimensional system
In the 4D Henon-Heiles system, it is well-known for certain parameters the attractor is a 2D torus. I am wondering how can we plot this actual torus (embedded in 3D) by somehow projecting all 4 ...
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Period-doubling bifurcation, quasi-periodicity and dimension of torus
This is more of a conceptual question but closely related with how non-linear dynamics simulations results should be interpreted.
I am confused about the relationship of "period" in the ...
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Convergence of Modified Crank-Nicolson Scheme
I'm dealing with a particular reaction-diffusion equation having the form
$$ c_t = \alpha \nabla^2 c + F(c,x,t). \tag{1}$$
where $F$ is nonlinear. I would like to solve (1) with a finite-difference ...
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Numerical solution for inviscid Burgers' equation seems to have no breaking time?
So I'm trying to use the Lax-Friedrichs method to solve the inviscid burgers' equation with initial condition $$u(x,0) = \sin(x)$$, using
$$u_m^{n+1} = \frac{1}{2}(u_{m+1}^n + u_{m-1}^n) - \frac{\...
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