Articles of numerical methods

Numerically solving the equation of a simple pendulum with Runge-Kutta.

I am trying to solve the equation $\dfrac{\mathrm{d}^2\theta}{\mathrm{d}t^2} + \dfrac{g}{L} \sin{\theta} = 0$ using Runge-Kutta. I have alread split it into the following equations $\dfrac{\mathrm{d}\omega}{\mathrm{d}t} = – \dfrac{g}{L} \sin{\theta}$ $\dfrac{\mathrm{d}\theta}{\mathrm{d}t} = \omega$ I have implemented it correctly for the most part however It doesn’t appear to care what value of theta I put in initially, […]

Howuse this $R_{l}=\frac{1}{n}\left(\frac{(-1)^l}{2n}+\sum\limits_{m=1}^{n-1}\frac{1}{m}\cos{\frac{ml\pi}{n}}\right)$ and MATLAB get this four fig?

we consider Tikhonov’s regularization method for $\delta =0.1, 0.01,0.001,$ and $\delta =0$ The Tikhonov’s regularization method you can see: and apply this investigated regulaization strategies to Symm’s integral equation $$ (K\psi)(t)=-\dfrac{1}{2\pi}\int_{0}^{2\pi}\psi(s)\ln\left(4\sin^2{\dfrac{t-s}{2}}\right)dx+\int_{0}^{2\pi}\psi(s)k(t,s)ds$$ where $\gamma(s)=(\cos{s},2\sin{s}),0\le s\le 2\pi$ for $0\le t\le 2\pi$ with the analytic function $$k(t,s)=-\dfrac{1}{2\pi}\ln{\dfrac{|\gamma(t)-\gamma(s)|^2}{4\sin^2{\dfrac{t-s}{2}}}},t\neq s$$ $$k(t,t)=-\dfrac{1}{\pi}\ln{|\gamma'(t)|},0\le t\le 2\pi$$ we use the trapezoidal rule gor periodic […]

Iteration matrix and convergence

Assuming $G_{JA}$ is an iteration matrix for Jacobi Algorithm: $G_{JA}(A) = I -D^{-1}A$ $D$ is the diagonal of $A$. The sufficient condition for convergence is spectral radius less than one($\rho(G) <1$). Now, what happen if the spectral radius is equal to 1? Is there a way to set the parameter(such as initial guess) that guarantee […]

Please explain the last step of this newton method for system of equations

The step of working out x$^1$. I know the above is the formula but do they actually work out the inverse of the derivative matrix, is there a quicker way to do this?

Two Questions Regarding Gaussian Quadrature

I’m learning about the Gauss-Hermite Quadrature method on my own so I apologize if these questions seem trivial, but since I haven’t found any online resources which specifically answer my questions I thought I’d ask it here. In particular, I don’t really understand the following things: 1) How do you know how large you should […]

absolute stability / inequality

i want to find the amountof $\theta \in[0,1]$ where it is absolute stable whith $y’=\lambda y$ ,$\lambda \in \mathbb C$ or $\lambda \in \mathbb R$ for $$u_{j+1}=u_j+h[\theta f(t_j,u_j)+(1-\theta )f(t_{j+1},u_{j+1})]$$ i got $$u_{j+1}=u_j+h[\theta \lambda u_j+(1-\theta)\lambda u_{j+1}]$$ $$u_{j+1}=u_j(1+h\theta \lambda)+u_{j+1}h(\lambda -\theta \lambda)$$ $$u_{j+1}=u_j \dfrac{(1+h\theta \lambda)}{(1+h\lambda(\theta -1)}$$right? for the absolute stability it needs to be $$\dfrac{|1+h\theta \lambda|}{|1+h\lambda (\theta -1)|}<1$$ […]

Inviscid Burger's Equation

I have a question about the following burger’s equation. $u_t + (\frac12u^2)_x = 0 $ with $u(x,0) = sin(x)$ on $[0,2\pi]$ and periodic boundary conditions. When I studied this equation numerically, I notice that once the shock forms, the shock stays still in the same place, yet the magnitude of the function decreaes as time […]

An apparently new method to compute the $n$th root of any complex number

I found  a series of articles (in Portuguese) by a Brazilian mathematician named Ludenir Santos, where presents a series of iterative methods, he said new, to extract nth roots of any complex number different from zero (and hence for real numbers), with any desired degree of accuracy.  In his first articles, he has two different methods […]

Weird observation about discretization of $\nabla^T\nabla$

For the discretization of gradient, if I set $$ \mathcal \nabla := \begin{bmatrix}1&-1&\\&1&-1\end{bmatrix},$$ then $$ \mathcal \nabla^T\mathcal\nabla := \begin{bmatrix}1&0\\-1&1\\0&-1\end{bmatrix}\begin{bmatrix}1&-1&\\&1&-1\end{bmatrix}=\begin{bmatrix}1&-1&0\\-1&2&-1\\0&-1&1\end{bmatrix}$$ It seems that $\mathcal \nabla^T\mathcal\nabla=-\Delta$, not $\mathcal \nabla^T\mathcal\nabla=\Delta$. Is this true? If yes, could you explain why it holds?

For what $x$ does $g(x)$ converge to a fixed point?

Show that the iteration: $x_{k+1}=2x_k-αx_{k}^2$ where $α > 0$ converges quadratically to $\frac{1}{α}$ for any $x_0$ such that $0 < x_0 < \frac{2}{α}$. I have been able to prove that it converges quadratically to $\frac{1}{α}$ since you can consider the fixed point(s) of $g(x)= 2x-αx^2$ which are $x = 0$ and $x = \frac{1}{α}$ and […]