Articles of numerical methods

Matlab numerical integration involving Bessel functions returns NaN

I need to numerically compute integrals such as this (some parameters omitted for simplicity): $$ \int_{0}^{\infty} e^{-x^2} I_{0}(x) K_{0}(x) \mathrm{d}x $$ where $I_{0}$ and $K_{0}$ denote the modified Bessel functions of the first and second kind, respectively. As $x \to \infty$, we have $K_{0} \to 0$ and $I_{0} \to \infty$, but the integrand goes to […]

Best approximation in a Hilbert space

Let $H$ be a $\mathbb C$-Hilbert space, $U\subseteq H$ be a closed subspace of $H$, $\operatorname P_U$ denote the orthogonal projection from $H$ onto $U$ and $x\in H$. I want to show that $$\tilde x=\underset{u\in U}{\operatorname{arg min}}\left\|u-x\right\|_H\Leftrightarrow\tilde x=\operatorname P_Ux\tag1\;.$$ “$\Leftarrow$”: $\tilde x=\operatorname P_Ux$ $\Rightarrow$ $$\langle\tilde x-u,u\rangle_H=0\;\;\;\text{for all }u\in U\tag2$$ and hence (since $t\tilde x\in U$) […]

Largange polynomial second order derivative

Well I came across a problem to find a generalized version ($n+1$ nodes) of first and second order derivatives for Lagrange interpolation polynomial. In some former post, I found an expression for deriving $L_j(x)$, where $L_j$ stands for Lagrange basis polynomial. The expression is as follows: \begin{align} L_j'(x) = \sum_{l\not = j} \frac{1}{x_j-x_l}\prod_{m\not = (j,l)} […]

Relative error of machine summation

Let $\mathbb{F}(b,t,L,U)$ (briefly $\mathbb{F}$) the set of all machine numbers. The definition is the usual, i.e. $\mathbb{F}$ is defined as \begin{equation*} \mathbb{F} := \left\lbrace (-1)^{s} m b^{e-t}\right\rbrace \end{equation*} where $b^{t-1} \leq m \leq b^{t}-1$, $L \leq e \leq U$ and $s \in \left\lbrace 0,1\right\rbrace$. The lower bound imposed on $m$ guarantees the uniqueness of the […]

Nonlinear simultaneous equations

I am looking at a system of nonlinear simultaneous equations. The two variables are u>0 and b>0. How can I solve this problem with computer packages, such as Matlab, Python, or Fortran? Thanks. $$\frac{40000-1.1u^{0.91}40000^{0.091}}{200}-b=0$$ $$640\pi\int_{0}^{b}t\left(\frac{40000-200t}{1.1u}\right)^{10}dt-4000000=0$$

Why does the midpoint method have error $O(h^2)$

This question already has an answer here: Numerical Approximation of Differential Equations with Midpoint Method 3 answers

Numerical solution to a system of second order differential equations

I’m writing a sort of physical simulator. I have $n$ bodies that move in a two dimensional space under the force of gravity (for instance it could be a simplified version of the solar system). Let’s call $m_1, \dots, m_n$ their masses, $(x_1, y_1), \dots (x_n,y_n)$ their positions, $(vx_1, vy_1), \dots (vx_n,vy_n)$ their velocities and […]

Symmetry Of Differentiation Matrix

I have a problem computing numerically the eigenvalues of Laplace-Beltrami operator. I use meshfree Radial Basis Functions (RBF) approach to construct differentiation matrix $D$. Testing my code on simple manifolds (e.g. on a unit sphere) shows that computed eigenvalues of $D$ often have non-zero imaginary parts, even though they should all be real. (For simplicity […]

Bifurcation Example Using Newton's Method

I am studying dynamical systems as part of a research project. I have been using Newton’s Method and studying the dynamic properties. Does anyone know where I could find a relatively simple example of bifurcation using Newton’s Method?

Power method for finding all eigenvectors

This is my homework. I was asked to find all eigenvectors of a symmetric and positive definite matrix by inverse power method with shifted. I encountered three problems: The eigenvalues to the matrix may not be distinct. In this case, how to find all eigenvectors corresponding to one eigenvalue? By the inverse power method, I […]