Questions on numerical analysis/numerical methods; methods for approximately solving various problems that often do not admit exact solutions.

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10
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0answers
125 views

Finding $1/x^2 + 1/x^3 + 1/x^5 + \dots $

The following function came up in my work: $$ f(x)=\sum_{p\text{ prime}}\frac{1}{x^p}=\frac{1}{x^2}+\frac{1}{x^3}+\frac{1}{x^5}+\frac{1}{x^7}+\frac{1}{x^{11}}+\cdots. $$ Naturally, this converges for ...
10
votes
0answers
795 views

Integral of rational function over $\mathbb{H}^4$

Suppose I have a rational function of $8$ coordinates $a,b,c,d,e,f,g,h$ that I want to integrate over $\mathbb{H}^4$: ...
9
votes
0answers
199 views

Could this approximation be made simpler ? Solve $n!=a^n 10^k$

I need to find the smallest value of $n$ such that $$\frac{a^n}{n!}\leq 10^{-k}$$ in which $a$ and $k$ are given (these can be large numbers). I set the problem as : solve for $n$ the equation $$n!...
8
votes
0answers
157 views

Open mathematical questions for which Numerical Search promising

I am fascinated by mathematical conjectures, especially those that are believed to be false. Now I wonder, are there some mathematical conjectures for which the numerical search would be promising? ...
8
votes
0answers
69 views

Stiff Nonlinear Differential Equations

As far as I know, the concept of stiffness is hard to define rigorously, but there are plenty of handwavy descriptions and motivating examples in the literature when it comes to linear differential ...
8
votes
0answers
200 views

fixed point iteration

I am trying to find the root of $f(x)=\arctan(x)$ by using successive iteration. There are some conditions to apply this in successive iteration . 1) The function has to be continuous. 2) $\sup|f'(...
7
votes
0answers
122 views

Tridiagonal matrix w/trigonometric eigenvalues

Let $N=2^n$ for a natural number $n$ and $B$ be the $N\times N$ square matrix of $0$'s and $1$'s $$ B=\begin{pmatrix} 0 & 1 & 0 & \ldots & 0 \\ 1 & 0 & 1 & \ldots ...
7
votes
0answers
167 views

Radial Basis Functions Interpolation

$ \let\oldcdot\cdot \renewcommand{\cdot}{\!\oldcdot\!} \newcommand{\e}{\varepsilon} \renewcommand{\p}{\varphi} \renewcommand{\p}{\varphi} \renewcommand{\vp}{\vec{\boldsymbol\p}(x)} \newcommand{\P}{\...
7
votes
0answers
79 views

Monotonic convergence of Newton's method for boundary value problems

I’m interested in solving nonlinear elliptic boundary value problems of the type $$ -a\Delta u + f\left(u\right) = 0, \\ u\big\vert_\Gamma = u_0 $$ by Newton’s method when its convergence is global ...
7
votes
0answers
244 views

What’s the best way to cut an apple?

Take the apple in one hand, and the knife in the other. In the first cut, the apple is divided in two pieces: a small one that drops into the plate and a big one that is still hold with the hand. This ...
6
votes
0answers
159 views

A new type of Arithmetic-Harmonic mean for $n$ numbers

Let's introduce the following iterative procedure. Take two numbers $x_0$ and $y_0$. $$a_0=\frac{x_0+y_0}{2}~~~~~~~~~~~b_0=\frac{2x_0y_0}{x_0+y_0}$$ $$x_1=\frac{x_0+a_0+b_0}{3}~~~~~~~~~~~y_1=\frac{...
6
votes
0answers
88 views

Implementation of a simulation of an incompressible Newtonian fluid with uniform density

Let $d\in\left\{2,3\right\}$ and $\Omega\subseteq\mathbb R^d$ be a bounded domain. I want to simulate an incompressible Newtonian fluid with uniform density $\rho$ and viscosity $\nu$. The evolution ...
6
votes
0answers
121 views

Condition for trigonometric inequality

I want to prove the following statement: Suppose $\frac{1}{4}(\cos(\theta_1)+\cos(\theta_2))^2+\lambda^2(a\sin(\theta_1)+b\sin(\theta_2))^2\leq 1$ holds for all $\theta_1,\theta_2\in[-\pi,\pi]$, then ...
6
votes
0answers
150 views

Solving numerically the equation of motion of D7 brane perturbation

I want to solve this equation $$ \partial_{\rho}^{2}\phi+\frac{3}{\rho}\partial_{\rho}\phi+\left(\frac{M^{2}}{(1+\rho^{2})^{2}}-\frac{l(l+2)}{\rho^{2}}\right)\phi=0 $$ numerically. I know that ...
6
votes
0answers
492 views

Runge's phenomen: interpolation error using Chebyshev nodes oscillates

We're trying to approximate the Runge function $f(x) = \dfrac{1}{1+25x^2}$ using Chebyshev nodes. When calculating the interpolation error, using different degrees ranging from 0 to 50, we get the ...
6
votes
0answers
486 views

Simulating from a Multivariate Gaussian without Cholesky

I'd like to draw a sample from a multivariate Gaussian distribution $\mathcal{N}(\mu, \Sigma)$, where $\mu$ is the mean vector (can assume it to be $\boldsymbol{0}$), and $\Sigma$ is a sparse positive ...
5
votes
0answers
58 views

Numerically solving a non-linear PDE by an ODE on the Fourier coefficients

I need to solve numerically a PDE of the form $$ u_t(x,t)=u_{xx}(x,t)+u_x(x,t)^2-a(x)u_x(x,t)-a_x(x) $$ with initial condition $u(x,0)=u_0(x)$. I can assume that both $u(\cdot,t)$ and $a(\cdot)$ are ...
5
votes
0answers
67 views

Efficient of Newton Polynomial Evaluation

The polynomial in Newton form having the coefficients $a_{0},a_{1},\ldots,a_{n}$ and centers $x_{1},x_{2},\ldots,x_{n}$ is the polynomial $$ p(x) = a_{0} + a_{1} (x-x_{1}) +a_{2} (x-x_{1})(x-x_{2}) + ...
5
votes
0answers
110 views

Jacobian of Stabilized Eikonal Equation $| \nabla u| = 1$

I am trying to implement a Finite Element Solution for the Stabilized Eikonal Equation : $$ |\nabla u| = 1 + \Gamma \Delta u, \quad \text{ where } \quad u = \text{ distance function }$$ $$ \text{...
5
votes
0answers
2k views

Simpson's Rule for Double Integrals

Simpson's Rule for double integrals: $$\int_a^b\int_c^df(x,y) dx dy$$ is given by $$S_{mn}=\frac{(b-a)(d-c)}{9mn} \sum_{i,j=0,0}^{m,n} W_{i+1,j+1} f(x_i,y_j) $$ where: $$W= \begin{pmatrix} 1&4&...
5
votes
0answers
138 views

Did I correctly derive the scheme for this PDE using the Crank Nicolson Method?

I'm taking an Applied Numerical Methods course this semester, and I was given the following homework problem: Basically, before I begin writing any sort of code, I would like to ensure that I have ...
5
votes
0answers
180 views

Diophantine approximation with additional constraints

I am trying to compute best rational approximations to various transcendental numbers $c$, subject to the following constraints: $$\frac {i j} {2^k} = c + \epsilon, \space\space2^n \le i, j \lt 2^{n+1}...
5
votes
0answers
276 views

What differential equation might model this almost-harmonic oscillator?

I need to precisely control the motion of a damped, driven (nearly) harmonic oscillator: $$ \ddot x(t) + \alpha\dot x(t) + \omega_0^2 x(t) \approx V(t) $$ I use the $\approx$ symbol because this is ...
5
votes
0answers
115 views

Invariant measure of Euler-Maruyama Discretisation of an Ito diffusion

Let $(X_t)_{t \geq 0}$ be a diffusion process with dynamics governed by the stochastic differential equation \begin{equation} dX_t = b(X_t)dt + \sigma(X_t)dW_t, ~~ X_0 = x_0, \end{equation} where $b,\...
5
votes
0answers
105 views

Another interesting integral related to the Omega constant

Another interesting integral related to the Omega constant is the following $$\int^\infty_0 \frac{1 + 2\cos x + x \sin x}{1 + 2x \sin x + x^2} dx = \frac{\pi}{1 + \Omega}.$$ Here $\Omega = {\rm W}_0(1)...
5
votes
0answers
1k views

3d-diffusion equation in spherical coordinates (numerical), boundary problem

There is one boundary problem $$\frac{\partial u}{\partial t}= \operatorname{div}\left(a^2 E \nabla u\left(r,\varphi,\psi \right) \right) $$ in a ball $$ B_{1}(0)=\left\{x \in \mathbb{R^3}: \left\| ...
5
votes
0answers
197 views

How do zeros on the complex plane affect the real number line?

Let's say there is a real-valued "signal" that you can only measure at discrete points $f(x)$. You have a theory that this signal is the result of an analytic function $f(z)$ on the complex plane but, ...
5
votes
0answers
461 views

How can I solve the Poisson PDE efficiently and fast in cylindrical coordinates?

I am trying to numerically solve the Possion PDE in cylindrical coordinate system. $$\Delta f = {1 \over \rho} {\partial \over \partial \rho} \left(\rho {\partial f \over \partial \rho} \right) + {1 ...
5
votes
0answers
569 views

Find roots of sum of sinusoids

Given this function and an initial point, find the next root: $$ \begin{align} f(t) & = -L\\ & {} + A \sin(\Theta_1 + \omega_1 t) \\ & {} +B \cos(\Theta_1 + \omega_1 t)\\ & {} - (...
5
votes
0answers
1k views

Convergence of Gauss-Newton method for piecewise linear functions

Notation for Gauss-Newton method Non-linear least squares problems are often solved by the Levenberg-Marquardt algorithm, which can be viewed as a Gauss–Newton method using a trust region approach. ...
5
votes
0answers
148 views

Shintani cone zeta function

Is there a procedure/algorithm for calculating sums of the form $$ \sum_{n_1,\ldots,n_r >0} \frac1{L_1(n_1,\ldots,n_r)^{m_1} \ldots L_r(n_1,\ldots,n_r)^{m_r}} $$ where $$ L_i(n_1,\ldots, n_r) :...
4
votes
0answers
46 views

Computing cube roots using three number means

I've asked a question some time ago about Computing square roots with arithmetic-harmonic mean but it turned out that this method is exactly the same as Newton's method (or Babylonian method) for ...
4
votes
0answers
19 views

What makes a geometric construction more or less stable?

As anyone who's actually done geometric construction of n-gons knows, not all construction methods are made equal. Some are very stable (the shape you get is always close to ideal even if you're not ...
4
votes
0answers
101 views

Large system of nonlinear equations

I am trying to solve a problem, which I find quite hard, like, headache-hard. I have to solve the following set of $M$ nonlinear equations: $$F(X)=\begin{bmatrix}f_1 (X)\\f_2 (X)\\...\\f_M (X)\\ \end{...
4
votes
0answers
114 views

Two Matlab ODE solvers, two different results

I am solving a system of ODEs using Matlab. One particular set of parameters caused the solver to fail, so I worked my way through the different solvers Matlab provides. I was surprised to find that ...
4
votes
0answers
50 views

Complex Chebyshev Polynomials

Chebyshev Polynomials can be used to compute a very nearly minimax polynomial approximation of an analytic function on $[-1,1]$. Is there a complex analog that can compute a nearly minimax polynomial ...
4
votes
0answers
39 views

Numerically iterating the dynamics of a constrained Newtonian system

This question is about the dynamics (in classical mechanics) of a rigidly linked chain of $N$ point masses, see figure. Let us say that the masses $m_1,\ldots,m_N$ have initial positions ($\mathbf{x}...
4
votes
0answers
45 views

Why does the pressure term complicate numerical methods for Navier-Stokes Equations?

I'm looking to code a solver for the Navier-Stokes equations. I will be using finite differences with the method of lines. Two questions: What is the significance of the pressure term in the full 3-...
4
votes
0answers
181 views

What does affine invariance mean in the context of the Newton's method?

The textbook Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (by Ascher, Mattheij, and Russell) states on page 329: [W]e observe that Newton's method is affine ...
4
votes
0answers
41 views

Squarefree products of a class of primes

Numbers which are the sum of two squares are the product of a square and a collection of distinct primes which are 1 or 2 mod 4. Landau proved that there are $\sim kx/\sqrt{\log x}$ such numbers up ...
4
votes
0answers
367 views

Optimized way to compute L1 distance matrix

I'm computing distances between two groups of multi-dimensional points giving a matrix of distances pairwise between points. For the L2 (euclidean) distance I can use optimized matrix multiplication ...
4
votes
0answers
563 views

Mean Absolute Deviation for a Stable Distribution as a Function of the Tail Exponent

Consider the standard Lévy-Stable (or Alpha Stable) distribution $S(\alpha,\beta, \mu, \sigma)$ where $\alpha$ is the tail exponent, $1 \leq \alpha \leq 2 $. Picking the symmetric case with $0$ mean (...
4
votes
0answers
197 views

How to compute Fourier coefficients using a cubic spline-corrected FFT?

I'm not particularly experienced in numerical analysis, and so I recently had quite a massive shock when I discovered that sampling a smooth function and computing the FFT of the result does not ...
4
votes
0answers
270 views

A late-diverging “approximating solution” for a system of functional equations

Peace be upon you, At the end of this question, I have shown that how computing MLE on an i.i.d Beta distributed data, results in the following system \begin{align*} &\begin{cases} \psi(\alpha)-\...
4
votes
0answers
107 views

Numerical integration of highly peaked 2D functions

I'm working on a problem that requires me to numerically integrate irregular and highly peaked functions $f:\mathbb{R}^2 \to \mathbb{R}, \, (x,y) \mapsto f(x,y)$ without a closed form. The method ...
4
votes
0answers
46 views

How to approximate derivative inside derivative

I am using a box-scheme for solving partial differential equation. The function is approximated with: $$ f_p=\psi\cdot\left(\theta\cdot f_{j}^{k+1}+(1-\theta)\cdot f_{j}^{k} \right) + (1-\psi)\cdot \...
4
votes
0answers
303 views

Test for equivalence of algebraic expressions

We are looking for the most efficient (most recent, or best) techniques to check if two algebraic expressions (elementary, Calculus-type functions) are equivalent (or if an expression is equivalent to ...
4
votes
0answers
186 views

Inexact Newton method.

Let's a nonlinear function $ f:[-1,+1]^N\subset\mathbb{R}^N\to\mathbb{R}^N,\; N\in\mathbb{N}, $ such that the the sequence generated by the method of Newton-Raphson $$ x_{n+1}=x_n-[Df(x_n)]^{-1}\cdot ...
4
votes
0answers
685 views

How to avoid numerical overflow while computing a sum of products?

Suppose we have $N$ vectors $\vec{x}_1, \vec{x}_2,\dots,\vec{x}_N$. $\vec{x}_i$ is a $M$-dimensional vector: $\vec{x}_i = \left[ x_{i1}\;\; x_{i2}\;\; \dots \;\;x_{iM}\right]^T$ with all $x_{ij}>0$....
4
votes
0answers
2k views

Computation of coefficients of Lagrange polynomials

For our homework we should write a program, that creates Lagrange base polynomials $L_k(x)$ based on a few sampling points $x_i$. Now i am eager to develop a formula to be able to compute the $\ell$-...