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

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7
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174 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 ...
5
votes
0answers
161 views

Propper algorithm to integrate ODE's numerically.

I have studied in a course several algorithms to integrate ODE's numerical: Runge-Kutta, Predictor-Corrector methods, Taylor... However the teacher failed to show which is the best for every ...
5
votes
0answers
91 views

What's the most efficient way to mow a lawn?

For $S\subseteq\Bbb R^2$ and $x\in\Bbb R$, define $E_x(S)=\{y\in\Bbb R^2:d(y,S)<x\}$. ($E_x(S)$ represents the expansion of $S$ by $x$.) Given a path $\gamma:[0,1]\to\Bbb R^2$, denote its length as ...
5
votes
0answers
87 views

Lagrange Interpolation

So the problem is this; For Lagrange Interpolation on the nodes $x_0 < x_1 < ... < x_{n-1} < x_n$ of the data ${(x_i, f(x_i))}^{n}_{i=0}$ the interpolating polynomial is $\sum ...
5
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96 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 ...
5
votes
0answers
100 views

Do there exist solutions for this equation?

We know that solutions exist for equations of the following variety: $$ye^y=x \iff y=W(x)$$ Where W is the Lambert W function. We can augment the problem slightly, and ask if there exist solutions ...
5
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476 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
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222 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 ...
5
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0answers
278 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
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1k views

Restricted Three-Body Problem

The movement of a spacecraft between Earth and the Moon is an example of the infamous Three Body Problem. It is said that a general analytical solution for TBP is not known because of the complexity ...
5
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0answers
126 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
91 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 ...
4
votes
0answers
56 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
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59 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 ...
4
votes
0answers
33 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
156 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
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0answers
145 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, ...
4
votes
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150 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
360 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 ...
4
votes
0answers
366 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)\\ & {} - ...
4
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0answers
306 views

Find the error approximation for the function $f(x) = \dfrac{1}{1-x}$

Problem Let $f(x) = \dfrac{1}{1-x}$, find the Taylor polynomial $P_n(x)$ about $x_0 = 0$. Find a value of $n$ such that the approximation is within $10^{-6}$ on $[0, 0.5]$. To find $P_n(x)$ is ...
4
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0answers
250 views

Computing complex principal value integral - sgn-function?

I currently face a less appealing integral which emerged computing the expectation of some random variable. It reads as (omitting all unnecessary constants except $\alpha\in(0,1)$) $$ PV ...
4
votes
0answers
782 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. ...
4
votes
0answers
272 views

Evaluating matrix-continued fractions?

I have a matrix-valued continued fraction defined in the following way: $\alpha_n$ and $\beta_n$ are matrices, and I am interested in the quantity $A_1$, where all the $A_n$, $n = 1, 2, \dots$ are ...
4
votes
0answers
88 views

Efficiently solving a large, sparse linear system $M(s)ab(s)=c(s)$ (determined by smooth functions) over some range of $s$

I'm looking at a differential equation on the edges of a graph (the application is neuroscience), and the Laplace transform of the solution on most of the edges has a general solution more-or-less of ...
4
votes
0answers
215 views

Block Toeplitz Matrix

What is the name of the following matrix: $$ \begin{pmatrix} a & b & 0 \\ c & d & 0 \\ 0 & a & b\\ 0& c& d& \\ b & 0 & a \\ d & 0 & c\end{pmatrix}\ $$ It looks like a Block Toeplitz matrix, but ...
3
votes
0answers
54 views

The definition and meaning of “machine epsilon” in MATLAB

I am taking a introductory course in numerical mathematics, using MATLAB and a numerical math text that refers to MATLAB often. In the text, the machine precision is defined as: The distance ...
3
votes
0answers
39 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 ...
3
votes
0answers
55 views

estimations of solutions

Let $\Omega=\mathbb{R}^2_+=\{(x,y) \in \mathbb{R}^2; y > 0\}$, $f \in L^2(\Omega)$, $\lambda \in \mathbb{R}^*_+$, $A=(a_{ij})_{1\leq i,j \leq 2}$, $a_{ij} \in \mathbb{R}$ and there exist $\alpha ...
3
votes
0answers
38 views

Find closest vector to a given vector from a particular set of vector

Let $x=\left(x_t\right)_{t=1}^n$ be a vector such that $$ x_t = \prod_{i=1}^t u_i, \tag{1} $$ where each parameters $u_i$ can take any of two value $$ u_i \in \left\{a,b \right\} = \left\{ 1.3, 0.8 ...
3
votes
0answers
33 views

Show that this initial-value problem has a unique solution

I am trying to show that the following initial-value problem $$\frac{dx}{dt} = - x + tx^{1/2}; \quad x(2) = 2$$ has a unique solution on $I = [2,3]$. By letting $f(t,x) = - x + tx^{1/2}$ and $(t_0 ...
3
votes
0answers
94 views

Loss of Significance problems - Taylor Expansion

(2) This question addresses the notion of loss of significance. You are encouraged to revisit the Taylor series expansion that you have learned in calculus, as you will need to apply it here. Explain ...
3
votes
0answers
75 views

Crank Nicolson Method PDE

I have the following PDE $0=\partial _t u+\frac{1}{2}\partial_{xx} u$, now I assume that $t\in[0,T], x\in[0,L]$ and initial data $u(T,x)=g(x), u(t,0)=a(t), u(t,L)=b(t)$ The grid $\{(ik,jh): ...
3
votes
0answers
63 views

Difference between Householder Reflections and Gram-Schmidt?

In numerical QR decomposition, when we calculate the orthonormal factor Q of a matrix, what is the difference in results if we use Householder Reflections to normalize the matrix or use Gram-Schmidt ...
3
votes
0answers
63 views

solve nonlinear second order ODE

I obtained Nonlinear second order differential equation as $y\cdot y''+y'^2-m\cdot y^{-a}y'^2+k=0$, Where $y'= \dfrac{dy}{dx}$, $y''=\dfrac{d^2y}{dx^2}$. I could not obtain the solution so please ...
3
votes
0answers
35 views

Finite Difference Methods for arbitrary elliptic PDE

I am looking for textbook references that describe lattice numerical methods for arbitrary elliptic PDEs, particularly finite difference schemes and particularly in 2d. The few references that I have ...
3
votes
0answers
75 views

In which space does my (weak) solution exist?

I am reading through some numerical analysis papers, and was wondering what characteristics of a problem define which function space the weak solution of a problem belongs to. For example, for the ...
3
votes
0answers
129 views

How to solve the advection equation with spiral motion

The advection equation is : $$\frac{\partial f(x,y,t)}{\partial t} + \nabla_{(x,y)} \cdot (A f)= 0$$ With initial condition $f(x,y,0) = f_0(x,y)$. If the vector $A$ is constant, ie. $A = ...
3
votes
0answers
50 views

Converting a linear program into standard form

In especially, I have a question about the demand that if I have $ Ax \leq b$, then I can convert this into $A'x'=b$ for some new $A'$ and $x'$. I have given the system of equations: $20x_1+30x_2 ...
3
votes
0answers
102 views

Differentiation in DFT

Let's consider some function $f: \mathbb R \to \mathbb R$ vanishing outside of some interval $(0,L)$. Then I can approximate the Fourier transform $\hat f$ by $$ \hat f (k) = \int_{\mathbb R} e^{-i k ...
3
votes
0answers
44 views

Timestepping PDE with positive eigenvalues

I'm trying to numerically solve a PDE, namely: $$ \partial_t \binom{u(x, t)}{v(x, t)} = x \left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right) \cdot \partial_x \binom{u(x, t)}{v(x, t)} ...
3
votes
0answers
96 views

Inadmissibility of Simpson's rule

Let $B_t$, $t\ge0$ be a standard Brownian motion and suppose $0<x_1<x_2<\cdots<x_n<1$. Then the conditional expectation $$ \mathbb E\left(\int_0^1 B_t\,dt \,\middle\vert\, B_0, ...
3
votes
0answers
59 views

Determining algebraically a point of intersection.

A student I was tutoring posed the question: "I know how to solve $$e^{-x} = \ln x$$ graphically, however how do you solve this algebraically?" I have been fiddling around with it for a while and I ...
3
votes
0answers
146 views

Runge-Kutta Error Analysis

Could anyone explain to me how to reduce the error propagated by using Runge-Kutta of order 4? Or can anyone give me a nice reference to it.
3
votes
0answers
248 views

How to solve the following system with parametric equation?

What is the maximum number of solutions for this system? Parameters $a_0, a_1, ..., a_{2013}$ could be any numbers. \begin{cases} y = a_0 + |x - a_1| + |x - a_2| + ... + |x - a_{2013}|\\ x^2 + y^2 = 1 ...
3
votes
0answers
49 views

finding the largest $p$ components of $x$

Given an $n \times n$ matrix $A$, and an $n \times 1$ vector $b$, the conventional way of computing an $n \times 1$ vector $x$ such that $x=Ax+b$ is to use the following iterations: ...
3
votes
0answers
108 views

How should c be chosen to ensure rapid convergence of $x_{ n+1}= x_ n+c(f( x_ n))$ to $\alpha$?

Consider the rootfinding problem $f(x)=0$ with root $α$, with $f´(x)≠0$. Convert it to the fixed-point problem $x=x+cf(x)≡g(x)$ with $c$ a nonzero constant. How should c be chosen to ensure rapid ...
3
votes
0answers
87 views

Gently push my non-Positive Definite matrix back into the set of Positive Definite matrices

I have a matrix $\eta$ that should be Positive Definite but it is not. Is there a numerical method to gently push my non-Positive Definite matrix back into the set of Positive Definite matrices? ...
3
votes
0answers
72 views

Calculate sums of logs in precision

I am encountering a situation where I cannot calculate exact sum of a seris of logorithms in calculating entropy. Suppose we have a series of numbers $p_i$ and we want to calculate $\sum_ilog(p_i)$, ...
3
votes
0answers
345 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 ...