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

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Why aren't numerical solutions (Euler method) to Lotka-Volterra system (all parameters equal to 1) periodic? [closed]

Why aren't numerical solutions (Euler method) to Lotka-Volterra system (all parameters equal to 1) periodic? Any help or just tips will be appreciated, thanks.
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1answer
22 views

Learning spectral methods in numerical analysis

I'm trying to learn the theory about spectral methods without any specific ties to a particular program like MATLAB. I tried to search for some lecture videos but it seems very limited and I'm not ...
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0answers
57 views

Confusion regarding dF/dx=0, F=constant

I thought I found a theorem Given a curve in the $(y,x)$ plane defined by DE $\frac{dy}{dx} = f(y(x),x)$ and if there exist a directional derivative of $F$ along this curve satisfies relation $g = ...
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0answers
31 views

Generate a random neutrally stable matrix

I need to generate random real matrices such that all eigenvalues have real part equal to 0 -- i.e. random neutrally stable matrices. What's the simplest way to do this? Note that I don't care about ...
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35 views

Is convex or non convex function?$J(u,c)=\int K(x).u.(f(x)-c)^2dx$

I have a function such as $$J(u,c)=\int K(x).u.(f(x)-c)^2dx$$ where $f(x):\Omega \to R$; c is constant; $0 \le u \le 1$; and K(.) is gaussian kernel. My question is that : Is J convex or non-convex ...
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1answer
25 views

Relationship between centralization and floating-point arithmetic

Suppose I have a problem of least squares where I have $n$ independents regressor variables $x_i$, and suppose I applied the process of centralization and scaling to this variable through ...
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1answer
63 views

Cholesky algorithm

Good afternoon everyone, I'm in need of a factoring algorithm cholesky and algorithms to solve upper and lower triangular systems, but I'm not finding any work in that octave. Recalling that need the ...
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1answer
49 views

fourth derivative

If I want to numerically compute a fourth derivative, I can adopt the central finite differences for the internal nodes, the backward finite differences for the last node, and the forward finite ...
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1answer
24 views

Find numerical minimum of a function with many parameters

I have a function $$f(\vec{r}_1\dots,\vec{r}_N)=\mathrm{The \ sum\ of\ square roots\ of\ the \ eigenvalues\ of\ }\Omega(\vec{r}_1\dots,\vec{r}_N)$$ And I want to find one of its local minima with ...
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1answer
35 views

System of linear equations: and a small perturbation

If $Ax=b$ and $Ax'=b'$ where $x'$ and $b'$ are $x$ and $b$ with a small perturbation, the following inequality will always hold: $ (\left\lVert x-x' \right\rVert/\left / \lVert x \right\rVert) \le ...
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1answer
38 views

Calculate the (variational) derivative of the following equation;

Consider $ E[u]= \int^1_0 \big(u'(x)\big)^2+\big(u(x)\big)^2-2f(x)u(x) dx.$ Calculate the variational derivation for a function $v$; in other words, calculate $\frac{d}{d\epsilon}E[u+\epsilon v]$ at ...
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34 views

Romberg Integration: accuracy

I'm applying the Romberg method to numerically integrate a data set of equally space, numerically determined values. I would like some estimate of the uncertainty (or accuracy or error) in my answer. ...
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1answer
33 views

Runge Kutta method order

I have a Runge-Kutta method given by the Butcher tableau: $$ \begin{array}{c|ccc} 0 & & & \\ 1/2 & 1/2 & & \\ 1/3 & 0 & 1/3 & \\\hline & -1/3& 1/3 &1 ...
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0answers
49 views

That is My class work problem. but I don't understand how to calculate this problem. Can u help me?

A Fourier analysis of the instantaneous value of a waveform can be represented by $$ y = (t + \pi/4) + \sin t + 18 \sin 3t $$ Apply the appropriate method to determine the value of $t$ near to $0.04$ ...
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0answers
25 views

Numerical Methods: Mid Point Higher Order ODEs

I am taking a Numerical Methods class and the professor told us to find out how to solve Higher Order Ordinary Differential Equations using the midpoint method. As of right now, I only know how to use ...
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0answers
36 views

Nonlinear Differential Equation with Pure Neumann Boundary

Four governing equations concerning the reaction occurred in the porous electrode are \begin{equation} \nabla \cdot i_1 + \nabla \cdot i_2=0 \end{equation} \begin{equation} i_2 = -\kappa \nabla ...
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1answer
17 views

How do I use these multi-step methods to solve the IVP?

Use the two second-order multi-step methods $$ω_{i+1} = ω_i + \frac{h}{2}(3f_i − f_{i−1})$$ and $$ω_{i+1} = ω_i +\frac{h}{2}(f_{i+1}+ f_i)$$ as a predictor-corrector method to compute an ...
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1answer
23 views

2D Richardson Extrapolation

I know Richardson extrapolation can be used to estimate a parameter at a single point, but is there a 2D analogous of it where it estimates a parameter over a surface? For example, I have a 1 m by 1 ...
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0answers
21 views

Simulating a SDE

I have a question about simulating a SDE. I want to simulate $dS=\alpha(K-S)dt+\sigma S dZ$ with use of a Euler-marayama scheme. The numerical scheme becomes: $S_{i+1}=S_{i}+\alpha(K-S_{i})dt+\sigma ...
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1answer
40 views

How to calculate the shielding time and determine the time step

The problem is illustrated as follows. A shielding plate scans over a target plate at a constant speed $v_{scan}$ and dynamically shadows the target plate to adjust the exposure time of the light ...
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1answer
60 views

Algorithms For Large-Scale $\ell_{\infty}$ Minimization

The general problem I want to solve is well studied: $$ \min_x \Vert Ax\Vert_\infty \;\;\; \mathrm{s.t.} \;\;\; Bx=c, $$ which is equivalent to the following linear program: $$ \min_{t,x} \, t ...
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0answers
45 views

Closed form for integral of an error function

My question is similar to that posted here. I have the following integral that I want to determine in a closed form. My uncertainty arises due to the addition term within the Error function: ...
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0answers
26 views

Smallest square problem, $A^*A$ singular?

In our numerics class, we have to solve the smallest square problem $Ax = b$ with $$A = \left( \begin{matrix} 1 & 3 &-4\\ 3 & 9 & -2\\ 4 & 12 & -6\\ 2 & 6 & 2 ...
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1answer
87 views

Evaluate integral over complex path numerically to show that $C_\infty$ is equivalent to $-I$

I would like to evaluate $$C_\infty = \int_{R = -a}^{R = a} H_0^{(1)}(z) e^{-izt} dz $$ where $H_0^{(1)}(z)$ is the Hankel function of the first kind, $a \rightarrow \infty$, and $$ z = R - i ...
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0answers
11 views

Finding common roots to a variable number of functions

I am trying to solve the following problem. Given $a\in\mathbb R^n$, $u\in\mathbb{R}^n$, $m\in\mathbb{N}^\star$, Find the/some common roots $(t_1,...,t_m)$ of the $\frac{m(m-1)}{2}$ ...
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2answers
39 views

Secant method with two ODE's of degree 2 - matlab

$$\frac{d^2r}{dt^2}-r\left(\frac{d\phi}{dt}\right)^2=G\cos\alpha-g\frac{R^2}{r^2}$$ $$r\frac{d^2\phi}{dt^2}+2\frac{dr}{dt}\frac{d\phi}{dt}=G\sin\alpha$$ The two ODE's above are given. I have written ...
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0answers
75 views

How do I solve first order non-linear system of PDE: $\partial f^i(x,y)/\partial z = F^i(f^1,f^2,…,f^n)$?

Suppose that I have a system of PDEs of the following form: \begin{eqnarray} \frac{\partial f^i(x,y)}{\partial z} = F^i(f^1,f^2,...,f^n), \qquad i = 1,..,n \end{eqnarray} Where $z = x + iy$, ...
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4answers
34 views

Soft absolute value

I'm looking for a "soft absolute value" function that is numerically stable. What I mean by that is that the function should have $\mp x$ asymptotes at $\mp\infty$ and behave smoothly in $[-1,1]$. ...
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1answer
40 views

Help Obtaining Numerical Approximation of Lambert W Solution

I am studying a particular generating function $$\frac{2e^x}{e^{2x}+1+2x}$$ and I thought I would try to solve the equation $$e^{2x}+1+2x=0$$ to determine for what value of $x$ if any the function ...
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0answers
38 views

Why do I get a big error when I compute this integral with Gauss-Legendre Quadrature?

I'm using Gauss-Legendre Quadrature to solve the following integral: $\int_0^{1}x^xdx$ After I've compared the result with the MatLab vpa(int(...)) of the same integral I've noticed that the ...
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1answer
36 views

fitting by linear combination of exponential functions

Suppose that we have a set of points $(x_1,y_1), \ldots (x_n,y_n)$, and we want to fit a function of the form $f(x) = ae^{2x} + be^x + c$ to those points. If we make $z=e^x$, then our function becomes ...
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1answer
53 views

Are my results realistic or is there an error somewhere?

The background is that I'm solving a problem in Numerical Analysis which I asked about here: Is my derivate correctly programmed? Now if I use the new code, then I get a result that is along the ...
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1answer
60 views

Is my derivate correctly programmed?

I'm solving a problem in numerical analysis, page 9 of this text (soundwaves under water) and I think that I'm getting the correct result but I'm not sure if I programmed my derivate correct. My ...
3
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1answer
48 views

Alpha max plus beta min algorithm for three numbers

There exists fast algorithm to approximate length of 2D vector - Alpha max plus beta min algorithm. It says that $\alpha\cdot\max(x,y)+\beta\cdot\min(x,y)\approx\sqrt{x^2+y^2}$ for some constants ...
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1answer
28 views

How to find a quadrature formula of a specific shape?

What are the steps one needs to follow to find a quadrature formula of a certain shape with maximal degree of precision. For example: Find a quadrature formula of the following shape $\int_1^2 ...
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1answer
38 views

How to find a Newton-Cotes formula with weights?

I want to build a Newton-Cotes formula with weights $\int_0^1f(x)x^\alpha dx = a_0f(0) + a_1f(1) + R(f), \alpha > -1$ But, I cannot find any example, moreover I don't really know where to ...
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0answers
34 views

Weighting on position within a range of numbers

Firstly, hello all. I'm normally to be found on StackOverflow but felt that this forum was more appropriate for my question. Count this is a coffee break teaser, rather than a fully challenging maths ...
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0answers
18 views

Multistep method Local Truncation Error

I was doing a practice exam for a Final I have coming up and I ran into this problem, and was unsure about how to approach b). Any advice would be greatly appreciated (As the final is in 5 hours)
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1answer
23 views

How to compute $\cos(x)$ within $n$ digit accuracy when $x = \sqrt{y}$ with $y \in \mathbb{N}$

How does one compute $\cos(x)$ within desired $n$ digit accuracy when $x = \sqrt{y}$ with $y \in \mathbb{N}$ and $x$ is not rational? The reason I am asking this question is that calculators ...
3
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1answer
79 views

Why are these functions called “kernels”?

In the last years while studying numerical analysis I came across different "kernels", like the Dirichlet Kernel $$D_n(x) = \sum_{k=-n}^n e^{ikx}$$ the Fejer-Kernel $$F_n(x) = \frac{1}{n} ...
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0answers
18 views

What is 'bursting' in least squares estimation, and what causes it?

I know as much that 'bursting' is some sort of unstable behavior of the least squares calculation, but more precisely what can one expect to see in the estimates in a bursting situation, what causes ...
3
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1answer
42 views

Show that the pivots of A are positive if and only if A is symmetric positive definite

I've been stuck on this question from a past exam for a while: Firstly is my understanding of the pivot correct? In this case I said our $2$ pivots would be a, and $c-b^2/a$ (Subtract $b/a$ * the ...
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0answers
30 views

Finite differences on a hexagonal/triangular lattice with Cartesian coordinates

So, I've been thinking recently about how to approximate the Laplacian operator using finite differences on a non-square lattice. For example, on a typical square lattice, in a Cartesian coordinate ...
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1answer
31 views

How to compute Newton-Cotes quadrature coefficients?

I'm struggling with the below problem. For the following approximation: $f''(x) \approx Af(x)+Bf(x+h)+Cf(x+2h)$ Find the coefficients A, B, C so that the degree of exactness is maximal and ...
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1answer
40 views

fourth order runge-kutta method and heavyside step function.

So I'm trying to model a hydrodynamic system that introduces a sudden "jump" in the value of a function at a specific time. The system is solved with a Runge-Kutta fourth order method. I have a ...
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1answer
29 views

When the false transient method could make an elliptic PDE easier to solve numerically?

I think that I do not fully understand the false transient method. This method consists on introducing a time derivative to an elliptic PDE to convert it to a parabolic PDE. However, it does not make ...
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2answers
40 views

converting numbers to degree

I have 0 to 1 that represent 0 to 360 degrees I know that 0.5 would represent 180 degrees but what formula would I use to get the other vales for 0 to 1. Thanks
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34 views

Galerkin method for the following integral equation

I have the following integral equation that I want to approximately solve for $u$ $$ u(x)=G(x_0,x)-\int\limits_{\partial D} \left\{ \frac{\partial G(y,x)}{\partial n(y)} +ik\beta(y) G (y,x) ...
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1answer
64 views

Gauss-Legendre Quadrature, computation of the abscissas and weights

I would like to write a program to calculate abscissas and weights of Gauss-Legendre Quadrature. I found the following source ...
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0answers
10 views

Backwards Stability of systems

Let $A$ be a nonsingular matrix, let $x_{k+1}$ be an approximation to the solution of $Ax=b$, and let $r^{k+1}=b-Ax^{k+1}$. Show that $x^{k+1}$ is $\epsilon$-backward stable approximate of ...