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

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1answer
30 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
30 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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1answer
21 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
25 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
19 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
31 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
16 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 ...
0
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1answer
19 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

Understanding golden section search

I don't understand it at all. The only thing I understand is that we have some interval in which we know a minimum lie and we know the function is unimodal. I also know that we are diving the interval ...
3
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0answers
16 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
36 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
54 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
41 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
21 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
78 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
8 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
28 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
50 views

Proof verification+proposition

Given 2 function $F(p,v)$ and $\frac{dF}{dv}=g(p,v)$ Differentiate F(p,v) with respect to v give $F_pf+F_v$ Formula 1 $$\frac{dF}{dv}=F_p\left(\frac{dp}{dv}\right)+F_v=g\\ ...
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0answers
17 views

How do I solve this system of PDEs numerically?

Suppose that I have a system of PDEs of the following form: \begin{eqnarray} (\frac{\partial}{\partial x} - i\frac{\partial}{\partial y}) f(x,y) = F(f,g,h) \\ (\frac{\partial}{\partial x} - ...
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4answers
29 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 $[-1,1]$. ...
1
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1answer
32 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
34 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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votes
1answer
23 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 ...
0
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1answer
48 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 ...
0
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1answer
57 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
votes
1answer
35 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 ...
0
votes
1answer
22 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
26 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 ...
2
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0answers
29 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
14 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)
3
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1answer
22 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
65 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
17 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
votes
1answer
37 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
6 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
23 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 ...
0
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1answer
31 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 ...
1
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1answer
23 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
36 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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0answers
30 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
45 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
8 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 ...
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0answers
23 views

Numerical scheme for system of PDEs

I'm trying to solve the following coupled PDE system for my master thesis: \begin{align} \kappa_0\frac{\partial p}{\partial t}&=- \nabla \cdot v \\ \rho_0\frac{\partial v }{\partial t} &= ...
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3answers
48 views

Linearizing an equation containing both $x$ and $\ln x$

The equation of interest is of the form: $$ k_1 \ln(y/x) = k_2 x $$ And I am wondering how can one linearize this equation for $x.$ Splitting the $\ln$ function would give something along: $$ k_1 \ln ...
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0answers
46 views

Convergence Newton method convex function

Is the Newton method convergent for convex differentiable functions and system of convex differentiable functions? Assuming you don't start with a stationary point and there exists a root. If not are ...
0
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1answer
31 views

numerical methods for ODEs

I am working on this equation: $$\frac{dx}{dt}=Ax+b$$ $$c'x=d$$ Where $x$ is a vector ,A is a constant matrix, b c are constant vectors. d is a constant number. i.e. $c_1x_1(t)+\cdots+c_nx_n(t)=d$ ...
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0answers
52 views

Proof that equation is non-convex function

I have a objective function as following $$E(\phi)=\int_{\Omega}(I(x)-m_1)^2H(\phi(x))dx+\int_{\Omega}(I(x)-m_2)^2(1-H(\phi(x)))dx+\int_{\Omega}|\nabla H(\phi(x)|dx$$ where $I$ is an image; $I: \Omega ...
0
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1answer
16 views

Numerically stable calculation of multinomial probabilities

I'm looking for a numerically stable method to compute expressions of the form $$\frac{(a+b+c+d)!}{a!b!c!d!}\left(\frac{1}{4}\right)^{a+b+c+d}$$ So far I've been using a compensated sum algorithm to ...
0
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1answer
24 views

System of equations from weighted Gaussian Quadrature

I've been working on a weighted Gaussian Quadrature problem for a Numerical Analysis class and have been having the hardest time. The problem boils down to solving a system of four equations: $$ ...