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

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23 views

Is it possible for the Simpson's method to converge faster than Rombergs method?

I have the following integral: $\int_{0}^{100} \frac{x^{3/2}}{\cosh{(x)}}dx$ I am running code for the Simpson's method and Romberg method to evaluate the integral numerically and the results show ...
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1answer
16 views

Explain instability in Numerics so that I can understand and answer this question that involves roots of a equation

I found this question in my math book: Instability. For small |a| the equation (x - k)^2 = a has nearly a double root. Why do these roots show instability? I read and belive I understood the ...
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0answers
19 views

Can someone show me how to properly code the secant method with what I have been given? [on hold]

Function: z(t)=sin((pi)*t)*exp(-t) Using the secant method matlab code (with while loop) find the location of the points where the given function passes through z = −0.5. Confirm that a tolerance of ...
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2answers
30 views

Why is the estimate of the order of error in Trapezoid converging to $2.5$?

The integral in question is: $\int_{0}^{\infty} \frac{x^{3/2}}{\cosh{(x)}}dx$ I coded a program to compute $p$, an estimate of the order of the error for the Trapezoid method of numerical ...
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0answers
10 views

Von Neumann stability analysis of spatial models

I am interested to perform a Von Neumann stability analysis applied to the finite volume formulations of the wave equation. The reason I am doing the analysis is to assess the spatial discretisation ...
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0answers
13 views

Clarification of matrix equation needed in recursive least squares example.

I was looking at the answer to the post entitled "simple example of recursive least squares" and I would like to post a question concerning the matrix equation that is presented in the answer. First ...
4
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0answers
27 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 ...
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16 views

How to choose grid for a numerical integral of complex function?

I need to numerically integrate a complex function $f(x)$ on R, i.e. to approximate $\int_{-\infty}^\infty{f(\xi)d\xi}$. Performance is crucial as the integration is repeated a high number of times ...
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22 views

Finding the Equations of Motion for the Leapfrog Integrator

I understand that the Leapfrog Integrator is used to find an integral for Newton's Laws of Motion and that the Equation of Motion are given by: $$\frac{dx}{dt} = v$$ and $$\frac{dv}{dt} = F(x) = ...
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1answer
23 views

System with arbitrary function of an unknown

How can I solve the following system $$ (u_x)^2 - (u_t)^2 = 1 \\ u_{xx} - u_{tt} = f(u) $$ where $f$ is an arbitrary function of $u$, $u$ and $f$ to be determined. I don't know any approach, ...
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0answers
56 views

Crank-Nickolson

I have these two equation $$ \frac{\partial q(t,x)}{\partial t} = \frac{\partial^2 q(t,x)}{\partial x^2} - \frac{L_1 a(t, x) q(t, x)}{1 + \frac{L_2}{L} (1 - q(t,x))}\\ \frac{\partial ...
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3answers
34 views

Absolute error in computing a sum [on hold]

I couldn't solve this trouble, I hope you can give me some ideas. In computing the sum of an infinite series $\sum_{n=1}^{\infty}\,x_n$, suppose that the answer is desired with an absolute error ...
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3answers
43 views

4th-Order Runge-Kutta Method

I am struggling with this question regarding the 4th Order Runge-Kutta Method. I wish to find an approximate solution to the ODE: $$\frac{dx}{dt} = f(x)$$ Using the 4th Order Runge Kutta method: ...
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1answer
19 views

Asymptotic error expansion of global error for single step methods

My question refers to the proof of the following theorem, but it may suffice to just skip the theorem and continue with the problematic taylor expansion $(\ast)$: Let $f(t,y)$ and the single step ...
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0answers
20 views

Quantify difference between regularity and irregularity

I am solving an equation numerically on a 1D-domain using the finite-element method. I am solving it using two different domains, one regular and one irregular. Naturally, the solution varies slightly ...
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1answer
22 views

$L^2$ product of Chebyshev polynomials and Legendre polynomials

The following was a problem in a recent numerical analysis exam: Let $k \in \mathbb{N}\setminus\{0\}$. Prove or disprove: $$ \int_{-1}^{1} cos\left(k \operatorname{arccos}(x)\right) \cdot ...
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0answers
35 views

Inverse of $A^\top BA+C^\top DC$?

I'm numerically solving a system of equations of the form: $$Mx = b$$ where: $$M = A^\top BA+C^\top DC,$$ $B$ and $D$ are block-diagonal, $A$ and $C$ are $n\times m$ matrices with $m \leq n+3$. ...
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1answer
17 views

Improving the performance of eigs for a large spd Problem

I have two large (think around $100.000\times 100.000$), sparse, real symmetric and positive definite matrices $A$ and $B$ and I want to find the smallest generalized eigenvalue $$Ax = \lambda_{\min} ...
2
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1answer
67 views

Find quickest line of interception to a moving object

First, a visual illustration of the problem: http://tube.geogebra.org/m/1512793 The goal is to mathematically predict the direction in which the player need to run to intercept the ball as fast as ...
0
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1answer
61 views

Newtons Method to approximate inflection point

Here's a question from my tutorial which I'm having difficulties with. Consider the function $$f(x) =\frac{e^x}{( 1+ x^2)}$$ a) Show that $f$ has an inflection point at $x = 1$ My answer: $$f'' ...
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0answers
26 views

Number of Arithmetic Operations in Gaussian-elimination/Gauss-Jordan Hybrid Method for Solving Linear Systems

I am stucked at this problem from the book Numerical Analysis 8-th Edition (Burden) (Exercise 6.1.16) : Consider the following Gaussian-elimination/Gauss-Jordan hybrid method for solving linear ...
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0answers
12 views

numerical solution for nash equillibrium

I have the following setup. $\pi_1=f_1(q,r)$ and $\pi_2=f_2(q,r)$ are the real valued payoff/profit functions of the two players. Player 1 gets to pick $q$ and player 2 gets to pick $r$. I also know ...
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1answer
21 views

Stability of (floating point) computed variance

Homework Question from Accuracy and Stability of Numerical Algorithms, 2nd Edition, by Nicholas J. Higham, page 33: So every time we store an number and do a operation, we introduce an error bounded ...
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12 views

How to Solve Using Recursive Least Squares Approach

We start with the initial point $\hat{P}_0\!=\left(x_0,y_0\right)$ and the function $f\!\left(x,y\right)=K$ where $K$ is a constant real number and where $f\!\left(x_0,y_0\right)\!{\ne}K$. We are ...
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1answer
36 views

Simpson's rule with precision?

I have an integral: $$\int_0^1sinx^2dx$$ Task is to solve this integral using Simpson's rule with precision $\frac{1}{2}10^{-4}$. I am not sure how should I do that. I have this formula for ...
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0answers
21 views

Trying to model a substance settling in water using an advection equation?

I am trying to model a substance dispersed in a container of water gradually settling at the bottom. I am considering only one dimension. The top is at $z = 1$, and the bottom is at $z = 0$. So at $t ...
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0answers
16 views

Is there a big difference between runge kutta 4th for ODEs vs SDEs?

I was working on 2nd, 4th order runge kutta method for stochastic differential equations. I saw 2nd formula for ODEs and SDEs. There is some difference between their formulas . Unfortunately I can't ...
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3answers
22 views

What should the initial guess be for the Bablyonian method of calculating square roots?

You can use any value as the initial guess for the Babylonian method of calculating a square root (other than 0), but the closer the guess to the root, the more accurate your result per iteration. Of ...
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0answers
19 views

Efficient approximation to integration of analytic expressions involving product of four bessel functions

I have to take many integrals of the form $$ \int_0^\infty \!dx\,\,e^{-x}\,x^{\gamma - 2\beta - 2\alpha} j_\alpha ( u_1 x)j_\alpha (u_2 x)j_{\beta}(u_3 x)j_\beta (u_4 x),$$ where $\gamma$ is an ...
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0answers
23 views

How can I modify this simple code to include the pressure term? (1-D Navier Stokes)

I have a mathematical model that involves a cylindrical container that is being modeled with a one dimensional simplification as the system is isotropic with respect to the z-axis. As part of the ...
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1answer
15 views

newton raphson method convergence problem

My problem is: An iterative method to find $n$-th root of a positive number $a$ is given by $x_{k+1}=\frac{1}{2} \left[x_k +\frac{a}{x_k^{n-1}}\right]$ Find the value of $n$ for which this ...
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0answers
20 views

Reducing or avoiding the Gibbs phenomenon.

What is your favourite method which would help reduce the Gibbs phenomenon in Fourier Series and Fourier Transforms. This could mean pre-processing or post-processing or altering the transform. With ...
2
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0answers
62 views

Contour integral mystery: why is the answer different from Maple/Matlab?

The mystery is that here is a fairly standard contour integral which can be done by the residue theorem. Yet when I tried to evaluate it using numerical softwares like Maple or Matlab, the answer is ...
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71 views
+300

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)} ...
1
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1answer
22 views

Godunov scheme for advection equation

I'm trying to solve the advection equation $$m_t+(\alpha m)_x=0$$ with $m(0,\cdot)=m_0$ numerically using the first order Godunov scheme. Hence I write ...
3
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1answer
58 views

Finite difference method works for $\frac{\partial u}{\partial t} = \frac{du}{dz}$ but not for $\frac{\partial u}{\partial t} = - \frac{du}{dz}$?

I am using the method of lines with forward differences to solve the transport equation $$\frac{\partial u}{\partial t} = \frac{du}{dz}$$ with initial condition $u(z, 0) = z$ and boundary condition ...
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1answer
17 views

How to find X, Y Coordinates on wooden plane

I'm working on a wooden cnc machine. How to calculate the x, x coordinates? I have a square 300mm x 300mm I want my point to be on a circle and to find the x, y to drill a hole. Can you help me?
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0answers
8 views

von Neumann stability of the implicit downwind scheme

I want to investigate von Neumann stability of the implicit downwind scheme for this PDE: $u_t-u_x=0$. I got $\frac{\Delta t}{h} \geq 1$. It seems odd. I also checked the CFL condition of implicit ...
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0answers
19 views

Fluid dynamics: mesh resolution close to the origin in spherical co-ord system

Suppose you have a spherical implosion calculation (e.g. ICF etc.) in which you have a material interface that you want to apply some sort of perturbation to. There are two possible configurations in ...
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1answer
42 views

Nonlinear differential equations: numerical solution [closed]

I have to find and plot a numerical solution for this second order differential equation: $$u''(x) + \frac{u'(x)}{x} - \frac{u(x)}{x^2} + u(x) - u^3(x) = 0$$ where $0 \leq x \leq +\infty$ and $$ ...
0
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0answers
59 views

How to I approximate $I = \int_{-1}^{1} \sqrt{1-x^2}\cos(x)dx$ s.t. the error is bounded?

Edit: Because the original question was pretty trivial, I want to ask the same question but with:$I = \int_{-1}^{1} \sqrt{1-x^2}\cos(x)dx$. How to I approximate $I = \int_{-1}^{1} ...
2
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0answers
13 views

variable transformation in optimization

I have an optimization problem with two sets of parameters, $x_i \in [0,1]$ and $y_k \in [-\frac{\pi}{2},\frac{\pi}{2}]$ where $i,k \in \{1...n\}$ are indices. One way to solve this problem is using ...
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0answers
13 views

How to obtain the stability function of $y_{n+1}=y_n+h[\theta f(y_n)+(1-\theta)f(y_{n+1})]$?

I am going through a past exam paper but I don't know how to obtain the stability function in this case, I know how to do it when I have the matrix or when I have an explicit method, but not in this ...
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0answers
10 views

how to project optimal parameters on to feasible region

Hi: I'm trying to understand the concept of projection and I created a toy example that might help me to do that. Suppose that I have a non-linear optimization with 3 parameters theta_1, theta_2 and ...
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0answers
41 views

Why does this “incorrect” Chebyshev function approximation work better than the correct one?

I recently had the need to approximate this function $$f\left(x\right)=\begin{cases} \log\left(\frac{\pi+2\arcsin\left(x\right)}{\pi}\right), & x<0\\ ...
1
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1answer
43 views

What method of numerical integration is this?

I am trying to update some old code that finds the area under a curve from $17$ evenly spaced discrete data points. I'd like to update it to calculate from $65$ data points. I'd like to use the same ...
0
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1answer
19 views

Determine the order of consistency of $y_{n+1}=y_n+(h/2)(y_n'+y_{n+1}')+(h^2/12)(y_n''-y_{n+1}'')$ (I want to improve my answer)

I can solve this problem but I was wondering if there is a quicker way to do it since time will be tight during the exam... I would really appreciate your tips and advice on how to calculate this in a ...
2
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1answer
31 views

What type (explicit, Runge-Kutta, Taylor series, one-step, etc.) is the numerical method $y_{n+1}=y_n+(h/2)(y_n'+y_{n+1}')+(h^2/12)(y_n''-y_{n+1}'')$?

This exam question is asked every year, but I am struggling to understand the difference between numerical methods even though I can solve all the exercises. Thanks a lot in advance for your help! ...
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0answers
24 views

Intuitive explanation for error in Newton's Divided Differences?

When interpolating a smooth function $f$ using $n+1$ points, the error in the interpolation is bounded by $e(x) \leq$ $f[x_0,\ldots,x_n,x] \cdot \prod_{i=0}^n(x-x_i)$. This seems kind of interesting ...