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

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3
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2answers
39 views

Use Taylor's Theorem with $n=2$ to prove that the inequality $1+x<e^x$ is valid for all $x\in \mathbb{R}$ except $x=0$.

Use Taylor's Theorem with $n=2$ to prove that the inequality $1+x<e^x$ is valid for all $x\in \mathbb{R}$ except $x=0$. Taylor's Theorem: $$ f(x)=\sum_{k=0}^n{1\over ...
4
votes
0answers
23 views

Interval arithmetic with different definitions of intervals

Interval arithmetic normally deals with intervals defined as $[a,b]$ with rules like $$[a,b]\cdot[c,d]=[a+c,b+d]$$ I am interested in interval arithmetic with different interval definitions such as ...
2
votes
1answer
33 views

Name for some kind of logarithmic norm/error

As known $(\mathbb R, +)$ and $(\mathbb R^{+}, \cdot)$ are isomorphic with $\exp:\mathbb R\to\mathbb R^{+}$ as an isomorphism. When I transfer the absolute value $|\cdot|$ on $(\mathbb R, +)$ via ...
0
votes
0answers
14 views

How to derive the error when approximating divergence using the Gauss divergence theorem?

I am trying to derive the error for approximatively computing the divergence of a vector field $\mathbf{a}$. The Gauss divergence theorem states $\int_V \nabla \cdot \mathbf{a} dx = \oint_{\partial ...
1
vote
0answers
18 views

Von Neumann stability analysis of non-linear systems

The von-neumann stability analysis is based on the time and space discretisation schemes, what if the schemes are non-linear and too complicated to analyse. Is there a way to look at the matrices of ...
0
votes
0answers
17 views

Finite difference for variable coefficient with Neumann Boundary

The equations is the same as this post, but with respect to the Neumann boundary. The physically correct boundary conditions for this equation are \begin{equation} A(x)\frac{\partial u(x)}{\partial ...
1
vote
0answers
11 views

condition number with component-wise norm for the sample variance any help is appreciated! :)

I'm looking through some notes and came across the following two statements in the notes where the author states it can be shown that one leads to the next. I've tried to show this using the ...
-3
votes
0answers
17 views

what is formula to this eqution [(256)16]1/32+[(169)6]1/12 [on hold]

how to solve this equation [(256)16]1/32+[(169)6]1/12 what is formula of this? What is the closed form expression for this? What is the right domain for this Hamiltonian 2? what is the right ...
0
votes
0answers
34 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 ...
0
votes
1answer
24 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 ...
-1
votes
0answers
21 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 ...
0
votes
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 ...
2
votes
1answer
2k views

Numerically solving a system of nonlinear ODEs with boundary conditions

I have a system of 6 second-order nonlinear ODEs involving 5 different functions of a variable $t$. Every function has a boundary condition at $0$. I've never taken a differential equations class and ...
0
votes
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 ...
1
vote
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} ...
1
vote
1answer
25 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, ...
4
votes
0answers
31 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 ...
0
votes
0answers
57 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 ...
1
vote
0answers
23 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) = ...
0
votes
0answers
17 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 ...
2
votes
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
votes
3answers
48 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: ...
1
vote
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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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 $m \leq n+3$. ...
1
vote
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 ...
0
votes
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'' ...
2
votes
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 ...
3
votes
1answer
678 views

Find the error bound

Hey guys I am unsure how to find the error bound. Use the langrange interpolating polynomial of degree 3 or less and four digit chopping arithmetic to approximate $\cos(.750)$ using the following ...
4
votes
1answer
524 views

Software for numerical solution of a non-linear ODE system?

I have been given a nonlinear system of ODEs which has arisen out of a colleague's engineering research: $$\begin{array}{rcl} \dot{x}_0&=&x_1\\ ...
0
votes
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 ...
4
votes
0answers
73 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)} ...
2
votes
1answer
100 views

Reference request: Newton-Kantorovich hypothesis for polynomials of integral coefficients

Kantorovich's theorem states that the Newton method for finding the roots of a nonlinear function is guaranteed to converge if a parameter $h$, determined by the values of the function and its ...
0
votes
2answers
671 views

How to compute the eigenvalue condition number of a matrix

How to compute the eigenvalue condition number, $\kappa(4,A)$, of a matrix $A$ $$A = \begin{bmatrix} 4 & 0 \\ 1000 & 2\end{bmatrix}$$ I am a bit stuck on how to proceed solving this problem ...
0
votes
0answers
13 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 ...
1
vote
1answer
345 views

Fixed-Point Iteration method unable to converge to any of a function's infinte roots

An equation is given to me which has to be solved by direct iteration method: $$sin(x) = {x+1 \over x-1}$$ or $$f(x)=\sin(x)-{x+1 \over x-1} = 0$$ I follow the following procedure with reasons ...
0
votes
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 ...
0
votes
0answers
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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
2answers
47 views

Looking for a Finite Difference scheme of the following form…

I'm having trouble deriving a finite difference scheme that calculates the second derivative of a function on the boundaries of a non-uniform grid and makes use of a known first derivative at the ...
1
vote
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 ...
10
votes
1answer
123 views

Is anyone talking about “ball bundles” of metric spaces?

In differential geometry: Each smooth manifold $M$ is equipped with a tangent bundle $TM,$ which is a manifold equipped with a projection back to $M$ Given a smooth map $f : M \rightarrow N$ between ...
5
votes
2answers
331 views

'(Pseudo)-random functions' by seeding of PRNGs?

I have an application that wants controllable random functions from $\mathbb{Z}^2$ and $\mathbb{Z}^3$ to $2^{32}$ , where by controllable I basically mean seedable by some parameters (say, on the ...
0
votes
1answer
49 views

Finite difference differentiation formula

I'm trying to understand how the co-efficients of finite differences are calculated. In particular I'm interested in the first derivative for a uniform grid of unit width. I found this document ...
0
votes
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 ...
2
votes
2answers
80 views

Numerical solution to a system of secon order differential equations

I'm writing a sort of physical simulator. I have $n$ bodies that move in a two dimensional space under the force of gravity (for instance it could be a simplified version of the solar system). Let's ...