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

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5 views

on a quantisation of the bell curve

The bell curve function: $e^{-x^2/2}$ is an eigenfunction of the Fourier transform (FT) on the real line. Is its quantisation/discretisation the binomial distribution (coefficients $n$ choose $k$) an ...
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0answers
27 views

Minimizing the error by finding optimum step-size

I need to recheck a proof for minimizing the error by finding optimum step-size. I re-checked the proof many times but still can't find a mistake although the number I am getting in Matlab is not ...
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0answers
11 views

Gaussian Legendre quadrature

I understand how to find the abscissas of the quadrature for certain orders, and also known the weight for each abscissa is defined as $w_i=\int_{-1}^1l_i(x)dx$ where $l_i(x)$ is the $i$th ...
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1answer
18 views

Ex ODE: $y'=4t \sqrt y- \lambda(y-(1+t^2)^2)$

How to solve the following equation? $y'=4t \sqrt y- \lambda(y-(1+t^2)^2)$ $y(0)=a$ Show those cases where a numerical method will solve this equation exactly. $(a,\lambda) \in {\Re}^2$
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0answers
19 views

Stability of gradient flow $x'(t) = -\nabla E(x)$, $E(x)$ is convex function. [on hold]

Let $E: R^d \to R$ be convex and continously differentiable. The IVP $x'(t) = -\nabla E(x), x(0) = x_0, t>0$ is called the gradient flow. Show that gradient flow has following properties a) ...
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2answers
25 views

Explicit Euler Method's Order

I am seeing everywhere that the order of the Explicit Euler Method is 2 but I can't prove it on my own. The textbook that I found the method says that the proof is very easy so it is up to the reader ...
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1answer
26 views

Using Newton's Method to solve $f(x)=x^2-2bx+b^2-d^2=0$

What would be the Newton's method in the form $x_{k+1}=g(x_k)$ to solve the equation $$f(x)=x^2-2bx+b^2-d^2=0$$ in which both $b>0,d>0$ are parameters? I also need to show that $|g'(x)|\le 1/2$ ...
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1answer
17 views

For what values of $a>0$ is the convergence faster/slower than first order? [on hold]

$g(x)=a+x-x^2$ and sequence $x_{n+1} = g(x_n)$ For what values of a does this converge faster/slower than first order. I have no idea where to start. Would someone be able to assist me please?
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0answers
32 views

Solve equation $x^3=0$ using Newton's method [on hold]

Use Newton's method in Matlab or Octave to solve the equation $$f(x)=x^3=0$$ Characterize the convergence as linear or quadratic by tabulating the number of correct bits at each step of the iteration. ...
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0answers
8 views

Bounding the perturbation between eigenvectors

Can somebody explain this part of the proof of a deduction from the Davis-Kahan $\sin \theta$ theorem? I understand how to get from: $||P_{u_1} - P_{v_1}|| \le \epsilon$ to $||P_{u_1}v_1 - v_1|| \le ...
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0answers
29 views

Bisection Method Numerical analysis Problem.

I am trying to solve the following $$(\frac{x}{2})^2-\sin x = 0$$ with initial starting points $a_o = 1.5, b_0 = 2$ and $n = 1(1)5$ using Bisection Method. From the little I have studied, I went ...
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1answer
82 views

Newton method solving the quadratic equation [on hold]

Solving the quadratic equation using newton method.
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2answers
78 views

Newton's method arctan

Why does it oscillate? I looked up the graph of it and I think it is convergent? And when the function is $0$, the solution is also 0. What is the difference of choosing diverse starting values? ...
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0answers
43 views

Deriving a New Iteration Method by Solving a Quadratic Equation

My Question: Derive a new iteration method for solving $f(x)=0$ by solving the quadratic equation $$f(x_k)+f'(x_k)(x-x_k)+\frac{1}{2}f''(x_k)(x-x_k)^2=0$$ Complete your algorithm by specifying ...
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0answers
34 views

Newton's Method Solving an Equation

What would be the Newton's method in the form $x_{k+1}=g(x_k)$ to solve the equation $$f(x)=x^2-2bx+b^2-d^2=0$$ in which both $b>0,d>0$ are parameters? Additionally, I need to show that ...
3
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1answer
23 views

Presicion check for the solution of equations in Numerical Analysis

In order to have precision of 5 decimal points in a Numerical Analysis method for the solution of an equation then: $$|x_{n+1}-\rho|\leq \frac{1}{2}\cdot 10^{5}$$ where $x_{n+1}$ the current ...
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1answer
26 views

Simulate random variable with PDF $x^2+\frac4 3x$ on $[0,1]$

Consider $X$ a random variable with the following density function: $f(x) =$\begin{cases} 0, & \text{x ∉ [0,1]} \\ x^2+\frac4 3*x, & \text{x \in [0,1]} \end{cases} I need to write a ...
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1answer
50 views

Best algorithm to compute the first eigenvector of symmetric matrix

Assume that we have a real symmetric matrix $\mathbf{A}\in\mathbb{R}^{n\times n}$ obtained as following : $$\mathbf{A}=\mathbf{N}-\mathbf{P},$$ with $\mathbf{N}\in\mathbb{R}^{n\times n}$ and ...
5
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2answers
117 views

Exact Expression for numerical Solution 0.9595767

I need you to do just what any math genuis in a shallow Hollywood movie does: looking at big tables of numbers and seeing exact structure! These $3 \times 3$ matrices are solutions to a well-posed ...
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1answer
29 views

Stable algorithm for computation of $\Phi(20)$, when $\displaystyle \Phi(x)=\frac{2}{\sqrt{\pi}}\sum_{k=0}^{\infty}(-1)^k\frac{x^{2k+1}}{k!(2k+1)}$

Let $\displaystyle \Phi(x)=\frac{2}{\sqrt{\pi}}\sum_{k=0}^{\infty}(-1)^k\frac{x^{2k+1}}{k!(2k+1)}$, i.e $\Phi$ is the MacLaurin series of the function $\displaystyle ...
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1answer
52 views

RK4 wrongly predicts null solution

When solving the hydrogen radial Schroedinger equation (with $r > 0$ the radial coordinate) for angular momentum $L=1$ and the modified radial wave function $P(r)=rR(r)$, $P(r)$ satisfies: ...
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0answers
22 views

Galerkin formulation of 1D linear Schrodinger equation

I am interested in the weak (Galerkin) formulation of the 1D Schrodinger equation: $i u_t−\beta u_{xx}=0$ and $u(x,0)=u_0(x)$. As usual, we do integration by parts which yields: $i (u_t,v)+\beta ...
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1answer
29 views

Runge-Kutta 4 failure?

Say we want to solve numerically $y'(x) = f(x) \cdot y$, with $y_0 = y(x=0) = 0$ and applying RK4 method with step $dx = h$: \begin{align} k_1 &= f(0) \cdot y(0) \cdot h = 0\\ k_2 &= f(0+h/2) ...
2
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1answer
82 views

Differential equation: $\ddot{y}(x) + \alpha\dot{y}^2(x) + \beta y(x) = 0$

I am interested in finding an approximate solution for this differential equation, since the exact analytic solution seems to not exist. I tried with Mathematica and it spits out nothing. ...
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3answers
61 views

Gödel's Incompleteness Theorem and Numerical Analysis

I am no expert in Mathematical Logic so I can't really express my question formally but I do hope that it will make sense. As far as I know the implication of Gödel's incompleteness theorem for ...
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1answer
21 views

numerical solution of a transcendental equation

I need some help with a program in MATHEMATICA or MAPPLE, that solves the following transcendental equation: $$\alpha+2x+2\sqrt{\beta+\alpha ...
1
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1answer
52 views

Compare analytic model with numerical, mass spring system.

So I'm trying to solve a problem here and I have been working on it all day, clearly i'm in need of some guidance. I have a rod of length $L$ and cross section area $A$, Young's modulus $E$ and ...
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1answer
48 views

Singular solutions of a system of nonlinear 2nd order ODEs

I'm faced with the following nonlinear 2nd order system of ODEs: $$ \phi''(r)+\frac{4r^3-1}{r^4-r}\phi'(r)+\frac{r^2 h(r)^2+2r(r^3-1)}{(r^3-1)^2}\phi(r)=0, \\ ...
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3answers
26 views

How do we find more appropriate constants for expansions of functions?

We all knonw that the expansion of $e^x$ is $$1+x+x^2/2+...$$. But what if I want to find more approximate expansion of $e^x$. I try that $$e^x-1-c_0(x)+(c_0+c_1)(x^2/2)-(c_0+c_1+c_2)(x^3/3)=0$$ and ...
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0answers
23 views

Calculating B-Splines and dimension of spline space

I've got the following assignment: Let $S$ be the space of piecewise polynomials of degree $3$ on the intervall $[-1;1]$ with knots $x_i = -1+\frac{i}{2}, 0 \leq i \leq 4$. (a) Calculate a basis of ...
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0answers
60 views

Solving equation using Newton's Method

Use Newton's method to solve the equation $$f(x)=\frac{1}{x}+\ln{x}-2=0$$ for $x>0$. Characterize the convergence as linear or quadratic by tabulating the number of correct bits at each step of the ...
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1answer
14 views

Conditioning in regards to matrix vector product

This program involves the matrix-vector computational primitive $y \leftarrow Ax$ where $x,y\in\mathbb{R}^n$ and $A\in \mathbb{R}^{n\times n}$. A is taken to be dense and banded in the two parts of ...
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3answers
115 views

How to find the root of a polynomial function closest to the initial guess?

I need some easy to implement and fast numerical method that finds the root of a nonlinear function (a polynomial in my case) closest to my initial guess. If I know that there is one root ...
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2answers
29 views

find the limit of the first one

To prove the first one, can I just say that $\lim_{n\to\infty} {a^n/n^p}$= $(\lim_{n\to\infty} a^n)(\lim_{n\to\infty} 1/n^p)$ and when the absolute value of $a$ is less than or equal to one, ...
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0answers
32 views

let $\alpha \in \Bbb{R} $ and $\cos(\alpha \pi) = \frac{1}{3}$, prove $\alpha $ is irrational [duplicate]

Let $\alpha \in \Bbb{R} $ and $\cos(\alpha \pi) = \dfrac{1}{3}$, prove $\alpha$ is irrational. (Proof by contradiction) If we consider $\cos \left(\dfrac{m\pi}{n} \right)=\cos \left(\dfrac{ m\pi ...
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0answers
78 views

I am a Math Hobbyist. I have made some simple discoveries in Math. How do I share it with the Math community out there? [closed]

I am a Computer Engineering graduate and have taken many courses in Math of course. While I was in the University, I got myself lost in the world of mathematics and I discovered stuff that I felt ...
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0answers
13 views

Statistical calculation for neural firing rates with negative rate on numerical simulation

I am now working on a biological neural network simulation (NEST-Simulator) project with a problem of calculating firing rates. Background: The data set as result of simulation is a set of events in ...
0
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1answer
14 views

Conjugate gradient projection

Let $V$ be a collectino of the search direction for the conjugate gradient applied on a quadractic minimisation problem. As a proof of orthogonality in conjugate gradient: $$ V^T V = I $$ Now ...
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0answers
24 views

Relating to representation of real numbers. [closed]

Can someone tell me which representation is better for representing real numbers: fixed point representation or floating point representation? If the answer is circumstance dependent, please specify ...
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1answer
78 views

Does a convex hull solution in 3 dimensions result in a minimum-area or maximum-volume solution?

The wikipedia entry for convex hull shows a 2-d example of a random set of points on x-y plane, and the "elastic band" solution that bounds the points with the convex hull solution. The definition of ...
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0answers
31 views

In a floating number system, are there always as many numbers between 0 and 1 as between 1 and $\infty$.

The question is as the following, where $\beta$ is the base, $t$ is precision (length of decimals), $e_{\min}$ is the minimum exponent, and $e_{\max}$ is the maximum exponent. I am not sure, ...
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0answers
16 views

One-dimensional deblurring

I just begun studying image deblurring on my own, and I have a question. Most books I found say that I can see the images as arrays, and that I can "vectorize" the arrays of the images by stacking the ...
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0answers
32 views

Show that $|g'(x)|\le\frac{1}{2}$ whenever $x^2>2|c|$

Consider the fixed point iteration $$ x_{n+1}=-b-\frac{c}{x_n}=g(x_n)$$ How would I show that $|g'(x)|\le\frac{1}{2}$ whenever $x^2>2|c|$?
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1answer
28 views

Fixed Point Iteration Proof

Given the fixed point iteration $$x_{n+1}=\frac{-x_n^2-c}{2b}$$ where $b$ and $c$ are fixed, $x_n\longrightarrow x$, what does $x$ solve? Additionally, what is the region for $(b,c)$ values where our ...
0
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1answer
31 views

Trapezoidal rule - truncation error

I am trying to prove that when solving numerically diff. eq.: $$ y'(t)=f(t,y(t)), \hspace{0.5cm} y(t_{0})=y_{0} $$ using trapezoidal rule, namely: $$ y_{n+1}=y_{n} + \frac{h}{2} \left( f(t_{n},y_{n}) ...
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1answer
50 views

Inner product vs. vector triad form

This program involves the matrix-vector computational primitive $y \leftarrow Ax$ where $x,y\in\mathbb{R}^n$ and $A\in \mathbb{R}^{n\times n}$. A is taken to be dense and banded in the two parts of ...
2
votes
1answer
42 views

Conceptual Differences among Galerkin Methods

I have a conceptual question about numerical methods for second-order elliptic partial differential equations. What is the difference among finite element, continuous finite element, discontinuous ...
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1answer
40 views

Calculation of coefficients of a function with respect to Legendre polynomials.

$$f(x) = \begin{cases} -1 &x \in [-1,0],\\ +1 &x \in (0,1]\\ \end{cases}$$ the formula for calculation of coefficients in terms of Legendre polynomial $L_k(x)$ is: $f_k= ...
1
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1answer
22 views

Numerically integration with a an infinite upper limit and non-zero lower limit

I have seen lots of quadrature formulas where we have definite limits or one of the limits is infinity and the other is zero. But what about the following case $$f(x) = \int_a^\infty e^{\frac{x}{t}} ...
2
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0answers
13 views

Gauss-Green cubature in 2d

Hello friends of maths, I've given an arbitrary polygonal cross section (in cartesian coordinates $y$ and $z$). On this cross section, there acts an arbitrary stress-field $\sigma = f(y,z)$ as ...