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

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12 views

Stirling's formula 12

Hello, how can we use Stirling's Formula to approximate a ratio? In Burden and Faires, it says the method is used to approximate f(x). Would we first have to use Stirling's formula to approximate ...
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0answers
12 views

Eigenvalue interpretation with direction field

I am running into some trouble with respect to some direction field plots of different eigenvalues. I am working with a system given as follows: $$ \overrightarrow{y'} = \begin{pmatrix} 0 & -5 ...
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0answers
27 views

Tridiagonal matrix w/trigonometric eigenvalues

Let $N=2^n$ for a natural number $n$ and $B$ be the $N\times N$ square matrix of $0$'s and $1$'s $$ B=\begin{pmatrix} 0 & 1 & 0 & \ldots & 0 \\ 1 & 0 & 1 & \ldots ...
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1answer
15 views

2-Norm of Non-Square Matrices

So, the 2-norm of an m x n matrix for m=>n is defined by the max singular value/square of the max eigenvalue. But, if it's not square, and you're only given a matrix A (no x-vector), what do you do if ...
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1answer
24 views

Taylor Expansion of sine and cosine

Here: Is there a formula for sine and cosine?, one of the answers mentioned using the Taylor expansion for approximating $\sin(x)$ and $\cos(x)$, and someone commented pointing out however that this ...
3
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1answer
23 views

$LU$ Factorization

Suppose the $A\in\mathbb{R}^{n\times n}$ is nonsingular and that $A = LU$ is its $LU$ factorization. Give an algorithm that can compute, $e_i^TA^{-1}e_j$,i.e., the $(i,j)$ element of $A^{-1}$ in ...
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1answer
16 views

Trapezoid rule for finding coefficient

If we know that $\int_{a}^b t(x)=h \sum_{k=1}^2 dk * t(a+kh)+O(h^m)$ where $h=\frac{b-a}{3}$, how do we find the coefficient d1, d2 and m in the equation? Answer says that d1=3/2, d2=3/2, m=3 I ...
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2answers
36 views

Numerical approximation of $\sin(11/8)$

How one can prove by pen and paper that $0.98089<\sin\left (\frac{11}{8}\right )<0.9809$? I was thinking some series proof but I'm not sure how to prove that the error is small enough. I also ...
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1answer
34 views

When does Newton Raphson Converge/Diverge?

Is there an analytical way to know an interval where all points when used in newton raphson will converge/diverge? I am aware that newton raphson is a special case of fixed point iteration where ...
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0answers
18 views

Relation between error of estimate and rate of convergence

How is bounds on estimated error of an iterative algorithm related to rate of convergence? Referring to references is appreciated.
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0answers
15 views

Solving 1-d wave equation as conservation law

So the long story short is the following. I break the 1-d wave equation ($\frac{\partial^2u}{\partial x^2}=\frac{\partial^2u}{\partial t^2} $) into a system of first order hyperbolic equations. I ...
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2answers
43 views

Sign of the error in Simpson's rule

Let $f : [a,b] \to \mathbb{R}$ be a $C^\infty$ function. The Riemann integral $I = \int_a^b f(x)\,dx$ can be approximated by using Simpson's rule: $$I \approx S = \frac{b-a}{6} \left[ f(a) + 4 ...
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1answer
23 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
34 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
13 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
21 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
28 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
28 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
22 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
37 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
30 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
83 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
44 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
votes
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
27 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
52 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
votes
2answers
118 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 ...
1
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1answer
30 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 ...
1
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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
25 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
votes
1answer
83 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 ...
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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 ...
0
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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
15 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 ...
3
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3answers
116 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
79 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
14 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 ...
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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 ...