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

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1answer
28 views

Quadratic Formula expressed with Taylor's Theorem

I am having trouble solving the problem below. I think I understand the first part by just doing a taylor expansion of $f(\delta - a)$ where $a=0$ and the function equals $\sqrt{1-\delta}$. But I do ...
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2answers
40 views

Show Newton's method can go wrong with two roots

If $f:\mathbb{R} \to \mathbb{R}$ is differentiable with at least two roots, I wish to show that Newton's method will not converge for some $x_0$. I know that $f'(x)$ has a zero, say at $z$. It ...
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0answers
46 views

Are numerical approximation always possible in ODE if an solution exists? [closed]

Are numerical approximation always possible in ODE if an solution exists? Numerical approximation defined as the method to find solution in any degree of accuracy I don't know which theorem talk ...
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1answer
10 views

Claim: The complex conjugate of $ \omega_N $

My textbook says that $ \omega^{N-k}_N = \bar \omega^k _N $, where the bar denotes the complex conjugate. Why is this true? Sidenote: I believe $\omega^{N-k}_N=(e^{-2\pi i}/N)^{N-k} $.
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7 views

Taking the fourier transform of a CFD problem

I have a 1D diffusion problem, linearized, and whose transients I do not need. I need to calculate $\Delta \theta = \theta - \theta_{o}$, the deviation from equilibrium of the concentration. I am ...
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1answer
45 views

Numerical differentiation: 2-point vs 5-point method

I want to compare the following two numerical differentiation schemes: 2-point numerical differentiation: \begin{equation} \dot{\omega}_t = \frac{1}{dt} \left [ \omega_{t} - \omega_{t-dt} \right ] + ...
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1answer
27 views

Numerically solve ODE with boundary conditions

If I want to solve the eigenvalue problem $-y''=\lambda y$ with either periodic or antiperiodic boundary conditions on $[0,2\pi]$, how can I enter the boundary conditions? I mean, in general I would ...
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1answer
27 views

Numerical error analysis

I stand before the following task and I do not know how to solve it. The input parameter $$a=10^6, b=10^6 + 10^{-2}$$ will be round internally to $$a^*, b^*$$ with $$a=a^*(1 + \epsilon_1)$$ ...
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1answer
42 views

Floating point arithmetic operations when row reducing matrices

A numerical note in my linear algebra text states the following: "In general, the forward phase of row reduction takes much longer than the backward phase. An algorithm for solving a system is usually ...
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2answers
50 views

Explaining roundoff error when row reducing matrices

In my linear algebra textbook (in the context of row reducing and obtaining a matrix in echelon or reduced echelon form), there is a numerical note that reads as follows: "A computer program usually ...
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1answer
27 views

Which matrix norm is used here and why is the rate of convergence linear?

I am reading the paper "Centroidal Voronoi Tessellations: Applications and Algorithms" by Qiang Du, and I have questions about the following. Here, Du talks about fixed point iteration of Lloyd and ...
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0answers
20 views

Technical Worries with Lagrange Form of Remainder

Here is a typical problem in a beginning class in numerical analysis. (My own words) Let $f: \mathbb{R} \to \mathbb{R}$ be a $C^2$ function on $[x_1,x_2]$. Use Taylor's Theorem to find the local ...
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1answer
39 views

Euler's Method -Calculation of order without noticing to which number $\frac{e^i}{e^{i+1}}$ converges

We have the initial value problem $$\left\{\begin{matrix} y'=y, 0 \leq t \leq 1, \\ y(0)=1 \end{matrix}\right.$$ for which the solution is $y(t)=e^t$. Applying the Euler's Method we get the ...
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1answer
29 views

Show that for all $ t \in [a,b] $ it holds that $ |y(t)-z(t)| \leq |y_0-z_0|$

Let $f: [a,b] \times \mathbb{R} \to \mathbb{R}$ be a continuous function so that: $$ \forall t \in [a,b] \ \forall y_1, y_2 \in \mathbb{R} (f(t,y_1)-f(t,y_2))(y_1-y_2) \leq 0 $$ (So, for each ...
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0answers
54 views

Finite difference solution of steady-state diffusion equation with variable material properties

I'm trying to use a finite difference method to solve the steady-state neutron diffusion equation in a nuclear reactor: $$ D(x) \nabla^2 \phi(x) + \left( \frac{\nu(x)}{k} \Sigma_f(x) - \Sigma_a(x) ...
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2answers
41 views

Numerical presentation of a term

Hello I have the following task: Let $$ y = \sin(x + \delta) -\ sin (x)$$ and $$\delta > 0$$ is very small. Write down a mathematical equivalent representation of this term which is stable. I'm ...
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1answer
30 views

Show that assumptions of theorem hold, determine the solution

Consider the initial value problem $$\left\{\begin{matrix} y'(t)=\sqrt{|y|}, 0 \leq t \leq 2\\ y(0)=1 \end{matrix}\right. \tag 1$$ Show that for this problem the assumptions of the following ...
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1answer
51 views

the Chord Method does not appear to be converging under the same cond as Newton's Method?

The Chord Method is: $x^{(k+1)} = x^{(k)} - {g(x^{(k)}) \over g'(x^{(0)})}$ The question is to compute the cube root of 2, using the Chord Method. Carry out the first few iterations, using x(0) = ...
2
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1answer
28 views

A Variant of Gradient Descent

Suppose I have some objective function $f(\beta)$ which I would like to minimize for $\beta$. A standard gradient descent would be $\beta^{(t+1)}=\beta^{(t)}-\alpha \nabla f(\beta^{(t)})$, where ...
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1answer
30 views

Evaluating a cumulative distribution function from normal distribution

How one can prove by using only pencil and paper that $$0.99815<\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{2.92}e^{-x^2/2}dx<0.99825?$$ I think there is a mistake in my book which says that ...
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0answers
34 views

Existence and uniqueness of solution of the ODE

Consider the initial value problem $(1)\left\{\begin{matrix} y'(t)=y^2 &, 0 \leq t \leq 2 \\ y(0)=1 & \end{matrix}\right.$. Verify that the following theorem: "Let $c>0$ and $f \in ...
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3answers
52 views

Find the approximations to within $10^{-4 }$ to all the real zeros of the following polynomial using Newton's method.

We have $P(x)=x^3-2x^2-5$. I know the formula of Newthon's method. That is given here. The problem is, how do I find the approximations to within $10^{-4}$ to all the real zeros of the following ...
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1answer
25 views

Function doesn't satisfy the local condition of Lipschitz at intervals that contain $0$

The local Lipschitz criterion is the following: Let $c>0$ and $f \in C([a,b] \times [y_0-c, y_0+c])$. If $f$ satisfies in $[a,b] \times [y_0-c,y_0+c]$ the Lipschitz criterion as for $y$, ...
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1answer
61 views

Prove or disprove - Newton's method convergence in higher dimensions

It's not an exercise for uni or anything like that, just something that's been bothering me a bit and I can't seem to find useful information on the web on the matter. When talking about real valued ...
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0answers
47 views

1D Advection Equation - Numerical Scheme error estimation

I have the following PDE $$ u_t + au_x = 0, \; x \in \mathbb{R}, 0\leq t \leq T $$ For which I am using the following numerical scheme, with $u_j^n$ the estimate for $u(x_j,t_n)$, where $x_j = jh$ ...
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0answers
24 views

Creating FEM mesh for image region — what is the most suitable shape function?

I wish to create a FEM mesh to solve an inverse elasticity problem, for an irregular domain. This domain is given by a medical image, so it is discretised and each square on the grid has one scalar ...
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0answers
55 views

Gram-Schmidt to prove recurrence relation of orthonormal polynomials

I have been told to apply Gram-Schmidt to $x\pi_k(x)$ to show that $$ \pi_n(x)=(a_nx + b_n)\pi_{n-1}(x)+c_n\pi_{n-2}(x) $$ with $n≥2$, $a_n$, $b_n$, and $c_n$ all constants that depend on $w(x)$. ...
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1answer
57 views

How to compute orthonormal polynomials from weight function?

I have a weight function $w(x)=e^{-x}$ with $x$ from $0$ (inclusive) to infinity. How would I compute the first four orthonormal polynomials with respect to this weight function?
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1answer
24 views

Using weight function to construct a minimum

I have a continuous, bounded function $f(x)$ and a weight function $w(x)$ on the same interval. If $$\{\phi_i(x)\}^N_{i=0}$$ is a family of basis functions for a linear space and $N>0$, I need to ...
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0answers
13 views

LU Decopmostions with block

So both $A_{11}$ and $\hat{A_{22}}$ have $LU$ decompositions say $A_{11}=L_{1}U_{1}$ and $\hat{A_{22}}=L_{2}U_{2}$. Show that $ \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} ...
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0answers
24 views

Finding Fourier series coefficients numerically

Given a known function $f$, I am wondering how fast (depending on $n$) we can numerically approximate the Fourier coefficients $\int_0^1 f(x) e^{2\pi i n x} \, \mathrm{d}x$, either for fixed $n$ or ...
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0answers
22 views

polynomials to Chebyshev polynomials

I was wondering how they got from the polynomial to a Chebyshev polynomial as outlined in the image. Anyone know? The link to the paper is: The actual paper, I'm referring to
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2answers
45 views

Numerical solution of the Volterra equation with an exponential factor

Given : $$u(x)=x+2 \int_0^x e^{x-t}u(t)dt$$ Solve the Volterra Equation numerically using Trapezoidal Rule in $(0,5)$ choosing $n=8$ and compare with the exact values. The Exact Solution I ...
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1answer
75 views

How to obtain a convergent solution iteratively for a linear system of equations?

I am working on a problem that requires an iterative procedure to solve a linear system of equations, the system of equations in matrix form is: $$\underbrace{\begin{bmatrix} r_{11} & r_{12} ...
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1answer
31 views

Theory of computation

For any language $A$, $B$ and $C$ such that $A\subseteq B \subseteq C$, if both $A$ and $C$ are decidable, then $B$ is decidable. True or False? How can I find this?
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44 views

Solution to the heat equation using a finite difference scheme

I have used a difference scheme and Fourier Analysis to find an expression for the solution to the heat Equation. My problem is plotting my solution. $w_{k,j,m}=(1-\Delta t\mu_k)^msin(k\pi x_j)$ ...
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1answer
33 views

What is the logic behind Jacobi iterative method?

The book I follow and on net also, all that I can find is the algorithm to find the solution, but I don't quite understand the physical significance or logic behind the algorithm. Can someone please ...
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0answers
18 views

Order of accuracy of an scheme

How does one find order of accuracy for the following scheme (see picture below). The picture shows two equations instead of the typical one equation which confuses me a lot. You can read about the ...
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0answers
14 views

showing LU decomposition takes n^3 operations

from this link: http://www.personal.psu.edu/jhm/f90/lectures/lu.html it says: "Time savings associated with things like matrix multiplies can be a very big deal in many applications. If you look at ...
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0answers
24 views

Irregular grid for finite differences PDE solution

I got a project in a class I'm taking. In this project, I need to solve fluid dynamics and heat equations. Up to here, the problem is not so complicated, however the chamber shape is the problem: as ...
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1answer
21 views

How to show the relative error of $ \frac{x_A}{y_A}$

First, this is how the relative error of $x_Ay_A$ (approximated errors) is computed as compared to $x_Ty_T$ (true errors) - $ \displaystyle Rel(x_Ay_A) = \frac{x_Ty_T - x_Ay_A}{x_Ty_T}$, Letting ...
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1answer
32 views

deriving an integral quadrature rule on a triangle

I'm trying to look for references on this but I've not found any. I'm probably using the wrong keywords ... Let's suppose that our domain of integration $\Omega$ is the triangle in $R^2$ with a ...
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0answers
42 views

Solution of non-linear Fredholm(Hammerstein) equation with non-degenerate kernel and reciprocal non-linearity

I have asked this question but got no response. I rephrase it so that anyone who knows operator theory and integral equations would help me out.....I faced a problem in physics which is a non-linear ...
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0answers
53 views

Nonsmooth optimization

Now I have a chance taking a course in nonsmooth optimization, the course outline writes: convex analysis, subdifferential calculus and proximal mapping. various numerical algorithms to solve ...
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1answer
18 views

Does the following limit exist as a result of the bisection method?

Does the following limit exist as a result of the bisection method? $$\lim_{n\rightarrow\infty}\dfrac{|r-c_{n+1}|}{|r-c_{n}|}$$ where $r$ is the root as a result of the method, and ...
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1answer
100 views

Lagrange Interpolating Polynomials - Error Bound

Let $f(x) = e^{2x} - x$, $x_0 = 1$, $x_1 = 1.25$, and $x_2 = 1.6$. Construct interpolation polynomials of degree at most one and at most two to approximate $f(1.4)$, and find an error bound for the ...
3
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1answer
50 views

Numerical computation of unlimited small Julia set details

I've read the claim of a fractal image application to be able to show infinite levels of zoom for Julia sets for the classic iteration formula $z_{i+1}:=z_i^2+c$. The application has a realtime ...
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1answer
44 views

Find a polynomial

I have to find a polynomial with the following characteristics for a problem. Find a polynomial $p(x)$ such that $$p(-1)=p'(-1)=p''(-1)=p(1)=p'(1)=p''(1)=0$$ I know and understand the process of ...
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0answers
17 views

stability assessment

I have been asked to assess the stability of my numerical solutions to two different sets of transient differential equations that govern the same phenomena. I am not sure how I can assess and compare ...
5
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1answer
123 views

Convergence of Conjugate Gradient Method for Positive Semi-Definite Matrix

Let $A\in\mathbb{R}^{N\times N}$ be a positive semi-definite matrix, given $b\in\mbox{Col}\left(A\right)$ we want to solve the equation system $Ax=b$ . To add some notation, we define ...