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

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How should c be chosen to ensure rapid convergence of $x_{ n+1}= x_ n+c(f( x_ n))$ to $\alpha$?

Consider the rootfinding problem $f(x)=0$ with root $α$, with $f´(x)≠0$. Convert it to the fixed-point problem $x=x+cf(x)≡g(x)$ with $c$ a nonzero constant. How should c be chosen to ensure rapid ...
3
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104 views

Numerical Analysis Gauss-Lobatto

I am trying to find the expression of the weights and nodes for the Gauss-Lobatto quadratures with 4 nodes. I am guessing this is a sum of weights? Does anyone here have experience working with this ...
3
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106 views

Gently push my non-Positive Definite matrix back into the set of Positive Definite matrices

I have a matrix $\eta$ that should be Positive Definite but it is not. Is there a numerical method to gently push my non-Positive Definite matrix back into the set of Positive Definite matrices? ...
3
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76 views

A basic question about randomly generated matrix

I have read in many research papers related with iteration methods to find the generalized inverses. Where to show efficiency of the methods randomly generated matrices of higher order have been ...
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298 views

Runge function error second factor

I'm currently learning about the Runge function. On Wikipedia, I read the following: Consider the function: $ \dfrac{1}{1+25x^2}$ Runge found that if this function is interpolated at ...
3
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353 views

linear interpolation error estimate for non-smooth function

Suppose I have two points $x_1,x_2$ between which I would like to have a linear interpolation $P_1$. I know the value of the function $f$ at $x_1,x_2$. The error at any point between the two will be ...
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338 views

Intervals for Newton's method

I have a function $$ F(x)= \frac{x^3 - 14x^2 + 7x + 203}{(x-3)(8-x)} $$ I need to use Newton's Method to find the max interval such that a number of constraints are valid. • $3 < a < b < 8$, ...
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289 views

Approximating a system of differential equations as a Bézier curve

I am looking for a general transform to approximate the solution to an n-dimensional system of differential equations and initial conditions as a cubic or quadratic Bézier curve. Sorry if my ...
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499 views

Approximating a function with a piecewise constant function

I have some distribution X of values (which I don't know exactly but I can sample many times). I also have a function $f : X \to Y$ which may be complicated. I want to approximate $f$ with a piecewise ...
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15 views

Bounds on the size of Voronoi cells

I am working on an algorithm for which bounds on the size of voronoi cells will come in handy. Suppose that the domain $D$ is partitioned according to the Voronoi cells $D_1,\dots,D_n$ with Voronoi ...
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19 views

Is it possible to design a strongly stable linear multistep method of order 7 which has stiff decay

I'm studying for a test and I'd like to know is it possible to design a strongly stable linear multistep method of order 7 which has stiff decay. I have no clue to verify the claim. Can anyone give me ...
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30 views

LU Decomposition vs. QR Decomposition for similar problems

Suppose I want to solve the 2D Poisson equation with Neumann boundary conditions. The solution is non-unique up to an additive constant. I have previously asked a related question here for the 1D ...
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52 views

Looking analytically if one formula is better than another one

I'm studying errors in functions on numerical methods. On my notes, I've written the Heron's Formula: Let $a\geq b\geq c$:$$A=\sqrt{p(p-a)(p-b)(p-c)}\ \ ,$$ where $$p=\frac{a+b+c}{2}.$$ This formula ...
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33 views

Kink, Antikink, solving the wave equation numerically.

I've got a task of solving the wave equation with a potential $$u_{tt}-u_{xx}+V'(u)=0,$$ where $$V(u) = \frac{u^2(1-u)^2(1+2u)^2}{2}$$ on Python. I'm not exactly sure how to do this; my lecturer ...
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23 views

Why use the logarithm of the relative error?

In my numerical analysis course, we had an assignment to use MATLAB to numerically solve the Poisson Equation $-\nabla\cdot\nabla u = 0$ in one dimension. We computed the numerical solution, plotted ...
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26 views

Gaussian and binary probability random variables numerical reconstruction

in my probability class I was given this question dealing with MATLAB code the purpose of which is to create and re estimate the random variables Z1, Z2, which reads as follows: rng('default') ...
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44 views

'Stable' Ways To Invert A Matrix

So lets say that I need to invert a matrix that is generally dense and is poorly conditioned. What are some ways I can get an accurate inverse? Here are my candidates: SVD Inverse Inverse Via ...
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23 views

Which error does one usually need to consider in numerical analysis?

When analyzing performance of a numerical method, I have considered and plotted $$\| x_n - x^* \|$$ where $x_n$ is the $n$-th iteration and $x^*$ is the true value (in our toy examples, it wasn't hard ...
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17 views

Method to Linearise PDE

I have a Monge-Ampere-type PDE I wish to solve using a finite difference method: $$(1-u_{xx})(1-u_{yy}) -u_{xy}^2 = f(x,y).$$ Is there generally a preferred method for linearising the system after ...
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22 views

A proof of the $QR$ algorithm when $A$ is symmetric and tridiagonal that doesn't involve shifting

Consider the following special case of the $QR$ algorithm: Let $A$ be symmetric and tridiagonal. Define the following sequence of matrices: $A_1 :=A$, and for $n \ge 1$ decompose $A_n=Q_n R_n$ ...
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28 views

Proof of Runge's phenomenon for a concrete case

Let $f(x)=\frac{1}{1+25x^2}$ and range is $[-1,1]$. Given $n+1$ equidistant points $x_0 = -1,x_1,...,x_n = 1$ and their values $f(x_0),f(x_1),..,f(x_n)$, perform polynomial interpolation by the $n+1$ ...
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23 views

Very Basic Numerical Methods Book for Freshman students

To cut a long story short; the nature of this degree (it's not a college degree) is such that numerical methods is treated shortly after Calc I (single-var) and linear algebra, but before multi-var ...
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25 views

Iterative methods: What happens when the spectral radius of a matrix is exactly 1?

I know that an iterative method (I'm using Jacobi and Gauss-Seidel in this case) will converge iff the spectral radius (max absolute value of eigenvalues) of its iterative matrix is strictly less than ...
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38 views

How to find values of x where $a_i x$ are nearly integers? $a_i \in \Bbb R$

I have a set $\{a_i \in \Bbb R | \ i <=7 \}$, and I'm looking for a way to find values of $x$ where given $\epsilon > 0$, $$\forall i \ \exists n_i \in \Bbb{Z} \ \ |a_i x - n_i| < \epsilon$$ ...
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45 views

make the Exponential interpolation vanish and show it has a unique solution

I have several questions concerning different parts of the question: a) Is it sufficient to show that ${1, e^x,...,e^{nx}}$ are linearly independent over the vector space of differentiable ...
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50 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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15 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 ...
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40 views

Error analysis for Runge Kutta, how to take Big O of 2 variables?

For the standard 4th order Runge Kutta: where the system is assumed to be smooth (so that the RHS has no discontinuous points) $\mathbf{y'} = \mathbf{F}(t,\mathbf{y})$ $\mathbf{y(t_0)} = ...
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63 views

The convergence of the fixed-point iteration for solving a cubic equation

I have a third-grade polynomial of the form $Ax^3+Bx+C$ and I want to find its roots. I cannot use Gauss to guess the first root and it is not trivial, so I try this: $0=Ax^3+Bx+C$ and for a given ...
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39 views

Numerical solution of $k\nabla^2 p(\vec{x}) = \nabla(\vec{f}(\vec{x})p(\vec{x}))$

I have the following equation: $$k\nabla^2 p(\vec{x}) = \nabla(\vec{f}(\vec{x})p(\vec{x}))$$ where the constant $k>0$ and vector field $\vec{f}(\vec{x})$ are both known. I wish to numerically ...
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53 views

Integrating sine with Monte Carlo / Metropolis algorithm

I'm learning Monte Carlo / Metropolis algorithm, so I made up a simple question and write some code to see if I really understand it. The question is simple: integrating sine over 0 to PI. The ...
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41 views

Newton's method of finding roots of an equation.

Consider Newton's method on finding the roots of $x^3-x=0$, how to show that $x_n$ converges to $1$ for any $x_0>1/\sqrt{3}$? My attempt: The Newton's method says ...
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63 views

Any good approximation for this integral?

I am interested in the following integral $$ \mathcal{I}=\int_{-\infty}^\infty\mathop{dz}\left[\frac{1}{\sqrt{a+b}(z^2)^{n/4}}-\frac{1}{\sqrt{a+b\cos^2\theta}(R^2+z^2)^{n/4}}\right], $$ where $R\ll ...
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24 views

Is there a faster algorithm than $O(n^2)$ for calculating “cofactors” $C_k = \prod\limits_{j\neq k}(c_k - c_j)$?

Is there a faster algorithm than $O(n^2)$ for calculating "cofactors" $C_k = \prod\limits_{j\neq k}(c_k - c_j)$ ? (presumably $O(n \log_2 n)$ if one exists) In other words, if I have factors ...
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33 views

Computing Cholesky Factorisation by Hand

It is a common exam problem to compute the Cholesky factorisation of a small (typically 4x4) matrix. I know that this can be done by first finding the matrix $U$ in the $LU$-decomposition (e.g. by the ...
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30 views

Why does this algorithm converge?

Consider the following problem. Let $p_1, \dots, p_n \in (0,1)$ such that $\sum p_i = 1$. Let $m > 0$ such that $$ q_i := p_i + m \frac{p_i \log(p_i)}{\sum p_k \log(p_k)} < 1 $$ Suppose ...
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18 views

Fitting nonlinear differential equations to correspond a predefined solution

When modeling temporal dynamics of a biological process I stumbled upon a set of differential equations having the following matrix form: ...
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32 views

FEM for a 1D heat equation system

I want to know how to implement the (nonhomogeneous) initial boundary value problem for a heat equation; $$u_{xx}=u_t ~~~x\in (-1,1),~t\in(0,1)$$ $$u(0,x)=u_0(x)$$ $$u(t,-1)=f(t), ~u_x(t,1)=0$$ Many ...
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52 views

Are two linear system equivalent?

Let $A$ and $M$ be square matrices of size $s$ and $n$ respectively, let $k_i \in\mathbb{R^n}$ be column vectors for all $i=1,\ldots,s$. Denote $K=\left[ \begin{matrix} {{k}_{1}} \\ \vdots ...
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44 views

Numerical solution of the stationary Navier-Stokes equations

Let $d\le 3$ and $\Omega\subseteq\mathbb R^d$ be a bounded domain. I'm considering an incompressible Newtonian fluid with uniform density $\rho_0$ and viscosity $\nu$. In this case, the stationary ...
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19 views

Why study the log of the absolute value of the error induced by a numerical method when solving a PDE?

Often when validating the use of a numerical PDE solution method, one considers the log of the absolute value of the error. In particular, instead of simply looking at the difference between the exact ...
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23 views

PDE Existence and Uniqueness through discretization

This is a question I have been thinking about, but I'm not sure where to look to find an answer. Have a PDE in space and time $(x,t)$. Have a time discretization of the PDE, this results in a ...
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0answers
32 views

Sturm-Liouville Variational Problem

I'm entirely clueless with this problem. No formal training in variational methods. Show that for function $\phi\left ( x \right )$ with $$\phi\left ( a \right )=\phi\left ( b \right )=0$$ and ...
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17 views

How to linearize this second order differential equation?

I want to linearize this second order non linear differential equation arround P($\pi/2 $,0) $\frac{d^2y}{dt}+\frac{dy}{dt}-9.8Sin(y)=5Sin(3t)$ I undestand how to linearize single variable functions ...
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43 views

Numerical intergration of a complex, oscillatory function (Bessel function, Singularities)

I am working on a university project at the moment and at some point I needed to calculate the intergral of the following function (Please refer to "Bakthiari et al. - Analysis of radiation from an ...
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65 views

Finding an error bound for Lagrange interpolation with evenly spaced nodes

I know that the error bound for Lagrange interpolation is usually $$\frac{M_{n+1}}{(n+1)!}\max_{x\in[a,b]}|(x-x_0)\cdots(x-x_n)|$$, where $M_i=\max_{x\in[a,b]}|f^{(i)}(x)|$. I'm trying to find the ...
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42 views

Transforming integrals of the form $\int \frac{f(x)}{\int g(x,y)dy}dx$ into multiple integration. Possible?

I'm currently dealing with integrals of this form, which I approximate numerically. I wonder if I can manage to perform some transformation resulting in a double integral or any other kind of multiple ...
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51 views

Problem in integrating the FitzHugh-Nagumo model

I am using the FitzHugh-Nagumo model to perform some simulations concerning neuronal networks dynamics. The model equations are: $\frac{dv}{dt}=a_3^3v^3+a_2v^2+a_1v+bw+I\\ \frac{dw}{dt}=1/\tau(v-w)$ ...
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51 views

Convergence of difference equation to differential equation

Starting with the difference equation: $$f(x(t+dt),t+dt)= (1-a)f(x(t),t) + a f(x(t+dt),t)$$ where $x(0)$ is given and positive, $a\in(0,1)$, $f(0,t)=0$, and $f$ is increasing in both arguments. ...
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36 views

Solving Schrödinger's Equation for the electronic energies of the Molecular Ion Hydrogen H2+ in the Elliptic coordinate system

Electronic Energies of Molecular Ion Hydrogen $H_2^{+}$ $r_1$ is the distance between the proton $1$ and the electron. $r_2$ is the distance between the proton $2$ and the electron. $R$ is the ...