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

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116 views

Inadmissibility of Simpson's rule

Let $B_t$, $t\ge0$ be a standard Brownian motion and suppose $0<x_1<x_2<\cdots<x_n<1$. Then the conditional expectation $$ \mathbb E\left(\int_0^1 B_t\,dt \,\middle\vert\, B_0, B_{x_1},...
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71 views

Determining algebraically a point of intersection.

A student I was tutoring posed the question: "I know how to solve $$e^{-x} = \ln x$$ graphically, however how do you solve this algebraically?" I have been fiddling around with it for a while and I ...
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245 views

Runge-Kutta Error Analysis

Could anyone explain to me how to reduce the error propagated by using Runge-Kutta of order 4? Or can anyone give me a nice reference to it.
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472 views

How to solve the following system with parametric equation?

What is the maximum number of solutions for this system? Parameters $a_0, a_1, ..., a_{2013}$ could be any numbers. \begin{cases} y = a_0 + |x - a_1| + |x - a_2| + ... + |x - a_{2013}|\\ x^2 + y^2 = 1 ...
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62 views

finding the largest $p$ components of $x$

Given an $n \times n$ matrix $A$, and an $n \times 1$ vector $b$, the conventional way of computing an $n \times 1$ vector $x$ such that $x=Ax+b$ is to use the following iterations: $$x_{k+1}=Ax_{k}+b....
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448 views

Understanding Fourier Transform and FFT

First off, I'm sorry if this is a repost. I am currently writing my thesis, and I've been thrown into some Fourier analysis, which I know nothing of. So, even if this question has been posted before, ...
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194 views

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 ...
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106 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 ...
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109 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? ...
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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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301 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 ...
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360 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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343 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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299 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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517 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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22 views

More precise trail function in Rayleigh–Ritz method

In order to obtain displacement field of an elasticity problem, say a plate structure, we approximate the solution using trigonometric series with unknown coefficients which satisfy the essential ...
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27 views

How to prove that one formula is numerically better than another

If $\mathbf{u}$ and $\mathbf{v}$ are vectors in real 3-dimensional space, here are two formulas for computing the angle between them: $$\theta = \operatorname{atan2}\left( \|\mathbf{u}\times\mathbf{v}...
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31 views

Implicit system differential equations

I came across a system of differential equations in the form: $\newcommand{\D}[1]{\frac{\mathrm{d}#1}{\mathrm{d}x}}$ \begin{align} f_1(x,y,z)\D{y}+f_2(x,y,z)\D{z}&=f_3(x,y,z),\\ f_4(x,y,z)\D{y}+...
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41 views

Numerical methods and KKT in NLP

I am studying numerical methods and NLP. I started with gradient based methods, newton methods and KKT conditions. I found the following sentence: A local minimum is found by solving KKT conditions, ...
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39 views

Numerical methods for ODE: Implicit, explicit, stability, stiffness

Hy everybody! I am new to the subject "numerical methods for ODE". I read some basic literature but since most of the concepts and methods are new to me, I wanted to ask you, if you could give me ...
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30 views

Numerical methods for ODE: Taylor vs. Interpolation approaches

Hy everybody! I am new to the subject "numerical methods for ODE". I read some basic literature but since most of the concepts and methods are new to me, I wanted to ask you, if you could give me ...
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31 views

Newton-Raphson method on manifolds

Has anyone explored the notion of the Newton-Raphson method on manifolds? Or to put it another way, on $\mathbb R^n$, is there a natural coordinate free way of defining an iterate of the Newton-...
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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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21 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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39 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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53 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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36 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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45 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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19 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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23 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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29 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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47 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 functions ...
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60 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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16 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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45 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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79 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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41 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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58 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 $x_{k+1}=x_k-\frac{x_k^3-x_k}{...
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65 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 $R\ll 1$,...
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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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34 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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33 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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19 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: $$\frac{d\mathbf{a}(t)}{dt}=\Gamma(t)\mathbf{a}(t)+C\mathbf{a}...
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36 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 ...