Questions tagged [numerical-methods]

Questions on numerical methods; methods for approximately solving various problems that often do not admit exact solutions. Such problems can be in various field. Numerical methods provide a way to solve problems quickly and easily compared to analytic solutions.

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14 views

Formula for Feigenbaum’s constant?

I have conjectured a formula to calculate Feigenbaum’s constant $\delta \approx 4.66920$. $\delta\stackrel{?}{=}$ $$4+\cfrac{1}{1+\cfrac{1+4}{2^2+\cfrac{1+4+6}{1+\cfrac{1+4+6+8}{3^2+\cfrac{1+4+6+8+...
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1answer
6 views

Finite Element boundary normal vector

I have a finite element and the meshs coordinates can be described using isoparametric shape functions as: $x(\xi, \eta) = \sum _i N_i(\xi, \eta)x_i$, $y(\xi, \eta) = \sum_i N_i(\xi, \eta)y_i$ I ...
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6 views

How choose (discrete) norm for Finite Volume Scheme?

I am doing some work on numerical methods for partial differential equations. I am right now using a finite volume method (FVM) to solve a conservation law, the heat equation. The PDE looks like this, ...
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23 views

How fast can we approximate an integral by sampling via a uniform partition?

Given $f:[0,1] \to \mathbb R$ a continuous function, it seems true that the quantity obtained by estimating the integral by taking the average $$ g_n := \int_0^1 f(x)\ dx - \frac 1 {2^n} \sum_{k=0}^{...
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28 views

Solve heat transport equation numerically with forward finite differences and explicit timestep

I am working on numerical solutions to the diffusion equation and came across a counter-intuitive phenomenon. Let's stay in 1D for this. The diffusion / heat transport equation is ($f$ my state ...
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32 views

How to efficiently compute the minimal polynomial of a number expressed in radicals?

Preamble: I want to calculate the minimal polynomial of a number of the form $$x=\sum_{i=1}^k \pm a_i^\frac 1{k_i}$$ Where the $a_i$ are algebraic numbers also of this form with a finite ...
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2answers
31 views

Can we use the inverse quadratic interpolation for this equation's non-real roots with these points?

$$f(x)=x^4+2x^3+5x^2+5x-3=0$$ I've chosen these values as guessed values: $$(x,f(x)) --> (-1+4i, 172+116i), (1+2i, -42+2i), (-3+i, 14-69i)$$ Am I doing something wrong or can't we obtain a non-...
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1answer
24 views

Laguerre's method explanation

Can anyone please explain the steps of Laguerre's method? I searched for it but I couldn't really understand them. I am a high school student and things in Wikipedia didn't really help me understand. ...
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23 views

Interpolation and approximate feedback for an linear quadratic regulator (LQR) problem

The problem is to perform the following numerical experiment: Suppose i have a second order controlled dynamical system: $$\dot{x}_1(t)=x_2(t)~,~x_1(0)=2\\\dot{x}_2(t)=-2x_1(t)+x_2(t)+u(t)~,~x_2(0)=-...
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15 views

Linear Stability Domain of Multistep Methods

We want to analyse the stability of numerical solutions of the following DE $$\begin{cases}y'=f(t,y) \\y(0)=y_0\end{cases}$$ For a given numerical method of step size $h$, we consider the sequence $...
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1answer
24 views

Why is this assumption correct in Laguerre's method?

From Wikipedia: We then make what Acton calls a 'drastic set of assumptions', that the root we are looking for, say, $x_1$ is a certain distance away from our guess $x$, and all the other roots are ...
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1answer
38 views

numerical solution of 1-d heat equation with mix boundary condition not converge

Consider the following heat equation: $$ u_t=u_{xx} \quad\text{ on }\quad(0,\pi) $$ with mix boundary condition $u'(0)+u(0)=0$ and $u'(\pi)+u(\pi)=0$. The eigenfunctions of the laplacian are $$ e^{-x}...
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8 views

maximize sum of rational polynomial functions

I have sum of rational polynomial functions and I want to find the maximum of the expression over the unit circle. Assume that the polynomials $a_k(z),b_k(z)$ is real on the unit circle, and all ...
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2answers
26 views

Recurrence relation in the bisection method

When beginning to talk about error bounds on the bisection method for root finding, my book states the following: Let $a_n$ $b_n$ and $c_n$ denote the $n$th computed values of $a,b,$ and $c$, ...
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1answer
23 views

Numerical solution of 1-dimensional heat equation

I need to solve the following heat equation problem $\frac{∂u}{∂t} =\frac{∂^2u}{∂x^2}, 0\leq x\leq 1,t\geq 0,$ $u|_{t=0}=f(x), 0\leq x\leq 1,$ $\frac{∂u}{∂x}|_{x=0}=1, t\geq 0,$ $u|_{x=1}=1, t\...
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1answer
26 views

Avoiding loss-of-significance errors by rewriting function.

I was given $f(x)=\sin{(a+x)}-\sin{a}$, for values of $x$ very close to 0. The problem seems to be subtracting two numbers very close to each other. Could I get around this by re-writing as $$f(x)=2\...
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36 views

nonlinear system of equations solved by iteration method does not converge

I've modeled an electrical machine by means of a magnetic equivalent circuit. I've reached to a system of equations like this $ [f(F_N)]_{N\times1}^q=[P_N]_{N\times N}^q*[F_N]_{N\times1}^q-[\phi_N]_{N\...
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1answer
78 views

Bounded root-finding methods: my modified Illinois method

tl;dr: I'm wondering if there's a name for the family of methods shown below, whether or not my method is known, and an analysis on how well it performs. Try some code online, close the tabs and see ...
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1answer
27 views

Normalize a vector (an array) - divide by total sum or absolute value?

When I think of normalizing a vector I mean divide each element with the absolute value of the whole vector, i.e. \begin{align} a &= (2, 4, 3, 1) \\ \hat{a} &= \frac{(2, 4, 3, 1)}{\sqrt{30}} \...
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1answer
30 views

Solving an equation using the floor function.

Let $x,a,b,c \in \mathbb{R}$. I would like to know for what values of $a$, $b$, and $c$: $$ \lfloor x^2 \rfloor = \lfloor a x^2 +b x + c \rfloor $$
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2answers
34 views

Let $A$ be non singular, prove that $\frac{1}{\|A^{-1}\|} = \min_{\|x\| = 1} \|Ax\|$

Let $A$ be non singular, prove that $\frac{1}{\|A^{-1}\|} = \min_{\|x\| = 1} \|Ax\|$ I know that $\|A\| = \sup_{\|x\| = 1} \|Ax\|$ but I have no idea how to proceed from here.
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17 views

Solving systems involving the round function [closed]

I would like to know the set of polynomials for which: $$\text{floor}(x^2) = \text{floor}(ax^2 + bx + c)$$ How do I go about solving such a system?
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35 views

Algorithms for numerical integration of functions which are zero in a “large part” of its domain

I need to evaluate a definite integral of a function which is zero almost everywhere, for instance something like a Gaussian with an artificial cutoff, such that any function value below some ...
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1answer
37 views

Solving a matrix equation in which the coefficient matrix is not diagonally dominant using Gauss-Seidel

I have to solve the following matrix equation using Gauss Seidel : $\begin{bmatrix}25 & 5 & 1\\64 & 8 & 1\\144&12&1 \end{bmatrix}$$\begin{bmatrix}x_1\\x_2\\x_3 \end{bmatrix}$= ...
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1answer
33 views

QL decomposition

I've read about the $QR$-decomposition, so I wonder is there an algorithm that does the factorization ${A = QL}$ of ${A} \in \mathbb{R}^{m \times n}$, $m \geq n$, such that ${Q} \in \mathbb{R}^{m \...
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12 views

Condition for inequalities

I have a collection of inequalities that I need to find a sufficient condition for: Let $d\in \mathbb{N}$ and consider natural $k$'s such that $k\geq d+2$. I want to find a condition of the form \...
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1answer
23 views

Recall the summation formula, write codes of 4 different ways to sum [closed]

As a beginner, I totally lost.
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1answer
24 views

Muller's method's initial approximations for non-real roots

What values should I use to obtain the non-real roots using Muller's method? The equation I am working on is: $$f(x) = x^4+2x^3+5x^2+5x-3$$ The equation has 2 real 2 non-real roots. I would like to ...
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0answers
24 views

Maximum value for delta in a linear system (relative error / absolute error)

$$A^{\sim}=\left(\begin{array}{ccc}{2+2 \delta} & {0} & {0} \\ {0} & {5-(1 / 2) \delta} & {0} \\ {0} & {0} & {3-\delta}\end{array}\right)$$ $$x^{\sim}=\left(\begin{array}{c}{2}...
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11 views

Convergence of a non-stable and non-consistent numerical method

Assume the multistep method $$ y_{n+3} - 2y_{n+2} - y_{n+1} + 2y_n = h(2f_{n+2} + 5f_{n+1}) $$ Zero-Stability: The first characteristic polynomial is $$ \rho(z) = z^3-2z^2-z+2 = (z-2)(z^2-1) $$ ...
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1answer
26 views

Solving A System Of PDE by Matlab

I have a system of PDE which I would like to solve it by Matlab(numerically or analytically). How can I do this? Are there any known analytical approaches to problems of this kind? $$\frac{\partial ...
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1answer
41 views

Proving an ERK is of order 4

restricting your attention to scalar autonomous equations $y'=f(y)$, prove that the ERK method with tableau \begin{array} {c|cccc} 0\\ \frac{1}{2} & \frac{1}{2}\\ \frac{1}{2} &0 &\frac{1}...
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1answer
29 views

Reformulation of matrix optimization. Is it equivalent?

Consider the linear least squares problem: $${\bf x_o}=\min_{\bf x}\|{\bf Mx-b}\|_2^2$$ can be solved by normal equations: $${\bf x_o} = ({\bf M}^T{\bf M})^{-1}{\bf M}^T{\bf b}$$ Assuming $\bf M$ ...
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22 views

Existence and uniqueness and numerical solution of a system of second order nonlinear ODE's

The following system of two non-linear second order ODEs comes up in my research: \begin{align*} & x''(t)+a_1 x'(t)+a_2 x(t)^\alpha+a_3\frac{1}{1+e^t}+a_4=0\\ & x(t)^\beta\Big(y''(t)+b_1 y'(t)...
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26 views

What does it mean to have n linearly independent eigenvectors?

I am learning Numerical PDE and came across this part in it. **Part in the material ** A system of n conservation law is called hyperbolic, if the following is fulfilled: Let $$ A_{i}(u):=D F_{i}(u)=...
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33 views

Modeling a microelectromechanical device, finite difference method

The following is a model of a microelectromechanical device. The device consists of two surfaces of which one is a rigid metal plate, and the other elastic membrane fixed only at the boundaries. ...
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1answer
23 views

Approximate the value of an integral using simulation

Calculate the following integral by generating $200$ of appropriate independent random variable. $\int\int_{x_1 \geq 0, x_2\geq0, 1<x_1+x_2<6}x_1^2(1+x_2^2)^{-2}dx_1dx_2$ From what I currently ...
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1answer
27 views

Proving numerical scheme inequality using mathematical induction

Considering the numerical scheme $$u^{n+1}=u^n+ku^n(1-u^n)$$ where $u'(t)=u(t)(1-u(t))$ and $u(0)=a$. Also, assume that $0<k<1$, and $0<a<1$. How can I prove that if $0 \le u^n \le 1$,...
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10 views

A question about multi step numerical methods and their characteristic polynomials

The following problem features in the book 'A First Course in the Numerical Analysis of Differential Equations' by Iserles. In this context, we have $$ \rho(w) = \sum_{s=0}^{m}a_sw^s, \quad \sigma(w)...
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19 views

How to investigate the contribution of a certain variable in multivariable analysis?

For example, I have a multivariable function $y=f(a,b,c)$. On my hand there is a dataset in form of $\{ a,b,c,y\}$ so I try to carry out regression analysis. The three variables $a,b,c$ are mutually ...
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0answers
15 views

Intuition into Taylor Series Local Truncation Error Derivation applied to Adam's-Bashforth-Moulton 2-Step Method

I am working on deriving the local truncation error (LTE) for what I think is a two-step Adams-Bashforth-Moulton IVP scheme by taylor series expansion. The given scheme is as follows. $y_{n+1}^* = ...
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1answer
43 views

How do I avoid significant rounding error in evaluating $(\ln(x) - \sin(\pi x))(1-x)^{-1}$?

How do I avoid significant rounding error in evaluating $$\frac{\ln(x) - \sin(\pi x) }{1-x}$$ This function causes error as $x\to 1$. How can this be avoided? I tried using taylor's expansion but I ...
4
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2answers
69 views

Numerical operations when numbers are very large?

Explain the best way to evaluate $f(x,y) = \sqrt{(x^2 + y^2)}$ numerically when $x$ or $y$ are very large. Does anyone have any insight to this? I'm lost. I usually know how to deal with these types ...
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0answers
19 views

Finite difference scheme for diffusion-advection equation

Consider the diffusion-advection equation given by $$ - \mu \Delta u + \mathbf{v} \cdot \nabla u = f \ \text{in} \ \Omega $$ with some appropiate boundary conditions. Here the velocity field $\mathbf{...
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23 views

Meaning of $\frac{\| {PA - LU} \|_{F}}{\| {A} \|_F}$ and $\frac{\| {PA - LU} \|_F}{\| {L}\|_F \| {U} \|_F}.$

I'm studying PLU decomposition and in one of the problem I was asked to implement the algorithm in MATLAB, then report $\frac{\| {PA - LU} \|_{F}}{\| {A} \|_F}$ and $\frac{\| {PA - LU} \|_F}{\| {L}\|...
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1answer
28 views

Matrix Norms - Burden and Faires(Page 439)

Burden and Faires define: If $\|\cdot\|$ is a vector norm in $\mathbb{R}^n$, then $$\|A\|=\max_{\|x\|=1}\|Ax\|.$$ In other words, the measure given to a matrix under such a norm describes how ...
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3answers
44 views

Positive definiteness of matrix in special case

Let $A \in \mathbb{R}^{n \times n}$, $C \in \mathbb{R}^{m \times m}$ be symmetric and let $B \in \mathbb{R}^{m \times n}$ such that $B^T \mathbf{x} \neq \mathbf{0}$ for all $\mathbf{x} \in \mathbb{R}^...
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28 views
+50

A-stability of an implicit three-stage Runge-Kutta method with two parameters

My question concerns the following extract, taken from page 881 of this paper by Muir and Chipman. The paper presents a three-stage implicit Runge-Kutta method with parameters $c_2$ and $c_3$. In an ...
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0answers
24 views

Finding the maximum likelihood estimator that can not be isolated (solving a system)

I have a likelihood function of three parameters $a,b,c$ where distributions of $x_i\mid x_{i-1}$ are normal with mean $x_{i-1}+a(b-x_{i-1})$ and variance $c$. Differentiating and equating the log-...
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7 views

Discretization of differential of a map on discrete manifolds.

Suppose $N$ and $M$ are two regular submanifolds of $\mathbb{R}^3$. Suppose you discretize these as meshes, and for simplicity assume they have same number of vertices and connectivity. Suppose $F:N\...

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