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 fields. Numerical methods provide a way to solve problems quickly and easily compared to analytic solutions.

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20 views

Why does the fixed point method rely on the derivative of the root for convergence or divergence?

I'm currently studying the fixed point iterative method, and I am confused as to how the derivative of g(x) at a point between a guess and a root can tell us if the method will converge or diverge for ...
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1answer
19 views

How to convert to exact arithmetic?

I'm working through a numerical analysis exercise where I use Newton's method over five iterations. Below is my code. ...
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37 views

Algorithm for computing Pi that doesn't get harder the further you go?

I'm looking for an algorithm for computing PI that doesn't get harder the more digits computed (e.g. using factorials). Additionally, digit computation must be feasible by giving a seed of a previous ...
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15 views

What is meant by a stiff/rigid ODE? [duplicate]

A differential equation is stiff if a numerical scheme requires a very small time-step in order to be stable for that equation. However I don't understand why it is called stiff (sometimes rigid). ...
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12 views

Order of Convergence when working with errors

I am looking at the numerical solutions of a problem when using the boundary element method, the exact solution is 0.25 I have 3 errors corresponding to using 20,40 and 80 boundary elements. I have ...
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1answer
38 views

Hamilton equations-Symplectic Euler method

We know that $\dot{q} = \frac{\partial H}{\partial p}$ and $\dot{p} = -\frac{\partial H}{\partial q}$, and we also know the values $Q$ and $P$ respectively of $q$ and $p$ at a later time step $\Delta ...
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18 views

Comparing GMRES to equivalent algorithms such as GCR

If I understood correctly both GMRES and GCR minimise the residual norm on $x_0+K_k$. Hence, they produce the same approximations over the iterations. The difference comes from the fact that GMRES ...
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8 views

Unstructured mesh generator with connectivity information

For solving PDE with self written code it is needed to preprocess the data from mesh generators. I finished reading the FVM part of the book im currently working with. The author - S.Mazumder, ...
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11 views

Newton form of polynomial interpolation β€” adding data points via divided difference

I wrote some code that implemented basic polynomial interpolation using Newton's representation of letting the interpolant polynomial be of the form $\displaystyle L(x) = \sum_{k=0}^n \alpha_i \prod_{...
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22 views

Proof check: Divided difference is exactly the coefficient of the interpolation polynomial (in Newton basis)

Below is my proof, which utilizes showing that the coefficient satisfies the recursive definition of the divided difference. Anyone can verify if anything is unjustified/wrong in the proof? I'm asking ...
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1answer
17 views

Need help to understand different formula for Trapezoidal's rule

My book (Numerical Analysis by Douglas) said the formula for Trapezoidal's rule is $$\int_a^b f(x)\, \Bbb dx \approx \frac h2 \left(f(a) + f(b) + 2\sum_{k=1}^{n-1} f(x_k)\right)$$ Here is the ...
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How can I prepare myself to learn Galerkin projections and related ROM models?

I may be assisting a research lab that develops reduced-order models. Now, I am an undergraduate student who has not yet taken PDEs, numerical analysis, nor topology. However, I would like to ...
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10 views

Slope Limiter Formula For The Second Order Fromm Method

I am currently writing a program in C that solves the 1D advection equation using the FVM method. I am trying to implement slope limiters for the second order schemes. I have already implemented Van ...
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13 views

Quadrature for Newton Polynomial is zero

I need to show that for $n$ even $$ \int_a^b \prod_{j=0}^n (t-t_j)dt = 0 $$ Now I know this is the Newton polynomial, but I don't now how to proof this correctly. I tried out $n=2$ and it was right. I ...
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22 views

Meaning of degenerate PDE

I am trying to understand the concept of degenerate PDE. The definitions I found depend on the type of equation but let say for elliptic (see here). Let's take an example: $$x\partial_x^4u + \...
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1answer
29 views

Absolute error formula

Trying to figure out a problem in my textbook. In my exercise in a textbook, a problem says that Determine the error in the approximation given that the actual length is $3.7437137$. And in the ...
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34 views

Arithmetic in Lambert-W number system

The principal branch of Lambert-W function is defined like so: $$W(x) = f^{-1}(x)\\ \text{where} f(x)=xe^x$$ So $W(e) = 1$, $W(2e^2)$ is 2, $W(3e^3)$ is 3 etc. We define Lambert-W number system to ...
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1answer
22 views

Lagrange polynomial interpolation maximum degree

I want to prove that no polynomial of degree $1$ that passes through $(0, \cos(0))$, $(0.6, \cos(0.6))$ and $(0.9, \cos(0.9))$. By the following theorem: Theorem 1. If $x_{0}, x_{1}, \ldots, x_{n}$ ...
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1answer
27 views

How does a constant magnitude in a system of ODEs affect the convergence of a numerical method?

Consider the following ODE system to be solved numerically: \begin{align} \dot{x}_1 &= -x_1 \\ \dot{x}_2 &= -2x_2 + \beta x_4 \\ \dot{x}_3 &= 2x_2 \\ \dot{x}_4 &= x_1 - \beta ...
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1answer
13 views

Approximated number of elastic in a box is 100 with 1 significant figure. How to express this in $x \pm y$ notation?

Question from stackexchange How many significant figures are there after 96 is correct to 1 significant figure? From this statement: It is given that the approximated number of elastic in a box is ...
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9 views

Measurements Numbers in Compressive Sensing

here is a question about compressive sensing. Let us denote the $k$-sparse signal $x\in \mathbb{R}^n$, measurement/sensing matrix $A\in \mathbb{R}^{m\times n}$ and the measurement $y = Ax \in \mathbb{...
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30 views

Legendre polynomials satisfying a recurrence relation.

Monic Legendre polynomials (which are orthogonal polynomials) on $[-1,1]$ are defined as follows: $$p_{0}(x)=1, \ p_{1}(x)=x$$ and $p_{n}(x)$ is a monic polynomial of degree $n$ such that $$\int_{-1}^{...
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26 views

Jacobi method for complex Hermitian matrices.

There is the article on wikipedia https://en.wikipedia.org/wiki/Jacobi_method_for_complex_Hermitian_matrices about Jacobi eigenvalue algorithm for Hermitian matrices. It says: Similar to the Givens ...
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21 views

relationship between Krylov subspaces and the Rayleigh quotient

Let $A \in \mathbb{R}^{n \times n}$ be symmetric with eigenvalues $\lambda_1 \geq ... \geq \lambda_n$. Then by Courant-Fischer $\lambda_1=\max_{x\neq 0} r(x)$ and $\lambda_n=\min_{x \neq 0} r(x)$, ...
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1answer
58 views

How to find all roots of nonlinear function - an example.

How would I find all the roots of the function $f(x) = \sin (10x) - 2x$? I know all sin functions have multiple roots and so this function can also have multiple roots but how would I find all these ...
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1answer
29 views
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Proof of relation between divided difference and interpolation polynomial coefficient

Given data points $\{(x_i,f(x_i)\}_{i=0}^{m}$, if we define divided difference recursively as: $$f[x_0,\cdots,x_{k+1}] = \frac{f[x_1,\cdots,x_{k+1}]-f[x_0,\cdots,x_k]}{x_{k+1}-x_0} \text{ with the ...
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43 views

Using a Partial Differential Equation to find the dynamics model of physical intearctions

I don't know a lot about Partial Differential Equations (PDEs) but I think they are used to find solutions to complex interaction problems. So my first question does solving a PDE enable us to obtain ...
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1answer
12 views

Exactness for special quadrature formula

Given the quadrature formula $Q(f):= w_0f(-a) + w_1f(a)$ and $I(f):= \int_{-1}^1 f(x) dx$ and $a\in [0,1]$. I need to find $a$ so that $Q(f)$ is a) exact for polynomials of degree $1$. b) exact for ...
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2answers
59 views

Numerical Approximation: Integral of Exponential to an Exponential power

I am trying to solve a self-consistent equation for the variable $S$. $$ \mathbf{S} = \int_0^t C_1 \mathbf{S} e^{-C_2 a-C_3\cdot\mathbf{S}\cdot e^{-C_4 a}} da \\ % 1 = \int_0^t C_1 e^{-C_2 a-C_3\cdot\...
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12 views

Showing $||F'(x)y||\geq c_3||y||$ in a Banach space

I'm looking at Lemma 2.1 in Reddien's paper "On Newton's Method for Singular Problems". I'm still fairly new to Banach spaces. So I'd appreciate a hint as to why Lemma 2.1 is an "easy&...
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1answer
21 views

Stable discretization for a given ODE

Given the ODE $$ \ddot{X} + \frac{3}{t}\dot{X} + F(X) = 0,$$ what would be a stable explizit discretization? One can rewrite the ODE as a first order equation of the form $$ \dot{Y}_1 = Y_2 \\ \dot{Y}...
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20 views

Fokker Planck Equation

I was told in the lecture like the following: "The Fokker-Planck equation is a deterministic partial differential equation that in general has to be solved numerically. For vector systems ...
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31 views

Prove sequence $(a_i)_{1\leq i \leq j}$ is increasing

Let $\alpha=2.1$. For a fixed $j$, let $(a_i)_{1\leq i \leq j}$ be a sequence of number, where $$a_i = i^\alpha \cdot \prod_{t=1, t\ne i}^j |t^\alpha - i^\alpha|.$$ Prove $(a_i)_{1\leq i \leq j}$ is ...
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2answers
27 views

A closed form expression for the finite-difference Euler's method

Consider the ODE $$f(u) \equiv u' = au+b, \>\>\> u(t_0 = 0)\equiv u^0 = u_0$$ For a timestep $k$, the forward-difference approximator for $u'$ gives the Euler update as $$ u^{n+1} = u^n + kf(...
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2answers
48 views

Example where $𝜅_𝑓 (x), 𝜅_𝑔 (y) \geq 10$ but $𝜅_β„Ž (x) ≀ 1$

Let $𝑓 : 𝑋 β†’ π‘Œ$ and $𝑔 : π‘Œ β†’ 𝑍$ be continuous functions between arbitrary vector spaces, and let $β„Ž = 𝑔 β—¦ 𝑓$ . Suppose $𝑦 = 𝑓 (x)$ and $𝑧 = 𝑔(y) = β„Ž(x)$. Give a concrete example where $ \...
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20 views

Magnus integrator

I'm trying to find a Matlab code about Magnus integrator (or any help in writing a code in Matlab) which used to solve the first order ODE: $Y'(t)=A(t)Y(t);Y(0)=Y_{0};$ Magnus integrator has the ...
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Choice of linear systems iterarive algorithms' convergence metrics

I'm studying the Jacobi iterative method, which solves a linear system of equations in the form $\mathbf A x = b$. As an example, I considered this simple C implementation, where the programmer seems ...
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3answers
94 views

Finding zeros of a function

I need to find when $f(x)=0$, where: \begin{equation} f(x)=1-\dfrac{k}{x}-\dfrac{k}{3}\dfrac{e^{-ax}}{x}+\dfrac{4k}{3}\dfrac{e^{-bx}}{x} \end{equation} Here, $k$, $a$ and $b$ are positive constants. I ...
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30 views

Odds of getting 5 heads in a row out of 45 tries [closed]

With a probability of .515 of landing heads, what are the odds that I get 5 heads in a row out of 45 tries?
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78 views

Does there exist a function for integral of $e^{-x^2}$?

I am wondering if the graph of $$f(x) = \int_{0}^{x}{e^{-t^2}}\,\mathrm{d}t$$ is possible to be written as some function without the Error Function, since it looks like an existing function when I ...
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2answers
53 views

Is there by chance an analytic solution of the following non-linear ODE: $x'(t) = a\, x(t) + b\, x^3 (t)$?

I'm in fact interested in a PDE for which I try to get some intuition (roughly, I interpret my PDE as a function of time with value in a space of functions of space variables). Does someone by luck ...
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1answer
47 views

Relative Condition number of composite function

I want to find a function $$h = g \circ f$$ such that condition number of $g$ and $f$ are greater than $10$, but the condition number of $h$ is less than $1$. I am trying to use polynomials like $x^{...
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4answers
79 views

Computing continued fraction expansions

My question concerns the numerical accuracy of a continued fraction expansion. A typical algorithm for computing a continued fraction can be written in Python as : ...
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0answers
33 views

Setting up Gauss-Seidel Method

Background Solve the following set of nonlinear equations by the Gauss-Seidel method: $$\left\{\begin{array}{ll} 27x+e^x \cos\ y -0.12z & =\ 3, \\ -0.2x^2+37y+3xz &=\ 6, \\ x^2-0.2y \sin x+...
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1answer
31 views

Butcher's Tableau to Diagonally Implicit Runge Kutta method

I'm trying to implement a function that would input a butcher's tableau for a Runge-Kutta method of order $s$, an ode and its jacobian, as well as the initial values and return the next step. If we ...
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1answer
23 views

Integration with unbounded spectral weight

I want to consider the solution of an equation in the space $L^2(e^{-2\alpha x} \mathbb{R}. )$ I am just wondering how to implement the weight when computing the norm. So far, I use Hermite spectral ...
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0answers
30 views

Looking for the title of this book in Numerical Methods [closed]

if anyone has ever come across this book on Numerical analysis I really need the Title, I only found some it's scanned pages
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3answers
112 views

When does a rectangular matrix A $\in$ $\mathbb C^{𝑚\times 𝑛}$ have the property such that $\|Ax\|_2=\|x\|_2$, where $x \in \mathbb C^𝑛 $

This is one of my homework questions which I am trying to solve I started with the claim that $\|Ax\|_2$ will be less than equals to $\|A\|_2$ $\|x\|_2$ and from here I can only get the condition for $...
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0answers
32 views

What is the period of the solution of a wave equation boundary value problem

In my studies of numerical PDEs, I was given this problem We consider a vibrating string that satisfies the wave equation $u_{tt}=u_{xx}$ on the unit interval with boundary conditions $u(0,t)=0$, $u(...
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0answers
21 views

Is the trapezoidal rule equivalent to Gauss-Chebyslev quadrature?

I encountered a paper in which the author derives a quadrature rule for integration of a function $f(x)$ over a domain $[0,L]$, $$\int_0^L f(x) dx = \sum_{i=1}^{N-1} w_i f(x_i).$$ I'll give a quick ...

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