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

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22 views

Could anybody please clarify the relationship between numerical stability and accuracy?

I was reading a paper and came up with this statement. Stability merely avoids uncontrolled error growth but cannot guarantee actual numerical accuracy. From what I understood from the concept of ...
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20 views

Newton's method for nth roots of complex numbers

Is it possible to use Newton's method to compute roots of complex numbers, say $\sqrt[n]{a+ib}$ to any desired accuracy? If yes,for what initial values will converge?
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12 views

Why Gauss-Seidel iteration is a projection method

Yousef Saad's iterative method for sparse linear systems says Jacobi, GS, SOR are all projection methods. for example GS is a projection method with $\mathcal K= \mathcal L =\{e_i\}$ (project on $\...
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20 views

Implicit method for ODE

I want to numerically solve the initial value problem of ordinary differential equation for function $u=u(t)$: $$ u'(t)=L(u). $$ I find an second-order implicit method: $$ u_{n+1}=u_n+\Delta t L(u_{n+...
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38 views

What constitutes sensitivity to initial conditions for a system of continuous ODEs?

I am currently working on a disease model (5 dimensions) which seems to exhibit sensitivity to initial conditions. I have used numerical analysis following the method described in Wolf's paper (link ...
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1answer
62 views

Prove an algorithm for logarithmic mean $\lim_{n \to \infty} a_n=\lim_{n \to \infty} b_n=\frac{a_0-b_0}{\ln a_0-\ln b_0}$

Take: $$a_0=x,~~~~b_0=y$$ $$a_{n+1}=\frac{a_n+\sqrt{a_nb_n}}{2},~~~~b_{n+1}=\frac{b_n+\sqrt{a_nb_n}}{2}$$ Then we obtain as a limit the logarithmic mean of $x,y$: $$\lim_{n \to \infty} a_n=\lim_{n ...
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53 views

What is the mathematical vernacular for $\Delta x$

I want to use the proper terminology when I discuss length scales associated with $\Delta x$, where $\Delta$ is the difference operator. In other words $\Delta x = |x_1-x_2|$. It is a measure of the ...
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2answers
116 views

An apparently new method to compute the $n$th root of any complex number

I found  a series of articles (in Portuguese) by a Brazilian mathematician named Ludenir Santos, where presents a series of iterative methods, he said new, to extract nth roots of any complex number ...
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11 views

Estimate the drift and diffusion function numerically

I have a 1D problem as following $$\frac{\partial f}{\partial t} = \frac{\partial}{\partial x} \Big[ \frac{1}{2} \frac{\partial (g(x) f)}{\partial x} -\mu(x)f \Big]$$ I have a time-series of ...
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68 views

Precise numerical integration of sine using the trapezoidal rule

I need to compute the integral $$\int_0^\pi \sin x \, \mathrm{d}x$$ by numerical integration. This is currently done by using the trapezoidal rule. The number of the trapezoidal "slices" $n$ is ...
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1answer
19 views

Intuitive Understanding Newton-Raphson method with second derivatives

From what I remember in school and Wikipedia, the Newton-Raphson method was always very intuitive to me since: $y=f'(x_n)·(x-x_n) + f(x_n)$ is basically saying take the slope for the small space in $...
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19 views

Principal Component Analysis to find correlations in data

first of all, I'm really not a fan of stochastic methods and so I didn't spent much time to get used to it in the past. But now I have run a PCA and got some questions regarding this. I already found ...
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1answer
12 views

Use QR decomposition to find LS decomposition, where S is orthonormal and L is a lower triangular matrix

Assuming that we have a black box that can find the QR decomposition of a matrix $A$, how can we use this black box to find a decomposition $A=LS$ where $L$ is a lower triangular matrix and $S$ is an ...
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0answers
14 views

Numerical Method for fitting parameters of an explicit integration to actual data

I have a heat transfer system described by, $$\{\dot{T}\} = [C^{-1}]\left([K]\{T\} + \{F\} \right)$$ where ${T}$ is a vector of the nodal temperatures of the system. From initial conditions I am able ...
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0answers
25 views

Numerical integration degree of precision

For a smooth function $f$ we have the integral$$\int^1_{-1} f(x) dx,$$Consider the numerical scheme that approximates this integral as$$\int^1_{-1} f(x)dx \approx w_1 f\left( \frac{-1}{2} \right) + ...
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17 views

Adams-Bashforth-Moulton two-step predictor-corrector method [duplicate]

enter image description here I have no idea how to do, can someone tell me the step? Cheers (By the way I cant post image)
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46 views

Computing cube roots using three number means

I've asked a question some time ago about Computing square roots with arithmetic-harmonic mean but it turned out that this method is exactly the same as Newton's method (or Babylonian method) for ...
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1answer
28 views

Preconditioner operator

hope you can help me. I have learned that a preconditioner is a matrix $P$ such that when it is applied to a system $A \mathbf{x} = \mathbf{b}$, the spectral properties of the matrix $P^{-1} A$ are ...
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2answers
99 views

Newton's Method and One definition?

We are some third year students on CS. We have encountered one definition that was not well formed for us. Could anyone describe it more simpler for us? (i.e: how this derivations is adopted?) ...
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159 views

A new type of Arithmetic-Harmonic mean for $n$ numbers

Let's introduce the following iterative procedure. Take two numbers $x_0$ and $y_0$. $$a_0=\frac{x_0+y_0}{2}~~~~~~~~~~~b_0=\frac{2x_0y_0}{x_0+y_0}$$ $$x_1=\frac{x_0+a_0+b_0}{3}~~~~~~~~~~~y_1=\frac{...
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1answer
32 views

The convergence of an infinite radical involving $\cos(\alpha/3)$

By using the triple angle formula for the cosine, $\cos 3\alpha$, we get the cubic equation $ 4x^3-3x = \cos \alpha $. Now, by expressing $ x $ as $ x = \frac{1}{2}\sqrt{3+\frac{\cos \alpha}{x}}$ ...
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1answer
53 views

Check numerically the definite-positiveness on linear subspaces

I have a given matrix $W\in \mathbb{R}^n$ with known fixed entries. I would like to check the definite-positiveness of $W$ on appropriate linear subspaces. Typically I would like to show (...
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1answer
37 views

interpolation polynomial error

We have points $x_0=a \lt x_1 \lt x_2 ....x_n=b $ and $\;w_{n+1}(x)=\prod_{k=0}^{n}{(x-x_k)}$. Let $h=max_{j=0...n}|x_j-x_{j-1}|$ Let $f \in C^{n+1}[a;b]$ and $p_n\in \mathbb P_n$ be the ...
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44 views

Adams-Bashforth-Moulton two-step predictor-corrector

Can someone help me this question please, this is from past years exam.
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1answer
21 views

Preconditioning : ILU($\emptyset$) factorization and SSOR relation?

Let's suppose we have a matrix A = D - L - U , where D,L,U are diagonal , strictly lower and strictly upper triangular,respectively. Generally the preconditioning matrix, according to SSOR(Symmetric ...
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20 views

Interplanetary Optimisation using a simulator with PyGMO or SciPy

I am currently trying to use a N-body gravity simulator to model a spacecraft trajectory and using the simulator as a BlackBox to optimise the trajectory. I am thinking of using basin hopping/ ...
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0answers
63 views

Measuring the degree of convergence of a stochastic process

Consider a set of random variables $(X_1,X_2,X_3,...X_k)$ that are i.i.d. $Bernoulli(p)$ While I do not know $p$, I can estimate it using $$ Y(k)=\frac{1}{k}\sum_{i=1}^k X_i $$ Notice that $Y(k)$ is ...
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1answer
35 views

Intersection of a helicoid and a line

I have a Helicoid described by the following parametric equations: $$x = u\cos(v)$$ $$y = cv$$ $$z = u\sin(v)$$ The helicoid revolves around the y-axis: Eliminating $u$ and $v$, we obtain the ...
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2answers
66 views

Error in trapezoidal rule via integral mean value theorem

During a class, I saw the following analysis of the error term in the trapezoidal rule For $f \in C^2([a,b])$, $\int_a^b f(x) \,dx - \frac{b-a}{2}[f(a)+f(b)] = -\frac{(b-a)^3}{12}f''(\eta)$ for ...
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1answer
36 views

Estimate counts with different sample sizes

Given an arbitrary time period, lets say one week, but it could be five days, one month etc.., I have a sample from a population. My sample consists of shoppers at a store. For week one my sample is ...
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32 views

Matrices over integer fields to solve complex polynomials.

Inspired by the fruitful answer to this question regarding numerically solving polynomial equations in terms of simpler fields (in that case representing real numbers as fractions of integers), I ...
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42 views

Solving differential equation with numerical method

How do I solve this differential equation numerically for $1<\beta<\inf$ knowing $\sigma$ ? $\frac{\gamma(u)}{du}=\beta\cdot\gamma(u)\cdot\gamma(\beta\cdot u)$ $\gamma(0)=\sigma$ Thank you ...
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Maxwell-type system: how to solve?

$\DeclareMathOperator{\curl}{curl}\renewcommand{\div}{\mathop{\mathrm{div}}}$ Consider the following system of equations for the unknown vector field $A$ in the unit 3d ball $B$ with boundary $S$: $$ ...
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0answers
28 views

Newton conjugate gradient algorithm

In this video, the professor describes an algorithm that can be used to find the minimum value of the cost function for linear regression. Here, the cost function is $f$, the gradient is $g_k$ where $...
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1answer
45 views

Proof Bisection method [closed]

Denote the successive intervals that arise in the bisection method by $[a_1\,, b_1], [a_2\,,\, b_2], [a_3\,,\, b_3],$ and so on. Let $c_n$ be the midpoint of $[a_n, b_n]$. Show that $|c_n − c_{n+1}| = ...
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28 views

Second-order differential equations methods

I'm looking for a method called 'Inexact Method' Idk if it goes by another name, here's what I do know: It's one of the two 2nd order differential equations methods. The other method is called '...
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1answer
33 views

To solve large systems of multivariate polynomial equations

Nicolas Courtois et al. proposed the eXtended Linearization(XL) method to solve the systems of multivariate polynomial equations and analyzed the time complexity. Polynomial when the number of (...
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1answer
49 views

Iterations with matrices over simple fields approximating solutions for more complicated fields.

Inspired by this question I started wondering if there exist some systematic way to construct approximation to any number one can find using matrices over a preferrably simpler field. In the question ...
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2answers
49 views

Numerical solutions of partial differential equations

I'm studying mathematical physics and working on numerical solutions of partial differential equations. I am having trouble understanding the way we solve partial dif. equations, e.g., $\frac{\...
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2answers
23 views

2nd Order Runge-Kutta Method

Could someone please help me with the next step of this 2nd order Runge-Kutta method. I am solving the ODE \begin{align*} x'=-\frac{x(t)}{2}, \ \ x(0)=2. \end{align*} I wish to use the second order ...
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1answer
28 views

Reconstructing a matrix

Before reading on, let me acknowledge that this problem is solveable generally, however I am interested in knowing if a certain form of solution exists. If I have a square complex unitary $n\times n$...
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36 views

Newton's method

Let $a > 0$. Starting from a convenient equation and using Newton's method, deduct a method to approximate $1/\sqrt a$ without division. How do you choose the starting value? Which is the stop ...
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3answers
51 views

For how many weeks can I group my students?

I thought of an interesting question that I don't know how to solve. I imagine there are numeric results out there somewhere, but I don't know if this question has a formal name; if anyone could link ...
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0answers
18 views

What makes a geometric construction more or less stable?

As anyone who's actually done geometric construction of n-gons knows, not all construction methods are made equal. Some are very stable (the shape you get is always close to ideal even if you're not ...
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2answers
70 views

Square root of x : $\sqrt{x}$ (Numerical Method)

$$f(x) = \sqrt{x}$$ has to be approximated by polynomial interpolation $p(x_n) = f(x_n)$ with the positions $\{x_n\} = \{1,4\}$. For such problem which method is the fastest? And find $p(2)$. My ...
0
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1answer
21 views

Condition number interpretation

I have a (nonlinear) problem with two variables, for which I computed a relative condition number as $$K_{rel}(x_1,x_2) = \max\{1, c\},$$ where I had $$\Bigg| \frac{f(x_1, x_2) - f(\tilde{x_1},\tilde{...
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0answers
33 views

Interpolation and Interpolationerror - how to compute ?

I want to compute the greatest $a>0$ for given $\epsilon>0$ such that $$max_{x\in [-a,a]}|f(x)-p_2(x)| < \epsilon$$ where $a$ is the distance between two grid points and the maximum is the ...
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29 views

Question about Moment generating functions, precision, rounding errors

Hello I have a question in regard to moment generating functions and something I noticed but wasnt to clear on. Say that a random variable $X$ has $$MGF=M_{X}(t)=\frac{4}{4-t^2}$$ for $-2 \lt t \lt 2$...