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

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33 views

For which starting values the iteration convergences?

Given: $g(x)=\frac{1}{2}(x+\frac{a}{x})$ for $a\in \mathbb R_{>0}$ Question: For which starting values $x_0>0$ does the iteration $x_{k+1}=g(x_k)$ converges? My thoughts: Should I find an ...
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0answers
24 views

Stability in partial differential equations

I have the following PDE, with parameters $a$ and $b$: $$ \frac{\partial c}{\partial t} = \frac{\partial}{\partial z} \left( a c + b \frac{\partial c}{\partial z}\right) $$ with, for now, just one ...
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2answers
22 views

How to represent non-linear operators computationally?

I have a finite dimensional vector space V, and want to compute a non-linear operator $R: V \rightarrow V$. I want to have a "general" form of this operator R. I think of the following series ...
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2answers
43 views

Calculating values of $1 - \cos(x)$ for $x$ near zero using computer arithmetic

Explain why calculating values of $1 - \cos(x)$ where $x$ near zero using the trigonometric identity $1 - \cos(x) = 2\sin^2\big(\frac{x}{2}\big)$ will result in more accurate results. Is it because ...
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0answers
13 views

Retail Inventory Prediction - More scientific approach?

This question concerns planning inventory for a retailer. Suppose you have access to all inventory and sales metrics for all styles within the department. You are planning an e-commerce order for ...
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1answer
35 views

Chaikin's Algorithm: Proof of Convergence

Chaikin's algorithm is, in some sense, similar to de Casteljau algorithm in that (in the limit) it produces a curve from a set of control points. There are claims all over the internet that Chaikin's ...
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0answers
21 views

Multivariable gradient descent with approximation of gradinet

This is not a statistics problem I have a vector $$X=[x_1,...,x_{10}]$$ and a cost function $$y=F(X)$$ and my aim in to find the best $X$ to minimize the cost function. It is impossible to ...
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1answer
57 views

Cubic function: Cardano's method

(Wikipedia link) So I am writing an essay on different ways to solve cubics. But I get stuck in the Cardano's method... Mainly is the part with Cardano's method's condition $\frac{q^2}{4} + ...
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1answer
49 views

How to numerically solve the Poisson equation given Neumann boundary conditions?

I want to solve the Poisson equation on a 2D domain given Neumann-type boundary conditions: The PDE: \begin{equation} \nabla^2 \; u(r,\theta) \;=\; f(r,\theta) \end{equation} The boundary ...
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0answers
32 views

What is the minimizer of the matrix norm and it's significance?

For $M_{n\times n}$ a p.s.d real matrix, if we minimize $||M^{\frac{1}{2}}x||_2$ over $x$ under a linear constraint on $x$ as in $Ax=b$, where $b$ is non-zero. what is the significance of this $x$? ...
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2answers
17 views

Lipschitz-like behaviour of quartic polynomials

I have observed the following phenomenon: Let the biquadratic $q(x)=x^4-Ax^2+B$ have four real roots and perturb it by a linear factor $p(x)=q(x)+mx$, so that $m$ not too large with respect to ...
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28 views

Is this idea correct?

Given a curve passing through point $(p_0, v_0)$ and defined in a standard way as $k(p, v) = k(p_0, v_0)$, i can find the 1st term,a 2nd term b and so on by expand k in taylor series and consider ...
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0answers
22 views

Issues with finite-difference implicit solution of Advection-Diffusion-Reaction eqn

Objective: I am trying to numerically solve $C(x,y,t)$ from the following advection-diffusion-reaction equation in 2D space (x,y) and time. I will be testing my numerical solution with an approximate ...
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1answer
59 views

Expected Utility Method and a Repeated Game Solution

I am trying to replicate Bruce B. de Mesquita's (BDM) results on political game theory for prediction. Based on where actors stand on issues, their capabilities, salience, BDM's method attempts to ...
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1answer
56 views

Numerically solving a system of partial integro-differential equations in Matlab [closed]

Given the following system of partial integro-differential equations - $\frac{dS(t)}{dt}=\Lambda-\mu S(t)-\beta S(t)F(t),\\ \frac{\partial I(t,\omega)}{\partial t}+\frac{\partial ...
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0answers
82 views

How can I resolve this improper integral?

I would like to resolve this integral numerically . However, I'm not sure about the best way to do it because it is an improper integral: $$ ...
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2answers
24 views

Estimating line paths in vector fields.

Assume I have a vector field sampled in discrete points. For simplicity let us assume it is sampled regularly on a Cartesian grid. I want to estimate flow lines through various points in this vector ...
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21 views

Monte Carlo (or other) Approximation to Infinite Summations

This question is sort of paired with a similar post I wrote in Stack Overflow, but not a repeat because I am asking about using another mathematical method. StackOverflow I am trying to approximate a ...
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0answers
25 views

Approximate solutions to differential equations

If one has a differential equation for $y(x)$. If this differential equation has two solutions one for $x\ll a$ and the other for $a\ll x$, where $a$ is constant real value. My question is at what ...
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0answers
13 views

Method of Modified Equations: algebra problem

I am having trouble following my professor's lecture notes on the method of modified equations and arriving at his final solution. Starting with the recurrence defining the numerical method: ...
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1answer
29 views

Assumption on Runge-Kutta-Method

I think I missed a crucial step concerning Runge-Kutta-methods. If a RK-method is given by its Butcher-table, is it necessary to have $\sum_{j = 1}^s a_{ij} = c_i$? In class we discussed, that a ...
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20 views

Easiest way to calculate Runge-Kutta stability function

For a Runge–Kutta method with Runge–Kutta matrix $A$ and weights $b_1, \ldots, b_s$ the stability function is defined as: $$R(\zeta) = 1 + \zeta \begin{pmatrix} b_1, \ldots, b_s \end{pmatrix}(I-\zeta ...
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1answer
26 views

Second derivative numerical estimate - stability and approach

I would like to know how to estimate second derivatives of a function sampled discretely with constant spacing. Let there be a function $f(x)$. I sample its values $\{f(x_i)\}$ at points $\{x_i\}$ ...
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1answer
54 views

Calculating the x-intercept of the line that passes through the points $ (x_0,y_0)$ and $(x_1,y_1)$

I got this problem from the book Numerical Analysis 8-th Edition (Burden): Suppose two points $(x_0,y_0)$ and $(x_1,y_1)$ are on a straight line with $y_1\neq y_0$, Two formulas are available to ...
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0answers
44 views

Matlab Coding finding zeros without using fzero or roots function

So i am a completely new at Matlab. I'm basically suppose to develop a function in Matlab that finds the zeros of a cubic polynomial. real and complex. I'm pasting below what I have so far. I started ...
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1answer
31 views

Prove the solution $u_i$ of equation $\lambda r_i+\frac{1}{\theta}(u_i-v_i)+\beta \left (\sum_{i=1}^Nu_i-1\right )$

I have an cost function such as $$E(U)=\lambda \sum_{i=1}^{N} \int_{\Omega}r_iu_idx+\frac{1}{2\theta}\sum_{i=1}^{N} \int_{\Omega}(v_i-u_i)^2dx+ \frac{\beta}{2} \int_{\Omega}\left ( ...
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1answer
50 views

What does the code do and what ODE is that?

This exercise I came across asks what kind of method of solving ODE is that: ...
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0answers
28 views

Why do Runge Kutta's method and Euler's are so different?

I am solving a $\underline {\dot A}=\underline A\cdot \underline x$ system of linear equations numerically. I have don'e this in the popular of methods of Euler and Runge Kutta. I have noticed a ...
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0answers
28 views

Numerical method for wave equation with nonlinear forcing in 1+1 D

I am looking for numerical method to solve the equation $$ \square \phi = \frac{\partial^2 \phi}{\partial x^2} - \frac{\partial^2 \phi}{\partial t^2} = \lambda \phi^3 \, $$ for real $\phi = ...
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2answers
219 views

Numerical methods (for ODE/PDE) that could take approximate solutions/good initial guesses, and further refine it to an certain accuracy

I am currently playing with an old analog computer, which could solve time-dependent ODE/PDEs pretty fast, without time-stepping; thus there is no convergence issues caused by time-stepping because of ...
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2answers
54 views

Numerical mathematics, Lagrange interpolation..

I am trying to solve this problem, but I don't have any idea. Maybe it doesn't look at first sight that Lagrange interpolation can be used, but I found this problem in that chapter of Numerical ...
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0answers
20 views

Polynomial interpolation- numerical analysis

Using polynomial interpolation show that the polynomials $p(x) = (x-1)(x-2)(x+1) $and $f(x) = x^3 - 2x^2 - x + 2$ are one and the same......its absurdly simple...... I tried to show by Newtonian ...
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0answers
25 views

Monotonic convergence of Newton's method for boundary value problems

I’m interested in solving nonlinear elliptic boundary value problems of the type $$ -a\Delta u + f(u) = 0, $$ $$ u|_\Gamma = u_0 $$ by Newton’s method when its convergence is global and monotonic. ...
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1answer
50 views

Is there an analytic or at least a numerical solution to an eqaution of the form $\sqrt{k_1\sqrt{x}+k_2}\;\Big(k_3x+k_4\sqrt{x}+k_5\Big)+k_6=0$?

So the problem comes from cosmology and I want to solve for the unknown function $a(t)$, which is the scale factor for the universe. So I have an integral involving $a$: $$ ...
2
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1answer
26 views

Use of ergodic theory in numerical simulations

Is ergodic theory used in numerical simulations? The kind of application I have in mind is: for $\alpha$ irrational, $( n\alpha \mod 1)_{n \geq 0}$ is equi-distributed on $[0,1]$, and I imagine that ...
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1answer
31 views

Distance between points

I am wondering how can I solve following problem. ...
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2answers
78 views

Estimating the value of an improper integral numerically

My question is how can I estimate the value of an improper integral from $[0,\infty)$ if I only have a programming routine that gives me the function evaluated at 100 data points, or 100 values of ...
0
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1answer
29 views

Gradient descent with linear perturbation

Given a convex, differentiable function $f$ (from a Hilbert space to $\mathbb{R}$) with a minimum (say $x^*$), I know you can find $x^*$ using gradient descent. Suppose now that you apply gradient ...
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0answers
28 views

Does this idea of explicit expansion to find a,b,c and so on work for all example of F(x,y)?

Does the method below work for all example? It could be more complicated since at here one can simply show $y_1=(x+h)^2=x_0^2+2hx_0+h^2=y_0+2hx_0+h^2$. However, i found it could be useful in proving ...
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0answers
34 views

3Dimensional runge kutta and Euler method(verification+proposition)

I been discussing this idea with a tutor for sometime. However it turn out that the proof is not comprehensible.Can someone please help to verify the the proof for 3D Euler method and runge kutta ...
2
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1answer
40 views

Stability analysis of Ralston's method

Ralston's method is given by: $$y_{n+1} = y_n + \frac{h}3(f(t_n,y_n)+2f(t_n+\frac34h, y_n + \frac34h f(t_n,y_n)))$$ carry out a stability analysis of this method to determine the condition for ...
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3answers
126 views

How to prove $\det(I+uv^\intercal)=1+v^\intercal u$

Let be $u,v\in\mathbb{R}^n$, then $\det(I+uv^\intercal)=1+v^\intercal u $ where $I$ denotes the identity matrix of order $n$. How to prove this? what I did: let be ...
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0answers
25 views

How to apply Runge-Kutta to an implicit scheme?

I see there are some differences in the solution as I increase the resolution of my grid. I'm using Operator Splitting to solve Diffusion Reaction equation \begin{equation} \frac{\partial ...
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0answers
14 views

Integral on an region defined by a regular grid of points

I'm trying to evaluate a multidimensional integrand $f$ that I know the major contribution is restricted to a specific region around its maxima (to be concrete, imagine a 2D gaussian function). What ...
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2answers
55 views

Newton Raphson Step Size

I am solving old exams and I came across the following question: Let $$ x_{n+1} = x_{n} - \alpha\frac{f(x_{n})}{f'(x_{n})} \;\;,\;\; f(x_{n}) \gt0 \;\;,\;\; f'(x_{n}) \neq0 $$ Is it true ...
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0answers
48 views

Numerical mathematics, Lagrange interpolation

I am trying to solve this problem, but I don't have any idea. Maybe it doesn't look at first sight that Lagrange interpolation can be used, but I found this problem in that chapter of Numerical ...
0
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0answers
9 views

Numerically/Computationally estimating parameters

I have a function $f(x)$ and I have an estimating function $\hat f(a,b,c,d;x)$ Say, I also have a scoring function $S(f,\hat f,x)$ (which could very well be mean square error) And I have some ...
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1answer
51 views

Quadrature formula on triangle

I am looking for a quadrature formula on the triangle, with points at the vertices and at the mid-edges, so 6 points, and that is exact for polynomials of degree at least 2, with weights strictly ...
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22 views

non-linear PDE finite difference approach

How to approach this equation using finite difference method ...
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1answer
183 views

Singularities of an integral

We have the integral : $$I(t)=-i\int_0^\infty \frac{\log\left[\frac{\sin(t\log\sqrt{1+ix})}{\log(1+ix)} \right ]-\log\left[\frac{\sin(t\log\sqrt{1-ix})}{\log(1-ix)} \right ]}{e^{2\pi x}-1} \, dx$$ I ...