# Tagged Questions

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

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### 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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### 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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### 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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### 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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### 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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### 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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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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### 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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### 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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### 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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### 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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### Distance between points

I am wondering how can I solve following problem. ...
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### 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 ...
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### 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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### 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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### 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 ...
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### 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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### 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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### 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 \frac{\partial ...
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### 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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### 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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### 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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### 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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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 ...
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 ...