# 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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### Finding $1/x^2 + 1/x^3 + 1/x^5 + \dots$

The following function came up in my work: $$f(x)=\sum_{p\text{ prime}}\frac{1}{x^p}=\frac{1}{x^2}+\frac{1}{x^3}+\frac{1}{x^5}+\frac{1}{x^7}+\frac{1}{x^{11}}+\cdots.$$ Naturally, this converges for ...
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### Piecewise-linear (or otherwise monotonic) interpolation as a matrix problem

Background: I'm hoping to find (or write) an algorithm to piecewise linear-interpolate large sets of unevenly sampled functions (10s of thousands of arrays of a thousand or so $x$ and $y$ pairs, where ...
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### Solving a system of N-1 Ist order ODEs by Euler's Method

In order to solve a system of N-1 first order ODEs by Euler's Method For N = 4; t=0, h= 0.1, x= 0.1 should the Euler formula be? $U_n(t+h) = U_n(t) + h F_n(x_n, t_n)$ for n = 1, 2,..,N-1 but we ...
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### Explicit and implicit RK methods with stiff problems

Even if there isn't a precise definition of stiff equation, i think we can sum up (for the sake of the question) the concept in two cases: A linear equation $u'=\lambda u$ with a negative $\lambda$; ...
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### How to understand the vector form of the Jacobi iteration?

When I read the book "Iterative Methods for Sparse Linear Systems" about Jacobi iteration, I can easily understand the component form for this iteration. However, since my background is computer ...
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### Numerical analysis of wave equation in polar coordinates:

Is there a simple solution to deal with the problem of radial symmetry when solving a pde numerically. If so can someone provide some references/resources that explain this. Any help would be greatly ...
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### Numerical method for fourier transform other than FFT/DFT

FFT relies on uniform samples, which cause aliasing, so FFT can be inaccurate in a certain case. Suppose you can obtain samples of $f(t): \mathbb{C} \to \mathbb{C}$ at any point ($t$ can also be a ...
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### Computing a double integral with applications to prime numbers

I was reading the preprint [1] which contains on p. 7 the following formula (for $4<s\le6$): $$f_1(s)=\frac{2e^\gamma}{s}\left\{\log(s-1)+\int_4^s\int_3^t\frac{\log(u-2)}{u-1}du\,dt \right\}$$ ...
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### How to define a variable which is an integral involving cauchy principal value inside?

How to define a variable which is an integral involving cauchy principal value inside in any computer programming language? I want to know how to break down the procedure step by step from a ...
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Given a weighted integral $$I(f)=\int_{-\infty}^\infty f(x)e^{-x^2}dx.$$ How can I calculate the Gauss quadrature for two points. I know how to calculate the quadrature wih Legendre polynomials, but ...
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Let us assume I have a "nice" curve and that I would like to introduce a small shock up/down of about 1% at a certain point along the curve. I am trying to find out what the best and most efficient ...
Define for real $x > 0$ the function: $$F(x)= 1 + x e^{x} Ei(-x),$$ where $Ei(x)$ is the exponential integral. I found in a physics papers (Amaldi, Fluctuations in ...
Consider two football teams $V$ and $L$ with strengths $W_V$ and $W_L$, respectively. Let's assume that the draw probability $\mathbb{P}(Draw)$ is known. Then this model is supposed to give estimates ...