# Tagged Questions

This tag concerns computational problems central to mathematical and scientific computating. The scope includes algorithms, numerical analysis, optimization, and linear algebra, computational topology, computational geometry, symbolic methods, and inverse problems.

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### How to exactly determine whether a sum of n-th roots of unity is zero

Define the set $R = \{e^{2\pi i k/n} | k=0,1,\ldots,n-1\}$ of $n$-th roots of unity. Let $S \subseteq R$ be a subset. How can I (algorithmically?) determine whether $\sum_{s\in S} s = 0$? I'm ...
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### Even natural numbers are sums of two primes with twins or of two primes without twins

I seems to be very few even numbers that can't be written as a sum of two primes with twins or as a sum of two primes without twins. That is, suppose that $\mathbb P'$ is the set of the primes not ...
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### Finding all real roots of an equation

I am looking for a computational method to find real roots of a function. The function looks like this: $$F(x) = \sum_{i=0}^n \frac{k_i}{\sqrt {(x - x_i)^2 + c_i^2}}.$$ I would like to use something ...
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### Is a possible use of a Mill's constant the encapsulation/encryption of messages?

I wonder if the way that Mill's constant is defined could provide a good data encapsulation and encryption method if instead of encapsulating primes, for instance a simple ASCII message is ...
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### Cylinder-ray intersections equation

I found an article involving infinite cylinder-ray intersections, and I don't know how they develop this equation: $$(q - p_a - (v_a, q - p_a)v_a)^2 - r^2 = 0$$ In the end of the first page I quote: ...
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### Book that helps recognizing underlying mathematical expression of numerical results

I strongly remember having seen a couple of years ago a huge book that only consisted out of an enormous table over hundreds of pages containing approx. 10⁵ numerical results in ascending order with ...
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### Analyzing Accuracy of a method to compute summation

How does one analyze the accuracy of the following method to evaluate the summation? Also, what is the difference between analyzing the accuracy and proving that it is backward stable? I think I ...
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### terms of taylor expansions of multiple variables at the origin

By the fundamental theorem of symmetric polynomials, $X_1,X_2,\cdots,X_n$ are polynomials of $e_1,\cdots,e_n$ and $$\mathbb{Z}[ e_1,\cdots,e_n]=\mathbb{Z}[X_1,X_2,\cdots,X_n].$$ We define a ...
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### Finite difference formula to approximate second derivative

I have one question which asks to derive a finite difference formula to approximate $f''(x)$ in the form of $$f''(x)\approx Af(x+2h)+Bf(x+h)+Cf(x)$$ with the method of undetermined coefficients. ...
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For h:= $\mathbb{R} \rightarrow \mathbb{R}, x \rightarrow e^{x}-x^2+1$ I know the formula as $$X_{n+1}=X_{n}-\frac{f(X_{n})}{f'(X_{n})}$$ so this would give me: $$X_{n+1}=X_{n}-\frac{e^{x}-x^{2}+1}{e^... 0answers 35 views ### R \subseteq \omega recursive iff \exists m \in \omega such that R=\{n \ | \ \bar{\omega} \models \phi[m,n] \}. The queston I'm trying to solve is use Kleene's enumeration theorem to show R \subseteq \omega recursive iff \exists m such that R=\{n \ | \ \bar{\omega} \models \phi[m,n] \} for some m \in \... 0answers 40 views ### Normalizing an elliptic curve to find integer solutions I have an elliptic curve$$ c_1y^2 + a_1xy + a_3 = c_2x^3 + a_2x^2 + a_4x + a_6 $$with integers a_1,a_2,a_3,a_4,a_6,c_1,c_2 and I would like to find all integer solutions of this elliptic curve. I ... 1answer 26 views ### Write column form elementary matrix in terms of element form elementary matrices Recall that any unit lower triangular matrix L\in\mathbb{R}^{n\times n} can be written in factored form as $$L = M_1 M_2\ldots M_{n-1}$$ where M_i = I + l_i e_i^{T} ... 5answers 113 views ### Minimize a^5+b^5+c^5+d^5+e^5 = p^4+q^4+r^4+s^4 = x^3+y^3+z^3 = m^2 + n^2 with distinct positive integers Find the minimum value of the following:$$a^5+b^5+c^5+d^5+e^5 = p^4+q^4+r^4+s^4 = x^3+y^3+z^3 = m^2 + n^2 where all numbers are different/distinct positive integers. I know the answer (see ...
In view of what we can compute by hand, on a piece of paper, without having to use a computer or a calculator, how far can we go with the evaluation of $\log$-functions and fractional powers? More ...