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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3answers
42 views

upper bound of a function $n^{1/\log(n)}$

I have the following expression $n^{1/\log(n)}, \quad where \quad n \in [1, 10,000]$. When I solve this numericall, I get the resultant value 2.718282 for all $n \in [2, 10,000]$. On this basis, I can ...
1
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0answers
26 views

proving euclid's algorithm stops after 2k iterations [duplicate]

Apparently, when initialized on $k$-bit integers $A$, $B$, Euclid's algorithm terminates after at most $2k$ iterations. This result is not immediately intuitive to me. I would appreciate help in ...
2
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0answers
38 views

How to find values of x where $a_i x$ are nearly integers? $a_i \in \Bbb R$

I have a set $\{a_i \in \Bbb R | \ i <=7 \}$, and I'm looking for a way to find values of $x$ where given $\epsilon > 0$, $$\forall i \ \exists n_i \in \Bbb{Z} \ \ |a_i x - n_i| < \epsilon$$ ...
0
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0answers
33 views

Find values of x where error term small between rounded and not calculation in non-linear functions f_{i}(x)

Here's my situation. I have a set theory background so I'm out of my league in applied, computational methods so I'd appreciate a hand-up. I have a set of five functions, $f_{i}(x) \ $ where $\ 0 <...
1
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3answers
64 views

n(n+1)/2 combinatorial proof (details in description)

Find the number of $2$-lists $(𝑎, 𝑏)$ we can form using the numbers $0,1,2,...,𝑛$ with $𝑎 < 𝑏$. a. Show that the number is $𝑛(𝑛 + 1)/2$ by considering the number of $2$-lists $(𝑎, 𝑏)$ in ...
0
votes
1answer
69 views

About Gröbner Bases

Recently I have come across a book of Gröbner Bases written by Adams & Loustaunau. The book is excellent and I have become interested in Gröbner bases after reading the book. I want to read more ...
0
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2answers
44 views

How to define numbers in a way that a number 'n' is equivalent to the function plus 'n'?

In lambda calculus, is it possible to define (or disprove the existence of) a number system alternative to church numerals such that each number is a function which on application, adds itself to it's ...
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2answers
42 views

Linear Algebra matrices question.

Let $A,B$ be 2 square matrices of the same size. And the following holds true $AB=A+B$ How do I prove that $(I-B)$ and $(I-A)$ are invertible
1
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1answer
38 views

Explicit piecewise linear approximation of a function of 4 variables

I have a table of numbers for fixed values of 4 parameters $x, y, z, t$, at this $x$ belongs to finite set of natural numbers, $y\in\{1;2\}$, $z\in\{5;10;15;20;25\}$ and $t\in\{1,2,3\}$. Is there a ...
1
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2answers
50 views

Big O notation: ratio of two $O(\cdot)$'s is $O(\cdot)$ of the ratio?

Is it true that if $f_1=O(g_1)$ and $f_2=O(g_2)$ then $$\frac{f_1}{f_2}=\frac{O(g_{1})}{O(g_{2})}=O\!\left(\frac{g_1}{g_2}\right)$$ ?
0
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0answers
30 views

Distance Geometry Problem (DGP) Programming Language Recommendation

We have been studying DGPs in clinic recently and I was hoping I might be able to get recommendations for computing languages in the processing of large network solutions. Specific computations ...
1
vote
1answer
42 views

How to write new algorithm of root finding by combining 2 or 3 standard algorithms(bisection, fixed, etc)

I just learned about Bisection Method, Fixed-Point Iteration Method, Newton- Raphson Method, and Secant Method. My prof wants us to be able to write new Algorithm of root finding by coming 2 or 3 ...
0
votes
1answer
37 views

Constructing a specific Rank-One Matrix

Given u $\in \mathbb{R}^{n}$ and v $\in \mathbb{R}^{m}$ with unit $L^{2}$ norm, i.e. $\|u\|_{2}$ = $\|v\|_{2}$ = 1. Construct a rank-one matrix B $\in \mathbb{R}^{mxn}$ such that $Bu = v$ and $\|B\|_{...
1
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2answers
99 views

Use $\log(x)$ to calculate $\log(x+1)$

Given that I know the value of $\log(x)$, I would like to calculate the value of $\log(x+1)$ on a computer. I know that I could use the Taylor expansion of $\log(1+x)$, but that uses $x$ rather than $...
0
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1answer
53 views

One wants to determine $1/(2+\sqrt 3)^4$ having access to an approximate value of $\sqrt 3$.

Can someone help me with this question please: One wants to determine $1/(2+\sqrt 3)^4$ having access to an approximate value of $\sqrt 3$. Compare the relative errors on direct computation and on ...
0
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0answers
35 views

A computation from an article in computational neurosciences (from physical review) which doesn't fit

I am reading this article (with this erratum) in computational neuroscience, and there is a computation there that simply doesn't fit.. Maybe one of you can see something that I am missing? For the ...
0
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0answers
42 views

Hilbert Matrix, Gaussian Elimination with varying pivot strategies, and computation error.

I'm doing a project for my Numerical Analysis class about computational error related to Gaussian elimination, gaussian elimination with partial pivoting, and gaussian elimination with scaled partial ...
0
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2answers
20 views

Error analysis on numerical solutiol of an equation

Say I am solving an equation numerically -- the derivatives in the equation I find by a finite difference scheme with an accuracy of the grid spacing $h$. Does this imply that the final solution I ...
2
votes
1answer
174 views

Non-calculator proof that $\pi^\pi -\pi \lt \frac{100}{3}$

I am looking for a few non-computational, non-calculator proof of the following inequality: $$\pi^\pi -\pi \lt \frac{100}{3}$$ I can't really seem to come up with a proof because of that killer $\pi^...
4
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6answers
560 views

Proving $\pi^3 \gt 31$

$$\large \pi^3 \gt 31$$ Using a calculator, $\pi^3/31 \approx 1.0002$, so I thought this may be challenging to do by hand. It is extremely easy with the use of any calculator, so I was wondering now:...
0
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0answers
7 views

How do I validate my ARMAX model?

Say I have some ouput $y_1, y_2, \ldots, y_N$ and inputs $x_1, x_2, \ldots, x_N$ which, by various time series methods, I've found to match an ARMAX(2,2,1) model. So I've found the estimations for ...
0
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0answers
20 views

Formula for MLM like system

I'm trying to figure out a formula for a system similar to a MLM system such that all members will receive 50/50 of the shares. So for example, X recieves 50% and A recieves 50%. When A recruits B and ...
0
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0answers
41 views

Inverse of sum of matrices

Let $A,B$ be invertible positive definite matrices of the same size. My goal is to efficiently compute $(xA + yB + zI)^{-1}$ for many triplets of positive real numbers $(x,y,z) \in \mathbb{R}^3$. ...
1
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1answer
78 views

Minimizing computations for evaluating two polynomial simultaneously

I want to evaluate two polynomials $f$ and $g$ simultaneously, on the same input (in a computer program). These polynomial have only coefficients $0, 1, a , b$ and their degree is less than 700. I ...
1
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1answer
71 views

Finite element method books

I know this question has been asked before; I just want to enquire if anybody has any suggestions to learn how to compute finite element problems, including plenty of examples. The topics I would ...
6
votes
2answers
98 views

For what values of $k$ does $(1+x)^{500+k}(1-x)^{500-k}$ exceed $10^9$?

Pretty simple question, for what values of $0\leq k \leq 500$ do we have $\max\{(1+x)^{500+k}(1-x)^{500-k}|x\in[0,1]\} \geq 10^9$ ? Some trivial observations: The problem is equivalent to finding ...
0
votes
1answer
40 views

More efficient method of computing the square root of $-1 \mod p$

I am currently doing collecting some preliminary data about elliptic curves over finite fields of order $p$ where $p$ is a prime congruent to 1 mod 4. Part of the data collection process requires me ...
-1
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1answer
30 views

c(x) = k for all positive k is primitive recursive [closed]

How can I show this function is whether primitive recursive or not? Do I need to use Godel number?
3
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0answers
56 views

Is there a systematic way of “discovering” an algebra from observations of its universe?

I am faced with the following situation: I have a finite set of some $m$ positive integers $Q^m \in \mathbb{N}$ These integers go through a series of $N$ possible black boxes that transform them. ...
0
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1answer
33 views

function computed by programs

I have two questions: Which is the function computed by the program $o^1_1(Succ, Succ)$? Which is the function computed by the program $\mu^1(\pi^2_1)$? where $o^n_m$ for the composition rule $...
0
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0answers
17 views

Solve Lotka-Volterra by hand? [duplicate]

I learn Lotka-Volterra model in computing mathematics textbook and solve it by different numerical methods. $$ \frac{1}{x}\frac{dx}{dt} = a - by$$ $$ \frac{1}{y}\frac{dy}{dt} = cx - d$$ where, $a,b,...
3
votes
2answers
91 views

Squeezing primes

Any positive odd number $n$ can be coded one binary digit smaller by the rule $\frac{n-1}{2}$ and that's obviously the best squeeze: a bijection from $\mathbb N$ such that $f(n)\geq n$. I'm looking ...
6
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3answers
116 views

Digits of $\pi$ using Integer Arithmetic

How can I compute the first few decimal digits of $\pi$ using only integer arithmetic? By 'integer arithmetic' I mean the operations of addition, subtraction, and multiplication with both operands as ...
5
votes
5answers
559 views

A valid floor function trick?

Given $x\in\mathbb R_+$ and $m,n\in\mathbb Z_+$, is it true that $$\bigg\lfloor\frac{\lfloor \frac{x}{m}\rfloor}{n}\bigg\rfloor=\bigg\lfloor \frac{x}{mn}\bigg\rfloor?$$ Thanks for at least three ...
9
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3answers
334 views

Three pythagorean triples

Are there any solutions for $a, b, c$ such that: $$a, b, c \in \Bbb N_1$$ $$\sqrt{a^2+(b+c)^2} \in \Bbb N_1$$ $$\sqrt{b^2+(a+c)^2} \in \Bbb N_1$$ $$\sqrt{c^2+(a+b)^2} \in \Bbb N_1$$
4
votes
4answers
93 views

What are all the concordant forms $n$ such that $a^2+b^2 = c^2,\,a^2+nb^2=d^2$ for $n<1000$?

Part I. The list of congruent numbers $n<10^4$ such that the system, $$a^2-nb^2 = c^2$$ $$a^2+nb^2 = d^2$$ has a solution in the positive integers is known (A003273) $$n = 5, 6, 7, 13, 14, 15, ...
1
vote
1answer
24 views

When is ${n \choose k} > (n-k)(k+1) + (n-k-1)k$?

I have two algorithms that output the same result for an input value of a non-negative integer k and a list of n elements, where $1 \leq k \leq n$. However, the two algorithms are very different in ...
1
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0answers
24 views

Trying to learn about kernel PCA but cannot understand some math.

I'm trying to learn about kernel PCA by reading through the paper of it's creators (I assume) "Nonlinear Component Analysis as a Kernel Eigenvalue Problem", Bernhard Schölkopf, Alexander Smola, Klaus-...
1
vote
1answer
85 views

How can I optimise a scalar function over a matrix?

I need to optimise the following scalar function with respect to a matrix $S$. $$ f(S) = \boldsymbol{y}^{T}\boldsymbol{X}w - \boldsymbol{1}_{n}^{T} \exp \left\{ \boldsymbol{X}w + \frac{1}{2} \...
2
votes
2answers
82 views

Which continued fraction for $e$ is the most computationally efficient?

I know that famous numbers like $\pi$ and $e$ have multiple representations as continued fractions and I'm fascinated with the variety of representations. My question: What continued fraction for $e$...
2
votes
3answers
71 views

Looking for fractals which are computationally demanding and preferrably parallelizable.

Oh hello guys. I am in the middle of challenging myself to putting my computer and math skills together, trying to build a small hobby computational cluster. Being interested in fractals for a long ...
0
votes
1answer
37 views

How to obtain fair competition between two teams

Consider a class with $4$ students having min goals as $\big\{ 1, 3, 4, 5 \big\} $ and max goals as $\big\{ 2, 5, 8, 6 \big\}$. Find the best way to divide the class in such a way that the match is ...
0
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0answers
22 views

Finite element boundary conditions

I have a boundary condition given by $\mathbf{n}\cdot \nabla m=\phi$, where $n$ is a vector normal to a surface, $m$ is a physical quantity (say mass) and $\phi$ is a constant. The boundary condition ...
0
votes
1answer
22 views

How can I implement Newton-Raphson's method with a function of one vector and one matrix?

I have a function $f(\mathbf{u}, \Sigma)$ where $\mathbf{u}$ is a $p \times 1$ vector and $\Sigma$ is a $p \times p$ real symmetric matrix (positive semi-definite). I somehow successfully computed ...
0
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2answers
81 views

Converting Java Code to Mathematic formula

I have algorithm , but I don't know how to convert it to mathematic formula. ...
0
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0answers
13 views

bio heat equation modification

I have the bio heat equation as described .here And the solution to it is, But to this I am trying to include the effect from exercise intensity as well. So the modified bio heat equation is ...
2
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0answers
18 views

Given a set of integers $S$, what is the maximum integer that is a product of one or more integers from $S$ not exceeding $X$?

Given a set of integers $S$, which will contain no more than $100$ integers. Now, what would be the fastest approach to find $M$ which is a product of one or more integers from $S$ (and multiple usage ...
0
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0answers
30 views

summation of even and odd numbers

In my video processing algoritm, I do some processing even and odd frames seperately. F = E(x) + O(x) where F is the video, E and O contains its even and odd ...
9
votes
1answer
146 views

How do I develop numerical routines for the evaluation of my own special functions?

This question has been cross-posted to ComputationalScience.SE here. When performing computational work, I often come across a univariate function, defined in terms of an integral or differential ...
0
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0answers
39 views

How to solve the Allen-Cahn equation with finite element method?

$$\frac{\partial\phi(\mathbf{x},t)}{\partial t}=\varepsilon^{2}\Delta\phi-F^{'}(\phi),\ \ \ \mathbf{x}\in \Omega,t>0$$ $$\frac{\partial \phi}{\partial\mathbf{n}}=0\ \ \text{on} \ \partial\Omega$$ $$...