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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17 views

Formula on the basis of discrete data

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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1answer
40 views

Smallest n such that x starting with digit 9 [on hold]

I have problem because my scientific calculator has a limit, my laptop is useless and wolfram alpha also can't help me. It is regarding the smallest natural number $n$ such that $x$ in the equation ...
1
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6answers
450 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 ...
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1answer
21 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 ...
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0answers
23 views

Reference in Latex [migrated]

For my thesis references, When I run latex, in pdf appear like below Aliabadi, M. H. The Boundary Element Method, Applications in Solids and Structures, Vol. 2, J. Wiley, New York, 2002. 4 Baudouin, ...
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2answers
46 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 $$\frac{f_1}{f_2}=\frac{O(g_{1})}{O(g_{2})}=O\!\left(\frac{g_1}{g_2}\right)$$ ?
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0answers
16 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 ...
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0answers
28 views

Looking for mathematical/combinatorial and computational explanation regarding adding values in a $5 \times 4$ (matrix?) with a constraint.

Given the following matrix (not sure if I should call it that): Matrix $5 \times 4$ I want to add all possible combinations of values such that each Horse gets but one value from each Bookie. What I ...
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0answers
18 views

Is there a way to reverse engineer an already large number to make it smaller? [closed]

Using the Ackermann function for example, it's quite easy to make massive numbers. My question is whether there's an existing algorithm that can take a large number and reverse engineer it to make an ...
1
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1answer
28 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 ...
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0answers
13 views

Statistical calculation for neural firing rates with negative rate on numerical simulation

I am now working on a biological neural network simulation (NEST-Simulator) project with a problem of calculating firing rates. Background: The data set as result of simulation is a set of events in ...
1
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2answers
163 views

How are 10-20 digit multiperfect and hemiperfect numbers efficiently computed?

This numericana item on multiperfect and hemiperfect numbers contains some impressively enormous numbers. How were these actually computed ? The associated OEIS pages (A007691 & A159907) just ...
0
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1answer
31 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 ...
4
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2answers
619 views

Software for numerical solution of a non-linear ODE system?

I have been given a nonlinear system of ODEs which has arisen out of a colleague's engineering research: $$\begin{array}{rcl} \dot{x}_0&=&x_1\\ ...
1
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2answers
86 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
49 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 ...
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0answers
34 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 ...
2
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1answer
165 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 ...
5
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1answer
564 views

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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0answers
17 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 ...
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2answers
17 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 ...
0
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1answer
950 views

Fast methods to check linearity of differentials? Generalizing linearity?

The L1 Mat-1.1010 -course here has taught me the linearity conditions $f(a x)=a f(x)$ and $f(a+b)=f(a)+f(b)$. I want to generalize it, some quite irrelevant slow investigation here. It requires time ...
7
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5answers
310 views

What is a nice way to compute $f(x) = x / (\exp(x) - 1)$?

I want it to be stable near $f(0) = 1$. Is there a nice function that does this already, like maybe a hyperbolic trig function or something like expm1, or should I just check if $x$ is near zero and ...
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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
12 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 ...
1
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1answer
44 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 ...
1
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1answer
66 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 ...
0
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0answers
29 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 $(x,y,z) \in \mathbb{R}^3$. ...
2
votes
2answers
451 views

Space spanned by matrices

I have a set of $5$ by $5 $matrices, $M_1,M_2,...,M_{19} ,M_{20}$. I want to try to find a basis from this set and also to find relationships between these matrices. This is how I think I should ...
2
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1answer
73 views

The Command “TzGoGo” in GAP

I am learning GAP and would like to ask one question about a command called "TzGoGo": If $P$ is a finite presentation of a group $G$, then will the eventual result of the command "TzGoGo(P)" be ...
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0answers
91 views

Generalizing Bellard's “exotic” formula for $\pi$ to $m=11$

Bellard's "exotic" pi formula has the form, $$a\pi+b = \sum_{n=1}^\infty \dfrac{P(n)}{{\displaystyle \binom{mn}{2n}2^{n-1}}}$$ where $a,b,m$ are integers and he uses $m=7$. However, it seems there ...
6
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2answers
94 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 ...
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1answer
37 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 ...
4
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2answers
96 views

Amicable numbers

Def: a pair natural numbers $a$, $b$, $a\ne b$ are an Amicable pair if $\sum_{d|a,a\ne d}d = b$ and $\sum_{d|b, b\ne d}d = a$. Ok. So I'm trying to optimize a calculation for finding the number of ...
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1answer
26 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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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 ...
3
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0answers
53 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
32 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 ...
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0answers
16 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, ...
5
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3answers
81 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
550 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 ...
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3answers
309 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
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2answers
52 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
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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
22 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, ...
2
votes
3answers
67 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 ...
1
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1answer
77 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} ...
4
votes
4answers
489 views

Computing partition numbers

Today a friend and myself came up with the question of computing partitions of numbers, i.e.: given a number $n$, what is the number $p(n)$ of was of different ways writing $n$ as a sum of non-zero ...
2
votes
1answer
59 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 ...
5
votes
1answer
154 views

Integer solutions to the equation $a^3+b^3+c^3=30$

The following problem was posed to me but I could not do much about it: Determine if there are any integer solutions to the equation $a^3+b^3+c^3=30$ I made a computer search that shows that ...