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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0answers
59 views

Finishing a problem using equalities

This is my problem: Let $a$, $b$ and $c$ be positive real numbers with $abc=1$. Prove that $$\frac{a^{n+2}}{a^n + (n-1)\,b^n} + \frac{b^{n+2}}{b^n + (n-1)\,c^n} + \frac{c^{n+2}}{c^n + ...
0
votes
1answer
40 views

Does 2 one-dimensional intervals touch?

As the title mentions I have 2 one-dimensional intervals given like so: $[a, a-b]$ $[x, x-y]$ where $a$ and $x$ are the start points, and $b$ and $y$ are the length of the intervals. The intervals ...
5
votes
1answer
192 views

Efficient free alternative to *Mathematica*

I am searching for a free alternative to Mathematica. By efficient, I mean that it should have every (or at least almost every) function that you can find in Mathematica, including for example Number ...
0
votes
1answer
53 views

Assertions about measures with computers

Let's consider the Lebesgue measure ($\mu$) over the closed interval $[0,1]$. As you know, $\mu(\mathbb{Q} \cap [0,1]) = 0$. In other way, as far as I know the computer just can represent accurately ...
1
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0answers
35 views

Use of Matlab to put equation into vector form

Is there a way to put the following equation of a line into vector form using Matlab? $\displaystyle y=\frac{cos(s_n)-cos(s_{n+1})}{sin(s_{n+1}-sin(s_n)}(x-sin(s_n))-cos(s_n)$
1
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0answers
102 views

Solving many independent non-linear systems simultaneously

I'm working on solving lots of systems of nonlinear equations. Luckily, the non-linear equation is the same, but the parameters are different: $$ f(\vec{x}_0; c_0) = 0\\ f(\vec{x}_1; c_1) = 0\\ ...
-1
votes
2answers
48 views

If $a+(1/(a-2))=4 $ then $(a-2)^2+(1/(a-2))^2$ is .

If $a+(1/(a-2))=4 $ then $(a-2)^2+(1/(a-2))^2$ is . Note: $a^2+(1/(a-2))^2=4^2$
2
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0answers
50 views

Computer program to simplify formulas

What is the computer program that attempts to simplify sums of binomial coefficients, factorials, etc.? Possibly Zeilberger wrote it, but I'm unsure. If so, possibly it was talked about in his A=B ...
2
votes
2answers
45 views

Will the Newton's method be convergent to the root of the following function: $f(x)=\frac{-x}{x^2-1}$?

Will the Newton's method be convergent to the root of the following function, if the starting point $x_0>1$ will be chosen? $$ f(x)=\frac{-x}{x^2-1} $$
2
votes
1answer
74 views

Graph pruning whilst ensuring connectivity

Problem: I have a graph (in this instance, it's represented by a matrix which is $\in \mathbb{R}^{n \times n}$). In the raw graph, all nodes are connected to every other node (except themselves) in ...
0
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2answers
72 views

Numerical integration tolerance pitfalls

Consider that we want to estimate $$\int_{\pi/2}^{\pi/2+8\pi}sin(x)dx$$ (the value of this integrate is obviously zero) with the Midpoint rule. We start with the endpoints $a=\pi/2$ and $b=\pi/2+8\pi$ ...
3
votes
1answer
111 views

Defining a subgroup of $GL(2,7)$ in GAP

Considering this resent post in which $|G|=42$, I am thinking of making this subgroup concrete in GAP environment. Maybe, if the structure of $G$ was known then, we would use an appropriate mapping ...
2
votes
2answers
51 views

Difference between the complex roots of $f(x)$ and $|f(x)|^2$

I suppose a basic question, but it's causing me more problems than I envisioned! I have some polynomial $f(x)$ for which the roots are complex, $x+iy$. How will these roots change if I now take ...
0
votes
1answer
189 views

Overflow and underflow of a probability value

I am evaluating the probability that the minimum of a process is a above a a barrier $\log(H)$. The probability is given by $$P_i=1-\exp\left(-2\frac{(\log(H)-x)(\log(H)-x_b)}{\tau\sigma^2}\right).$$ ...
1
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1answer
186 views

Calculating the Average Number of Games Required to Reach a Theoretical True Elo Skill Rating from a given Initial Elo Rating

The USCF uses the following formula for Elo rating adjustments: $$R'=R_0+K(S-E)$$ $$E=\frac{1}{1+10^{(R_n-R_0)/400}}$$ Where $R'$ is the new rating $R_0$ is the initial rating $K$ is a ...
1
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0answers
39 views

How to reconstruct geometric object that a Frobenius group acts on

A Frobenius group has equivalent definitions: a transitive permutation group on a finite set such that no non-trivial element fixes more than one point and some non-trivial element fixes a point. ...
0
votes
1answer
184 views

How to discretize mixed partial derivatives?

How to discretize $\frac{\partial^3 f}{\partial x\partial y^2}$ at mesh point $(i,j)$? We should use mesh points which are nearest to $(i,j)$.
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0answers
68 views

Setting up Kernel to Numerically Solve Fredholm Equation of Second Kind

I am looking to confirm if what I am doing is the proper procedure. I writing a program to discretely solve a Homogeneous Fredholm Equation of Second Kind that is set up as follows: $ \int ...
1
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0answers
31 views

Transformation between Ideal and Warped Surface

I work on manufacturing metal panels with holes drilled in them. Suppose I have an ideal 3D surface from CAD. I want to compare it to the actual part using reference points to compare between the two. ...
8
votes
4answers
912 views

What is the most efficient way to calculate the sine of a rational number?

I'm happy that we can use some trig identities like $$\sin\left(\frac{\theta}{2}\right) \equiv \pm \sqrt{\frac{1-\cos(\theta)}{2}}$$ and $$\sin(\alpha \pm\beta) \equiv \sin(\alpha) \cos(\beta)\pm ...
1
vote
1answer
85 views

Isomorphism between two magmas with one.

Do we have a method to find one (or all) isomorphism between two given magmas with one using GAP? Edit If we have Loop or Latin square (with one) instead of Magma then do we have the method?
0
votes
4answers
89 views

Min and Max of $ f(x,y)=\frac{x-y}{a-x-y}$

I'd like to find max and min of $$ f(x,y)=\frac{x-y}{a-x-y}$$ where $0\le x<y\le a/2$. Any one can suggest? Thank you
3
votes
1answer
190 views

How to numerically solve the eigenvalues of the laplacian in a triangular domain with Dirichlet boundary condition?

Consider an arbitrary triangle. Now impose the Dirichlet boundary condition. How to solve the eigenvalues and eigenvectors of the Laplacian $-\nabla^2 = - \frac{\partial^2}{\partial x^2} - ...
4
votes
1answer
121 views

GAP code to get Multiplication Table.

I have a finite set $S=\{0,1,2,\ldots,n-1\}$ and binary operation $\star$ on $S$ defined by $$x\star y= \left\{ \begin{array}{l l l} \frac{3(x+y)}{2} ~~\text{modulo} ~~n& \qquad \mbox{if $x$ ...
0
votes
1answer
72 views

what is the coefficient of following expression

what is the co-efficient of $x^{50}$ in the expansion of $$\frac{1}{(1-x^{1.7})(1-x^{1.8})(1-x^{2.6})(1-x^{3.0})(1-x^{4.0})(1-x^{6.7})(1-x^{7.5})(1-x^{8.2})}$$ can you please explain me the logic
19
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4answers
457 views

How could I improve this approximation?

In a computer application, I need to solve trillions of times an equation which can be reduced to $$f(x)=\sin(x)-a x=0$$ Newton methods (quadratic and higher orders) are used for the solution. ...
0
votes
1answer
54 views

What is and what are the use for an “ AINV preconditioner ” or “ SAINV ”?

In an article that I'm reading there is a mention to this "thing" and I absolutely don't know anything about it, for me it could be anything. I noticed that this thing is somehow related to the math ...
0
votes
1answer
51 views

Converting x number of petaFLOPS into a base 2 number

I would like a few different formulas or methods for doing a couple of conversions and calculations: 1) How can I convert petaFLOPS into a base $2$ number representing how many operations per second ...
1
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0answers
64 views

Time needed to algebraically solve system of $15$ nonlinear equations with parameters

How long can I expect it will take to algebraically solve a system of $15$ nonlinear equations (without any numbers, only parameters), if I feed it into a computing software? I'm asking for symbolic ...
7
votes
2answers
192 views

How was the $3x+1$ problem checked up to $5 \times 2^{60}$?

The Wikipedia article for the Collatz conjecture states that: The conjecture has been checked by computer for all starting values up to $5 \times 2^{60} \approx 5.764 \times 10^{18}$. It gives ...
5
votes
3answers
572 views

The J programming language: is it useful for mathematics?

I just stumbled upon the J programming language, which has the description: J is particularly strong in the mathematical, statistical, and logical analysis of data. It is a powerful tool in ...
0
votes
1answer
68 views

Numerical evaluation of a complex integral

I have to evaluate numerically $f(z)$ via the Cauchy representation (so via a complex integral), in other words, I have to calculare $f(z)$ performing a complex integral: $\dfrac{1}{2\pi ...
0
votes
1answer
52 views

Change of variables in function $T(n)$.

I've been given this recurrence to solve: $T(n) = T(\sqrt n) + \theta(lglgn)$ And I'm told that the way to solve it is to let $m = lgn$, so that the recurrence can be rewritten as follows: $S(m) = ...
3
votes
1answer
180 views

Changing streams in PhD

I've a masters degree from a reputed Indian university in pure mathematics, with a specialization in Algebraic Number Theory. However, I'd like to apply for a PhD in computational math/theoretical ...
1
vote
1answer
77 views

minimize distance

consider a two dimensional system. two points are given whose co-ordinates are $(h1,h2)$ and $(k1,k2)$. I want to minimize the distance between these two points with the condition that person has to ...
1
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1answer
70 views

another counting problem

There are $k$ warriors that participate in the Wars, which have happened for the past $n$ years. Each year there has been a victor. Further, a particular warrior $W$ has won the Wars an even number of ...
1
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1answer
69 views

A problem on GCD

I want to calculate $f(n)$ where $f(n)$ is given by $$f(n) = \sum_{i=1}^n \dfrac{n}{gcd(n,i)}$$ and $2\leq n\leq 10^{12}$. Can someone tell me the fastest algorithm to calculate this. thanks
0
votes
1answer
70 views

Deleting subsets in the list of sets

If we have a list like $[[2,1],[5,2,1],[6,5,2,1],[9,10]]$ and I want to find the result in the form of $[[6,5,2,1], [9,10]]$ and want to delete all those list that are contained in another. How may I ...
1
vote
2answers
114 views

How can I solve this problem without having to do it by hand?

I'm dealing with the following problem in computational programming. I'm trying to find a way to build an algorithm that can quickly resolve the following problem statement without forcing me to do it ...
0
votes
1answer
78 views

How can I solve this problem without doing it by hand? [duplicate]

I'm dealing with the following problem in computational programming. I'm trying to find a way to build an algorithm that can quickly resolve the following problem statement without forcing me to do it ...
1
vote
2answers
106 views

Is there any way to solve this problem without having to do it by hand? [duplicate]

I'm dealing with the following problem in computational programming. I'm trying to find a way to build an algorithm that can quickly resolve the following problem statement. Is there any way to group ...
3
votes
0answers
26 views

Has there been work on computational group theory applications to computing colimits of crosses n-cubes of groups?

I'm trying to compute homotopy groups of a few spaces using crossed n-cubes of groups. I'm able to describe a few colimits in terms of quotients of induced crossed modules and nonabelian tensor ...
3
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0answers
494 views

Nerve Theorem: Is the finite union of closed convex sets triangulable?

My Question: Let $A_1, \ldots, A_k \subseteq \mathbb{R}^n$ be closed convex sets. Is the union $\bigcup_{i=1}^k A_i$ triangulable$^1$? If so, why? Background: I'm trying to better understand the ...
0
votes
1answer
232 views

Is there any algorithm (if possible, I need the codes) for Jordan normal form decomposition for large matrices in practice?

Although it is an ill-posed problem as B Kågström said in "An algorithm for numerical computation of the Jordan normal form of a complex matrix", I wonder what people do when they need to do Jordan ...
0
votes
1answer
36 views

Eliminate $p$ from these 2 equations.

$$ X \ = \ 2 \left[ \dfrac {h_1pv_1} {(1-p^2v_1^2)^{1/2}} + \dfrac {h_2pv_2} {(1-p^2v_2^2)^{1/2}} \right] \\ T_2 \ = \ 2 \left[ \dfrac {h_1/v_1} {(1-p^2v_1^2)^{1/2}} + \dfrac {h_2/v_2} ...
2
votes
1answer
119 views

Going through all Bit Strings with no 11 in it (no consecutive 1s)

My question is very simple: How can i (efficiently) go through all Bitstrings which don't contain two consecutive 1s? So for instance, all Bitstrings of length 3 with no consecutive 1s are: 000, 001, ...
1
vote
3answers
131 views

For how many consecutive numbers Collatz conjecture was checked?

I heard here that Collatz conjecture was checked at least for every first $5 \cdot 10^{18}$ natural numbers, but I cannot find any source or actual information about this. Can anyone help to find out ...
1
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2answers
231 views

Questions about the field scientific computing

I have heard about the field of Applied and Computational Mathematics, Scientific Computing and want to get some information. Is this a combination of computer science and mathematics? What subjects ...
1
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2answers
66 views

How to compute the sine of a complex number in floating-point arithmetic?

What is the most efficient way to numerically compute the sine of a complex number? Suppose I want to calculate the sine of a complex number a + bi on a computer. Suppose that a and b are both ...
1
vote
1answer
42 views

Approximating zeros on an interval

I'm writing a program for my AP Calculus class, and I'm trying to write an equation solver that approximates the zeros of functions. Right now it can take symbolic derivatives and evaluate functions. ...