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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0
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1answer
102 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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2answers
46 views

Maple not able to calculate Bernstein polynomial

Hope you can help me on this one. Please look at this simple Maple code: Obviously $B(1)=g(1)=4 \neq 0$. Why is Maple not able to compute this right? Am I doing something wrong? Kind regards PS: ...
0
votes
1answer
17 views

Finding the bounds for a truncation error

I have two series, $S$ and $T$ which approximate $\pi$ such that $$S_n = 4 \sum_{i=1}^n \cfrac{-1^{i+1}}{2i-1}$$ and $$T_n = \Big(12 \sum_{i=1}^n \cfrac{-1^{1+i}}{k^2} \Big) ^{\frac{1}{2}}$$ It is ...
0
votes
0answers
12 views

What does “empirical error” mean in this context?

I recently sat an exam for computational mathematics. The question asked for us to: "Write the empirical error in $\mathcal{O}(n^{-p})$ where $p$ is some integer" We were given a series $$S = 4(1 - ...
-1
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0answers
161 views

need mathematical expression for the below [closed]

Want to convert below algorithm into a mathematical model:- General points 1. Let there be a Connected Directed Graph. G = (V, E) V vertices or nodes E edges. This graph can be seen as a network ...
4
votes
0answers
71 views

On a unique(?) binomial property of $3003$

Given the triangular number, $$T_k = \frac{k(k+1)}{2}$$ and remembering that, $$\binom{n}{m}=\binom{n}{n-m}$$ Excluding $a_0=1$, we then have the six-fold (at least) equalities, $$\begin{aligned} ...
1
vote
2answers
36 views

(Fast way to) Get a combination given its position in (reverse-)lexicographic order

This question is the inverse of the Fast way to get a position of combination (without repetitions). Given all $\left(\!\!\binom{n}{k}\!\!\right)$ combinations without repetitions (in either ...
-2
votes
0answers
40 views

What is elnekiti's triangle? (edited) [closed]

Elementary ceĺular automata shows amazing complex systems such as pascal's triangle is similar to " wolfram rule 90 " , so i looked over youtube searching for extra content and i found this video Here ...
12
votes
1answer
333 views

Krylov-like method for solving systems of polynomials?

To iteratively solve large linear systems, many current state-of-the-art methods work by finding approximate solutions in successively larger (Krylov) subspaces. Are there similar iterative methods ...
2
votes
3answers
56 views

What is the value of this function?

Consider the three-variable function defined at the following way for all natural numbers $n,x,y$ : $f(0,x,y) = x+y $ $f(n,x,0) = x$ $f(n,x,y) = f(n-1, $ $ $ $f(n,x,y-1) , $ $ $ $f(n,x,y-1)+y ) $. ...
4
votes
1answer
502 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\\ ...
0
votes
0answers
30 views

How to calculate $det(X^T X)$ efficiently, update one column of X each time

$X_{1} = (A, b)$, where $X_{1}$ is a $n\times p$ matrix, $A$ is a $n\times (p-1)$ and $b$ is $n\times1$. First calculate $\det(X_{1}^T X_{1})$, then update $b$ with $c$, st. $X_{2} = (A, c)$ and ...
2
votes
3answers
105 views

Fast way to get a position of combination (without repetitions)

This question has an inverse: (Fast way to) Get a combination given its position in (reverse-)lexicographic order What would be the most efficient way to ...
1
vote
1answer
59 views

Expected Utility Method and a Repeated Game Solution

I am trying to replicate Bruce B. de Mesquita's (BDM) results on political game theory for prediction. Based on where actors stand on issues, their capabilities, salience, BDM's method attempts to ...
0
votes
0answers
48 views

Need to solve for an angle, do I need to use numerical methods??

I need to run a simulation on MATLAB where I need to solve two equations for two unknowns. The equations look something like this, $$x_{comp} = G\cdot \cos^2 b \cdot \cos(a+b).$$ $$z_{comp} = ...
1
vote
2answers
322 views

Space spanned by matrices

I have a set of 5 by 5 matrices, M1,M2,...,M19 ,M20. 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 approach the ...
0
votes
1answer
46 views

Constructive Induction to derive and prove the formula for a geometric sequence

$\sum_{i=1}^nr^i = a \cdot (b^n) + c$ Base Case: n=1, this holds Inductive Hypothesis: Assume for $n = k$, $k \ge 1$ that $\sum_{i=1}^k r^i = a * (b^k) + c$. Inductive Step: Prove for $n = k+1$ ...
0
votes
0answers
25 views

In bipartite directed graph, how to efficiently find nontrivial edge values such that for each node, sum of edge values is zero?

Suppose I have a set of sources U and a set of sinks V and a set of directed edges E going from elements of U to elements of V. I want to find a vector z of size |E| (each value of z is assigned to ...
7
votes
1answer
83 views

Are the unit partial quotients of $\pi, \log(2), \zeta(3) $ and other constants $all$ governed by $H=0.415\dots$?

Khinchin showed that given the simple continued fraction of a real number, $$r = a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1} {\ddots}}}$$ then it is almost always true that the partial quotients $a_i$ ...
6
votes
5answers
260 views

Computing as many digits as possible of $\sqrt{2}$ with a pen and a paper in 5 minutes

You have to compute as many digits as possible of $\sqrt{2}$ with a pen and a paper (an eraser if you're lucky...) in 5 minutes. What will you do? What is your justification for doing it? The ...
0
votes
1answer
29 views

Seeking the Recommendation on Complexity Theory books

S.E advisers, I am a rising college junior in US with a major in mathematics and an aspiring applied mathematician in the fields of theoretical computing. I just recently got a research project on ...
0
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0answers
15 views

How can I measure the dispersion of a discrete time system?

I have a discrete time system: x[n+1]=A*x[n]+B[n], with B[n] as an excitation signal. How can I measure the dispersion of this system? I guess this is related to some property of A, which depicts the ...
2
votes
2answers
85 views

Verify input is the sum of other numbers

I have a relationship: 4000k + 2500j + 400g = n, k >= 0, 0 <= j <= k, 0 <= g <= j I have to, given n, verify ...
2
votes
1answer
71 views

How do computers generate primes so quickly?

From what I understand, when a computer encrypts a file using an encryption standard like RSA, one of the steps is to generate two large primes, and multiply them together. I have created RSA keys on ...
0
votes
0answers
19 views

Analytical expressions for the orthogonalization of a specific set of vectors

I would like to know whether analytical or closed-form expressions could be obtained for the orthogonalization of a set of vectors in the following setting. Let $x_t$ be a vector indexed as a time ...
0
votes
0answers
23 views

Unknown variable in formula - binomial coefficient? [duplicate]

I'm currently researching a filter and don't quite understand one of the equations used there, since it contains a variable I don't know how to calculate: $$(1)\,\,a^{m, k}_s = \frac{c^{k, ...
0
votes
0answers
19 views

hyperoperation sequence with non-integer values of n

This probably has a very simple answer of some sort, but I'm not a mathematician. For the hyperoperation sequence: $$G(n,a,b)$$ ...there are obvious defined values for positive integer values of $n$ ...
0
votes
0answers
29 views

deconvolution of exp($x^2$)

I would like to know whether we can get the function of type exp($x^2$) by convoluting any functions. That is which function convolution gives exp($x^2$). Thanks in advance
1
vote
1answer
25 views

computing a fixed interest rate

I've been struggling for hours now with understanding a Topcoder problem, Autoloan , but i cannot grasp the way of computing it from a mathematical point of view. The excerpt goes as follows: A ...
0
votes
1answer
17 views

Calculate height for rows of rectangles within a given width

I have an array of rectangles, all of the same height, but with different widths. Imagine they are on a single line with a uniform gap between them as shown below... |XXX| |X| |XXXXX| |XXX| ...
1
vote
2answers
62 views

Can car traffic be managed by mathematical formula? [closed]

How car traffic is managed? Is it managed by mathematical algorithm? Or by human(operator)? If it's by operator, can it be managed mathematically? Or is it by physics? By what theories/formula? ...
0
votes
2answers
127 views

Computation complexity with simple algebra expression reduction

I'm watching this computer science video on computational time complexity of a function where they introduce some maths and it doesn't make sense to me. I'm not even sure what the name for this maths ...
4
votes
3answers
3k views

Derivative of Associated Legendre polynomials at $x = \pm 1$

I'm creating meshes for spherical harmonics, and I need a normal at a given point. Whenever I'm at the poles, $\cos{\theta} = \pm 1$, and I do not know how to find the derivative there. All the ...
0
votes
1answer
729 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 ...
2
votes
1answer
24 views

Approximate recursively defined error in fixed point iteration

Problem: With an initial guess of $x_0$, the fixed point iteration is given by $$x_{k+1} = e^{-x_k}, \mbox{ for } k=0,1,2,...$$ If $x^*$ is the exact solution, then the approximation error is $$e_n = ...
1
vote
2answers
76 views

Is there any sort algorithm quicker than Quicksort given a random array of integers?

How can we proof (mathematically) that any complexity of sorting algorithm that sorts a random array of integers is no better than $O(n\log n)$?
8
votes
0answers
74 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 ...
1
vote
0answers
17 views

Is there a way to delineate the parameter of highest influence in a system of differential equations?

So I have a system of nonlinear ordinary differential equations dependent on parameters. These equations can traditionally be solved numerically with robust methods and the solution is well defined. ...
0
votes
0answers
6 views

State-of-Art library or a method for paralell matrix inversion?

Do you have reference to a computer library or a paper regarding a state-of-art method to obtain inverse of a matrix in parallel? Thanks, Mojo.
0
votes
0answers
16 views

FFT differential equations

Given a generical differential equation what is the procedure to solve it using fft command. Can anyone explain me how to do it? For example: $$\frac{d^2y}{dt^2}+10\cdot \frac{d\:y}{dt}=-5\cdot ...
0
votes
0answers
9 views

How to solve the Helmholtz equation in a triangular region?

Suppose we take the Dirichlet boundary condition, namely the function must vanish on the boundary of the triangle. How about a general n-polygon?
5
votes
1answer
469 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: ...
-1
votes
1answer
28 views

What is transpose multiplier and forward multiplier?

For linear system X = A*s, we define the forward and transpose multiplies Af and At as follows: Af = @(s) A*s; At = @(s) A'*s; I want to know what is forward ...
1
vote
0answers
37 views

Simplifying the Generalized Eigenvalue Problem

Let $\Sigma_1$, $\Sigma_2$ be symmetric positive-definite real $n\times n$ matrices. We want to solve the generalized eigenvalue problem $$ \Sigma_1V=\Lambda\Sigma_2V, $$ where $\Lambda$ is the ...
1
vote
1answer
48 views

inverse of $AQ^{-1}A'$

Suppose that $A$ is a $m\times n$ full row rank sparse matrix, and $Q$ is an $n\times n$ symmetric positive definite sparse matrix with $m<n$. Besides, $m$ is about $10^5$, and $n$ is about $10^6$. ...
0
votes
0answers
18 views

Using Orthogonal Collocation to solve Coupled Ordinary Differential Equations

I am trying to solve six first order coupled ODE's, two of these are associated with a heat balance of a catalyst pellet, and four are mass balances. I have been trying to solve these equations using ...
10
votes
1answer
914 views

How do I prove the partial denominators formula of the Bauer-Muir transformation of a generalized continued fraction?

Notation: $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ is the Gauss Notation for generalized continued fractions. Description of the Bauer-Muir transformation ...
3
votes
1answer
53 views

Prime numbers distribution theorem

I'm trying to understand Gauss' theorem: $$ \frac{\pi(x) }{x/\ln x} \to 1 $$ for large $x$. I've taken the list of first 1000 prime numbers from Utah university site, saved them to file ...
4
votes
2answers
287 views

What is the fastest computational graph theory package?

What is the fastest computational graph theory package with respect to executing algorithms and computing graph theoretic data? I am aware of this related question, which requests graph theory ...
1
vote
0answers
22 views

periodic boundary conditions and the FEM

I am trying to set up the mass matrix for a 1D system which I want to solve using finite elements. So the mass matrix is defined as $$ M = \int{NN^T}dL, $$ where $N$ is the finite element linear ...