Computational mathematics involves mathematical research in areas of science where computing plays a central and essential role, emphasizing algorithms, numerical methods, and symbolic methods.

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10
votes
1answer
500 views

Why are there mathematicians that do not use computers?

I was watching a video on Andrew Wiles and his proof of Fermat's Last Theorem and I quite liked the video, especially the complexity of the proof only to prove a simple concept which can be understood ...
2
votes
1answer
137 views

Is there a computer programm or CAS (maybe GAP?) that can calculate with projective (indecomposable) A-modules (A is a finite dimensional k-algebra)?

I have the following question(s): I have an "Algebra-With-One" $R$ as a subalgebra of a full matrix algebra in GAP. Furthermore, I have 5 primitive orthogonal idempotents $e_1,...,e_5$, which sum up ...
6
votes
1answer
196 views

how to calculate this complementary Bessel function?

I am trying to calculate this complementary Bessel function $$\Psi(a,b,\gamma)=\int_0^\infty\Phi({a\over \sqrt{u}}+b\sqrt{u}){u^{\gamma-1}e^{-u}\over \Gamma(\gamma)}du$$ where $\Phi$ is the standard ...
2
votes
0answers
83 views

Solving a particular system of Diophantine equations in $n$ variables (Frobenius equations)

I have a particular system of linear Diophantine equations in $n$ variables for which I need to find all nonnegative integer solutions. Specifically, they are Frobenius equations, meaning the ...
1
vote
1answer
65 views

Is there a good strategy for computing eigenspace corresponding to $1$ of a matrix with entries of trigonom

For example, say $A= \left ( \begin{matrix} \cos x & -\sin x & 0 \\ \cos y \sin x & \cos x \cos y & -\sin y \\ \sin x \sin y & \sin y \cos x & \cos y \end{matrix} \right)$. ...
-2
votes
2answers
78 views

Is there any good strategy for computing null space of a matrix with entries $\cos x$ and $\sin x$?

For example, say $A= \left ( \begin{matrix} \cos x & -\sin x & 0 \\ \cos y \sin x & \cos x \cos y & -\sin y \\ \sin x \sin y & \sin y \cos x & \cos y \end{matrix} \right)$. ...
2
votes
1answer
284 views

Numerical Integration over 2D Regions with Discontinuous Functions

I've run into a tricky problem, and I haven't really thought up a good solution. I have to compute many integrals of the form $$\iint\limits_{B((x_k,y_k),\epsilon))} f(x,y) dy dx$$ where ...
0
votes
0answers
35 views

Polytime programming

Given a linear system of the form: $$x_r = a$$ $$x_j = b$$ $$c_1x_1 + c_2x_2 ... c_nx_n = n$$ $$x_1 + x_2 + x_3 ... x_n = k $$ $$0 \leq a,b,x_1, x_2, x_3 ... x_n \leq 1$$ $$k \geq 0$$ How quickly ...
4
votes
4answers
318 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
2answers
95 views

Uniform grid on a disc

Do there exist any known methods of drawing a uniform grid on a disk ? I am looking for a map that converts a grid on a square to a grid on a disk.
2
votes
1answer
92 views

What does noncomputable really mean?

I believe I understand the definition of a noncomputable problem from an introductory computer science class, but I don't understand what it really means. One of my hypothesis was that a ...
6
votes
2answers
141 views

What is the average weight of a minimal spanning tree of $n$ randomly selected points in the unit cube?

Suppose we pick $n$ random points in the unit cube in $\mathbb{R}_3$, $p_1=\left(x_1,y_1,z_1\right),$ $p_2=\left(x_2,y_2,z_2\right),$ etc. (So, $x_i,y_i,z_i$ are $3n$ uniformly distributed random ...
2
votes
0answers
33 views

Computing a particular finite set of quaternion matrices.

Let $B = \left(\frac{-1,-11}{\mathbb{Q}}\right)$ be a choice of quaternion algebra ramifying at $11$ and consider the maximal order ...
0
votes
0answers
107 views

Effects of numerical integration stepsize on impulse inputs (e.g., delta function)

Some models of neurons treat synaptic input (from other neurons) as a single impulse, such as the Dirac delta. But doesn't this make the magnitude of that impulse a function of numerical integration ...
0
votes
1answer
33 views

Find all points where the gradient of a high-dimensional function are equal to zero in some domain, numerically

I was wondering if anybody was aware of a numerical method to find all points where the gradient of a high-dimensional function are equal to zero in some domain. Thanks
5
votes
1answer
330 views

cylinder-ray intersections equation

Can You Pleas Help with this one I found an article http://www.mrl.nyu.edu/~dzorin/rendering/lectures/lecture3/lecture3.pdf for Infinite cylinder-ray intersections And I don't know how they develop ...
1
vote
2answers
170 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
0answers
60 views

Constrained computational optimization of a functional of a vector valued function.

I am trying to increase the efficiency of a program I have written that must run in real time. I am asking this question in a broad sense, since I'm not sure what tools are available to me. I am ...
1
vote
0answers
13 views

Tools for optimizing asymptotic bounds.

Is there any tool for this task ? Given the asymptotic bound in term of $n$ and other paramaters $t_1,\dots,t_r$, then return the value for each $t_i$ which optimizes the expression in term of $n$, ...
4
votes
3answers
2k 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 ...
4
votes
1answer
53 views

About parallel time computation

I am studying a paper where it is mentioned that Newton iteration may be used to compute the inverse of $n \times n$, well- conditioned matrix in parallel time $o(\log^2n)$ and that this computation ...
5
votes
3answers
543 views

What free software can I use to solve a system of linear equations containing an unknown?

Question: What free software can I use to solve a system of linear equations $M\mathbf{x}=\mathbf{y}$ where the entries of $\mathbf{y}$ vary with an unknown quantity $n$? Presumably I could do ...
0
votes
0answers
112 views

Simpson's rule characteristics

I just wanted to ask a quick question in regards to simpson's rule for integration. I have been reading up on the trapezoidal rule, and have found the notations and have an understanding such that: ...
9
votes
2answers
206 views

Efficient computation of $\sum_{k=1}^n \lfloor \frac{n}{k}\rfloor$

I realize there is probably not a closed form, but is there an efficient way to calculate the following expression? $$\sum_{k=1}^n \left\lfloor \frac{n}{k}\right\rfloor$$ I've noticed $$\sum_{k=1}^n ...
6
votes
1answer
403 views

How to find an expression whose value is 190

Given a set of numbers (in this case): 3, 7, 7, 100, 50 Either: prove it is impossible to form the number k = 190 using ( ) + - * / operators between sub set of the these numbers ex: 1000 = ((3 + ...
0
votes
1answer
94 views

how can I find the following problem using laplace transform?

For example here is the problem: $(t^2 \cos{\omega t})u(t)$ I have to find it using laplace transform; here is what I think it is, I have $t^2$$(\cos{\omega t})u(t)$ which I think I can solve them ...
1
vote
6answers
727 views

What is the value of $2^{3000}$ [closed]

What is the value of $2^{3000}$? How to calculate it using a programming language like C#?
11
votes
1answer
146 views

Evaluation of a slow continued fraction

Puzzle question... I know how to solve it, and will post my solution if needed; but those who wish may participate in the spirit of coming up with elegant solutions rather than trying to teach me how ...
1
vote
2answers
103 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 ...
4
votes
1answer
193 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
1answer
83 views

How do I determine if two of my software's representation of algebraic numbers are equal?

I have software which stores information about algebraic numbers with absolute precision. If you build it up by creating instances of a Python representation of an integer, float, Decimal, or string, ...
0
votes
2answers
58 views

Does rationalizing the denominator lead to more or less round-off error?

I evaluated $\frac{1}{\sqrt{2}}$ and $\frac{\sqrt{2}}{2}$ in Matlab, and got a slight difference: $0.707106781186547$ and $0.707106781186548$, respectively. Which is more accurate, the one with the ...
1
vote
1answer
56 views

Matrix completion: supplementary questions

Continuation of the question here, what is going to happen if we change the some of the conditions. I write it as a quote from here and change the appropriate places which are underlined: I need ...
2
votes
3answers
130 views

Drawing graphs (vertices and edges) with or without technology

Given a collection of vertices $V$ and a collection of edges $E \subseteq V\times V$, is there an algorithm or program that will allow you to draw a nice graph? The placing of the vertices is very ...
-1
votes
1answer
196 views

Best graphing program for Mac or PC?

I just bought the highest end iMac, with a student discount, of course, and was wondering what is the best graphing program out there. A program that can graph any equation that I throw at it AND one ...
3
votes
2answers
326 views

How to get the minimum angle between two crossing lines?

I'm not a student, I'm just a programmer trying to solve a problem ... I just need the practical way to calculate the smallest angle between two lines that intersect. The value, of course, must always ...
1
vote
1answer
124 views

Matrix completion

I need to find an algorithm (if exists) of the following matrix completion problem. I need to construct $n^2$ positive semi-definite matrices, say $\{P_i\}_{i=1}^n$. Entries of these matrices are ...
1
vote
1answer
386 views

Minimizing mean squared error

I want to find a $d$ that minimizes the value of the expression below. I think the first step is to find the derivative w.r.t. $d$ (is that correct? If not, what is the first step?). If so, I'm having ...
6
votes
3answers
200 views

Mathematical Limitations of Computer Experiments

One problem that has always bothered me is the limitations of computers in studying math. With a chaotic dynamical system, for example, we know mathematically that they possess trajectories that never ...
1
vote
1answer
64 views

Discrete numerical derivative with respect to d/d(n*x)

How can I generate a stencil for a d/d(n*x) operator? I am writing a program that needs a method to calculate line derivatives in an image. If we want to calculate the simplest forward derivative ...
-1
votes
1answer
95 views

Qubit state finding [closed]

Suppose we have two qubits in the state $x|00\rangle+y|11\rangle $. What is the resulting state of the second qubit in that case? Use and to denote and respectively.
1
vote
2answers
102 views

Can product of all pairwise sums be computed faster than the naive method?

Let $S$ be a set of integers. $|S|=n$. Can we find the product $\prod_{a,b\in S} a+b$ faster than naively add all pairs then multiply them one by one? By faster, I mean use less than $O(n^2)$ ...
1
vote
1answer
68 views

How to find a close form expression in terms generating functions for the triple summation

Given $$\sum\limits_{i=0}^\infty a_i z^i=A(z)$$ and $$\sum\limits_{i=1}^\infty b_i z^i=B(z)$$ and $$\sum\limits_{i=0}^\infty c_i z^i=C(z)$$ Find $\sum\limits_{i=1}^\infty\sum\limits_{j=0}^\infty a_j ...
4
votes
8answers
5k views

Fastest Square Root Algorithm

What is the fastest algorithm for finding the square root of a number? I created one that can find the square root of "987654321" to 16 decimal places in just 20 iterations (I'm not ready to release ...
2
votes
1answer
55 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 ...
3
votes
3answers
560 views

Understanding recursive definitions of a language.

I am having difficulty understanding the recursive definition of a language. The problem asked how to write this non recursively. But I want to understand just how a recursive definition of a ...
0
votes
1answer
42 views

Probability : Dividing a list into 2 classes

I have a list of integer numbers ($n$). I am dividing it into two parts $n_1$ (smaller) and $n_2$ (bigger) such that the length of $n_1 \ge a*n$; $a$ is positive and $a \lt 0.5$. What is the ...
5
votes
1answer
1k views

A search for integers which can be written as a sum of two squares in multiple ways

As part of a number theory hobby project, I'm looking for a computational way to enumerate all integers $n$ which can be written as a sum of two integer squares in three or more ways. The range of ...
7
votes
1answer
138 views

Fractional part of exp(x)

I have a real number $x$ (for concreteness, say $10^4<x<10^6$) and would like to find $e^x-\lfloor e^x\rfloor$ to reasonable precision (10-20 decimal places). What is the most efficient method? ...
2
votes
0answers
220 views

How to convert a hologram into an image?

Suppose one knows in full detail the phase and intensity of monochromatic light in a plane. This is basically what a hologram records, at least for some section of a plane. By using this as the ...