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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5
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3answers
475 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
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0answers
108 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
190 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
353 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
91 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 ...
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6answers
722 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
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1answer
137 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
1answer
81 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
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1answer
179 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
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1answer
82 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
55 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
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1answer
55 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
128 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
186 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
270 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
118 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
341 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
194 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 ...
0
votes
0answers
158 views

Solving an overdetermined system of inequalities using null-space arguments

The solutions to a linear system of equations: $$A\cdot x = b$$ (where $x$ is a $(n\times 1)$ column vector, $b$ is a $(m\times 1)$ column vector and $A$ is $(m\times n)$ matrix) can all be ...
1
vote
1answer
62 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
91 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
98 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
66 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
4k 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
54 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
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3answers
524 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
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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 ...
6
votes
1answer
128 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
212 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 ...
2
votes
2answers
95 views

Number of ways to move 1 or more elements from one list to the previous list until one list remains

Given N elements, divided into at most N groups, which are then labeled 1 thru N, move all of the elements into the group labeled 1. By moving 1 to all of the elements, in group i to i-1. This means ...
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2answers
1k views

Questions that can be solved using Excel.

I recently started to realize that Excel is a powerful tool that can solve many problems. What interesting mathematics problems are there can be solved using excel? I am looking for a set of ...
1
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1answer
128 views

Where does the input x in Turing Machine subroutines come from in solving reductions to undecidable problems?

I'm taking an introduction to computation theory class and we went over the chapter on undecidable problems and proving undecidability through reductions. I can't seem to grasp some of the simplest ...
2
votes
1answer
120 views

Oracle turing machine

I am learning computational complexity and this is a question of my assignment that I have issues trying to solve/understand. An oracle Turing Machine M with oracle A is a Turing Machine with an ...
2
votes
1answer
206 views

Complement of NP-Complete

If a language L is NP-complete, with respect to polynomial time reducibility, does L ≤ co-L in polynomial time?
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0answers
23 views

Model to reward adaptative communication

I'm currently working in different simulations focused on the study of artificial evolution of communications in robots, and I have stumbled against the problem of its mathematic formulation. The ...
0
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2answers
108 views

floating point binary arithmetic

Prove that the decimal number $\displaystyle \frac{1}{5}$ cannot be represented by a finite expansion in the binary system.
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2answers
174 views

Finite representation in the binary $\implies$ finite representation in the decimal system

Any number that has a finite representation in the binary system have a finite representation in the decimal system. Why?
1
vote
1answer
120 views

Simplified way of finding a complex number raised to another complex number

This question here has the answer but I'm still in school and I don't understand any of it. I'm writing a computer program that takes a complex number a + ib and raises it to c + id and I need to ...
1
vote
1answer
322 views

Exponential Probability Monte Carlo simulation

I need to write a Matlab program to estimate the quantity $\theta = \mathrm{Pr}(X < 1)$, where $X$ is an exponential random variable with mean $1$. I am doing this for multiple monte carlo ...
1
vote
0answers
169 views

Power sums, fast algorithm

I know some schemes to compute power sums (I mean $1^k + 2^k + ... + n^k$) (here I assume that every integer multiplication can be done in $O(1)$ time for simplicity): one using just fast algorithm ...
2
votes
1answer
60 views

Is regularity is preserved under reversal?

When talking about languages and regular languages. Can I say that reversal preserved regularity since if the language L is regular, we can generate it by right linear grammar. Therefore, the ...
1
vote
2answers
92 views

Solve right linear set equations

Trying to solve this set of equations. I'm feeling like I'm making it so complicated. Of course + is union. Am I on the right track? A = 0B + 1D B = 0C + 1A C = 0A + 1B + λ D = OD + 1C + λ A ...
1
vote
1answer
47 views

Show Two f.s.a. machines accept different input

I'm trying to solve it for two hours already. I know it somehow related to the pumping lemma Let $M_1 = \langle Q_1,S,f_1,s_1,F_1\rangle$ and $M_2 = \langle Q_2,S,f_2,s_2,F_2\rangle$ be two machines, ...
1
vote
1answer
811 views

Show two finite state machines are equivalent

Suppose $M_1 = \langle Q_1,S,R,f_1,g_1\rangle$ and $M_2 = \langle Q_2,S,R,f_2,g_2\rangle$ are two strongly connected machines. I need to show that $M_1 \equiv M_2$ iff there exist a state $p \in Q_1$ ...
2
votes
1answer
434 views

Computational Complexity of Modular Exponentiation

The following was posted from a lecture: "($a^n \bmod N$) has a runtime complexity of $\mathcal{O}(n*|a|*|N|)$ using the brute force method. $Z_1 = a \bmod N$ $Z_2 = (aZ_1) \bmod N$ $Z_3 = (aZ_2) ...
2
votes
1answer
541 views

radial basis function and neural networks

actually i need a simple explanation consider it for dummies about what is Radial basis function are?and what is the relation between radial basis function and neural networks ?and is there's any ...
2
votes
2answers
122 views

Approximating $\pi$ in Binary

I am interested in creating a Java program that generates digits of $\pi$ (in Binary though). To be clear, the number I'm looking for begins: $11.00100100 \dots$ I am unsure of the most efficient way ...
2
votes
0answers
322 views

Logistic regression algorithm in Casio and Texas Instruments calculators

When using logistic regression on a Casio or Texas Instruments calculator, the output is of the form $$f(x) = \frac{c}{1+ae^{-bx}} $$ The problem I have (when teaching in a class where both types of ...
0
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0answers
176 views

A (contour?) integration (even if by using Mathematica!)

I need to be able to calculate integrals of this type where the sum over $R$ is the sum over representations of a Lie group $G$ on whom $dU$ is the Haar measure and $\chi _ R ()$ is the character of ...