Questions tagged [computer-algebra]
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35
questions
2
votes
1answer
54 views
Basic Set Theory problem
So i'm relatively new to computer science and have been learning set theory and am stumped on a question in it.
The question specifies that we're only looking at subsets of universe U = {0,...,n-1}. ...
2
votes
1answer
26 views
When an algorithm says Summation this, and Integral that, what does it mean in coding terms?
I'm a student at a college with only two units of mathematics and I don't know if I'm asking in the right place so please bear with me.
I'm currently reading GPU Gems by nvidia and I have a question....
2
votes
2answers
27 views
An approximate quantity of multiplications in $\mathbb{F}_p$ amounting the same bit complexity as one inversion in $\mathbb{F}_p$
Consider a prime finite field $\mathbb{F}_p$ of quite large characteristics $p$, for example $\log_2(p) \approx 256$ bits. I would like to know an approximate quantity of multiplications in $\mathbb{F}...
2
votes
0answers
44 views
An efficient algorithm to find a linear transformation between two ternary quadratic forms
Let $\mathbb{F}_p$ be a prime finite field for $p > 2$. Consider two ternary quadratic forms
$$Q_1\!: x^2 - a_1(t)y^2 - b_1(t)z^2,\\
Q_2\!: x^2 - a_2(t)y^2 - b_2(t)z^2$$
over the field $\mathbb{F}...
0
votes
1answer
92 views
Simplifying SOP: implementing OR with NAND
I am learning how to implement basic logic gates using NAND. I have learnt that you can use De Morgan's theorem as such:
$a+b = \bar{\bar a} + \bar{\bar b} = \overline{(\bar a *\bar b)}$
In other ...
0
votes
0answers
19 views
What is the computational cost of automatic differentiation compared to symbolic and numerical differentiation?
Automatic differentiation is a set of techniques to numerically evaluate the derivative of a function.
Quoting from Wikipedia (emphasis mine):
These classical methods run into problems: symbolic ...
2
votes
0answers
31 views
Nearest codeword in a translation-invariant code over $\mathbb{Z}^d$
Let $c_1,...,c_n,c':\mathbb{Z^d}\rightarrow \{0,1\}$ all have finite support.
Let $C$ be the linear, shift-invariant code generated by $c_1,..,c_n$.
It is possible to calculate the nearest codeword $...
3
votes
1answer
59 views
How to compute quotient subgroup efficiently?
Let $G$ be a finite group given by the table representation and a normal subgroup $H$ of $G$ is given. I want to compute $G/H$ that is quotient group.
Model of computation is RAM
For all pair of $a$...
1
vote
0answers
48 views
Provably correct algorithm/CAS for checking term equalities
Within my research of term rewriting systems (TRS) I stumbled upon a paper (Siekmann, J., and P. Szabó. “The Undecidability of the DA-Unification Problem.” The Journal of Symbolic Logic, vol. 54, no. ...
6
votes
1answer
1k views
cannot construct the infinite type [closed]
I am trying to learn haskell and could not configure it out, why following code snippet can not get compiled:
...
1
vote
0answers
18 views
Is there evidence to suggest Macsyma was directed at the Diophantine equations in the Entscheidungsproblem?
I'm reading the book The Annotated Turing by Charles Petzold. In it he mentions the Diophantine equations - which was a joy to read.
This then lead to Hilbert's 10th problem - finding an algorithm ...
4
votes
1answer
300 views
Using Brzozowski's derivatives method to construct a minimal DFA
so I am currently learning about dfa and nfa and i came across the following question which requires me to use Brzozowski's derivatives method to construct a minimal DFA recognizing the language ...
5
votes
1answer
568 views
How to calculate sum of binomial coefficients efficiently?
I want to compute the sum
$$\binom{n}{0}+\binom{n}{2}+\binom{n}{4}+\binom{n}{6}+\dots+\binom{n}{k} \bmod 10^9+7$$
where $n$ and $k$ can be up to $10^{14}$ and $k\le n$.
I found several links on ...
5
votes
2answers
299 views
Constant problem for discrete functions
Is there an algorithm to decide whether a closed-form expression over integer variables using, say, $\{+,-,\times,\div,\text{^},\lfloor\text{lg}\rfloor,!,()\}$, or some other useful set of operators, ...
0
votes
1answer
51 views
The symbolic differentiation of univariate expressions
I was reading "Doug McIlroy: McCarthy Presents Lisp" and the phrase "symbolic differentiation of univariate expressions" triggered a faint memory of a demonstration of differentiation done in haskell ...
9
votes
2answers
131 views
Decidability of checking an antiderivative?
Let's suppose I have two functions $F$ and $G$ and I'm interested in determining whether
$$F(x) = \int G(x)dx.$$
Let's suppose that my functions are composed of elementary functions (polynomials, ...
1
vote
1answer
700 views
Difficult Question to Understand (Computer Artitechture) [closed]
You are designing an elevator controller for a building with 25 floors. The controller has two inputs: UP and DOWN. It produces an output indicating the floor that the elevator is on. There is no ...
1
vote
0answers
26 views
How to simplify a rational function with floating point real coefficients (GCF)
How does one compute the simplified form of a rational function where the coefficients of the polynomial are floating point numbers (real, though I expect using complex numbers would be the same).
...
3
votes
1answer
197 views
Properties of Reverse Polish Notation expressions that are algebraically invariant
The RPN expressions
a b + c *
and
f d e + *
are algebraically equivalent, though the names of the variables are different ...
4
votes
1answer
298 views
Abstract algebra and programming languages
Quite often, I stumble upon abstract algebra concepts like initial algebra, free algebra, and similar while reading papers on programming languages. For instance, in papers on algebraic data types, ...
1
vote
1answer
112 views
Introduction to number theory [closed]
What is the best book for a beginner in Introduction to number theory?
I am new to this field and getting deeper into cryptography, so I think reading some intro books about number theory can be of ...
2
votes
1answer
69 views
Multiplication of two or more algebraic quantities [closed]
Recently, I was facing the problem how to multiply to two or more algebraic quantities in c++. For example, if the two algebraic quantities are
$$x^2-2x+3, \text{and } x-5$$
then the result of ...
3
votes
4answers
4k views
Why do Computers use Hex Number System at assembly language?
Why do computer use Hex Number System at assembly language? Why don't they use any other number system like binary, octal, decimal? What thing forced computer designer to use hex system at assembly? ...
1
vote
1answer
49 views
Is everything in CS either a numeric method or a symbolic method?
Or maybe also a combination of the two, but not something else, whether numeric, nor symbolic. Do they cover the whole field?
4
votes
1answer
54 views
How does mathematical software evaluate symbolic sums?
Wolfram alpha is able to compute this sum:
$$ \sum_{j=1}^n \binom{j}{2} = \frac{1}{6}(n-1)n(n+1). $$
How can Wolfram alpha do it? What kind of algorithm does it use?
1
vote
1answer
73 views
An algorithm for making 2 carts meet [closed]
Say I have 2 carts on an infinite railroad, each cart is initially under a lamp.
There are only 2 lamps, and they are at a fixed location, hence they don't change their location. The distance between ...
4
votes
1answer
509 views
How does automatic differentiation work?
What is the intuitive idea behind automatic differentiation?
If I have a program which computes $f(x, y)=x^2+yx$, which steps lead to the program which computes the derivative $df/dx$ of f?
...
9
votes
0answers
615 views
Does Automatic Differentiation handle conditional branches, if yes how?
I'm trying to understand how Automatic Differentiation (AD) works.
For simple algebraic operation, I get the chain rule thing.
But, when the code contains conditional statement like
...
2
votes
2answers
92 views
Solve modulus with constraints for multiple equations
I'm trying write a program to solve equations from the following form:
$$ \begin{align}
a \bmod x &= t_1 \\
b \bmod x &= t_2 \\
\end{align} $$
where $a$, $b$, $t_1$ and $t_2$ are known ...
4
votes
4answers
575 views
Using a computer algebra system to optimize mathematical expressions
This is something I've been wondering for years. Software like Mathematica is great at manipulating expressions into simplified, factorized, and other forms. I'm wondering if there's a way, ...
4
votes
2answers
241 views
Computer Algebra: Algorithms for solving equations symbolically
As a hobby, I have written a basic computer algebra system. My CAS handles expressions as trees. I have advanced it to the point where it can simplify expressions symbolically (i.e., sin(pi/2) returns ...
11
votes
2answers
233 views
Decidability of a problem concerning polynomials
I have come across the following interesting problem: let $p,q$ be polynomials over the field of real numbers, and let us suppose that their coefficients are all integer (that is, there is a finite ...
5
votes
3answers
1k views
Constructing a data structure for a computer algebra system
In thinking about how to approach this problem I think several things will be required, some tivial:
An expression tree where non-leaf node is an operation (not sure if that part is redundant), but ...
8
votes
0answers
228 views
Complexity of computer algebra for systems of trigonometric equations
As discussed in this question, I drafted a spec algorithm that hinges on finding a specific root of a system of trigonometric equations satisfying the following recurrence:
$\qquad f_{p_0} = 0\\
\...
14
votes
2answers
6k views
Complexity of computing matrix powers
I am interested in calculating the $n$'th power of a $n\times n$ matrix $A$. Suppose we have an algorithm for matrix multiplication which runs in $\mathcal{O}(M(n))$ time. Then, one can easily ...
