Questions tagged [symbolic-computation]

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Admissibility condition of wavelet functions

After a badly formulated question, I decided to make a new post searching for help. The basic problem is the follows: I have a wavelet function $\psi(t)$ (real or complex) and would like to compute (a)...
Luciano Magrini's user avatar
3 votes
0 answers
80 views

Solving an underdetermined system of linear equations among the rationals, in the neighbourhood of an approximate solution

I have a system of linear equations $Ax=b$. Extremely underdetermined, for concreteness $x \in \mathbb{R}^{17,000}, b \in \mathbb{Z}^{156}$. $A$ is sparse, integer, full rank. I have a very precise ...
Dániel Varga's user avatar
3 votes
1 answer
128 views

Is the smallest root of this quartic always the closest point on the Hyperbola? [closed]

Let $a>b>0$. Suppose we want to minimize $$ f(x)=(x-a)^2+(1/x-b)^2, $$ over $x>0$. Equating $f'(x)=0$ leads to the quartic equation $$ g(x)=x^4-ax^3+bx-1=0. \tag{1} $$ Question: Is the ...
Asaf Shachar's user avatar
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1 vote
1 answer
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Analytic expression for the min value of $g(t):= \sqrt{(t-1)^2 + a^2}+ b|t|$ subject to $|t-1| \le c$

Disclaimer. Not sure this is MO-level but would really appreciate some help with this. Thanks in advance. Moved from SE. Let $a,b,c \ge 0$, and define a function $g:\mathbb R \to \mathbb R$ by $g(t) :=...
dohmatob's user avatar
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2 votes
1 answer
116 views

Analytic value of $\alpha := \sup_{(x,y) \in C} ax+by$, where $C := \{(x,y) \in \mathbb R^2 \mid x^2 + y^2 \le 1,\,x^2 + c y^2 \le R^2\}$

Let $a,b \in \mathbb R$, $R \ge 0$, and $c > 0$. Define $C := \{(x,y) \in \mathbb R^2 \mid x^2 + y^2 \le 1,\,x^2 + c y^2 \le R^2\}$, and set $$ \alpha := \sup_{(x,y) \in C} ax + b y. $$ Question. ...
dohmatob's user avatar
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11 votes
1 answer
641 views

Non-trivial solutions for $-(c^2-d^2)(a^2-b^2)=2(ad-bc)(bd+ac)$?

Consider the quartic system in four variables $a,b,c,d\in\mathbb R$: $$-(c^2-d^2)(a^2-b^2)=2(ad-bc)(bd+ac).$$ Does this system admit rational solution with $$abcd(c^2-d^2)(a^2-b^2)(a^2-c^2)(b^2-d^2)\...
Turbo's user avatar
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1 vote
0 answers
41 views

Computer verification for hyperbolic trigonometry

I am currently writing a paper that requires some lengthy computations using basic hyperbolic trigonometry. So, several hyperbolic figures appear, and we apply the law of sines and so on in order to ...
user44172's user avatar
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3 votes
1 answer
226 views

Symbolic powers of a prime ideal of height one

Can someone please give me an example of a Noetherian normal local domain of dimension two such that there exists a prime ideal $P$ of height one having the property $P^{(n)}$ is not a principal ideal ...
Jatin Majithia's user avatar
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0 answers
184 views

Advice on how to deal with an elementary "long-to-prove" statement

I am not entirely sure if this question totally fits here. If it doesn't, I apologise in advance. In a paper I've been working on, we have a very elegant result which, when forgetting about the ...
Luis Ferroni's user avatar
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2 votes
0 answers
146 views

How to decide if an algebraic number is a root of a given polynomial?

Let $p$ be a polynomial with rational coefficients and $\alpha = \sqrt[n]{q}e^{i2k\pi/m}$ for some natural numbers $n,m,k$ and a rational number $q > 0$. Is there an effective algorithm for ...
Jára Cimrman's user avatar
4 votes
0 answers
203 views

Effective bounds for a Bertini-type result

Suppose $X$ is a projective subvariety of $\mathbb{P}^n$ of codimension $r$ over $\mathbb{C}$, defined set-theoretically by $r$ homogeneous polynomials $P_1,\dots,P_r$ of degree at most $d$. By ...
Zeyu's user avatar
  • 537
7 votes
2 answers
339 views

Books/Lecture notes which contrast Risch algorithm with basic standard procedure of finding an antiderivative

I vaguely remember a book/some lecture notes which introduce integration algorithms such as Risch algorithm by first giving a list of quasi-algorithmic way of evaluating symbolic integrals. (For ...
Ma Joad's user avatar
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2 votes
1 answer
164 views

Existence of solutions of polynomials systems (and their "rough" shape) over $\mathbb{R}$ & friends with positive-dimensional ideals

This is a follow-up (but self-contained) question to my previous one. There I asked about state-of-the-art methods to solve multivariate polynomials systems over non-algebraically closed fields in ...
user43263's user avatar
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3 votes
0 answers
715 views

Differences between GAP and MAGMA [closed]

GAP and MAGMA are computer algebra systems. What are the objective differences between the two? Which capabilities are not shared? How do they compare on facilities for working with character tables?...
Philip's user avatar
  • 147
2 votes
1 answer
76 views

CAS implementing free algebras with involution

Is there any software that easily allows to make symbolic computations with involutions and homomorphisms? I need to define a product in an associative algebra with an (abstract) involution and ...
Jose Brox's user avatar
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26 votes
5 answers
6k views

Minimal polynomial of cos(π/n)

I know that $\cos(\pi/n)$ is a root of the Chebyshev polynomial $(T_n + 1)$, in fact it is the largest root of that polynomial, but often that polynomial factors. For example, if $n = 2 k$ then $\cos(\...
pavpanchekha's user avatar
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5 votes
1 answer
266 views

speeding up Gosper and WZ algorithms

In our ongoing work to speed up symbolic summation and other similar algorithms in Sagemath, we notice that naive implementations of Gosper and Wilf-Zeilberger (a.k.a. WZ) algorithms are usually quite ...
Dima Pasechnik's user avatar
7 votes
2 answers
2k views

Fast Symbolic Linear Algebra CAS?

I am a regular user of Mathematica, Julia, and MATLAB but I am looking for something different. The problem I am trying to solve in Mathematica only requires (dense) linear algebra to specify but is ...
Chris Rackauckas's user avatar
6 votes
1 answer
414 views

Finite field analogue of Chebotaryov theorem on roots of unity?

Chebotarev's theorem on roots of unity says that all the minors of a prime-length DFT matrix over the complex numbers are nonzero. I was wondering if there was an analogue for finite fields. More ...
user41530's user avatar
  • 115
3 votes
0 answers
121 views

Algorithmic quantifier elimination over p-adic fields

It is known that the first-order theory of p-adic fields is decidable, and that the p-adics admit elimination of quantifiers. What is the state of the art in algorithmic aspects of quantifier ...
352506's user avatar
  • 1,001
6 votes
2 answers
811 views

How to compute the normals to Costa's minimal surface?

I am trying to draw Costa's minimal surface in high resolution using the PovRay raytracer. For this I need to compute points on the surface as well as the normals. It is relatively easy to compute the ...
Andrej Bauer's user avatar
1 vote
0 answers
197 views

Testing functional equivalence

We are looking for the most efficient (most recent, or best) techniques to check if two algebraic expressions (elementary, Calculus-type functions) are equivalent (or if an expression is equivalent to ...
Denis Serbin's user avatar
2 votes
1 answer
1k views

Finding zeros of a multi-variable nonlinear trigonometric function

I am trying to calculate analytic solution (or locus) of zeros of a very large multi-variable function which is consisted of thousands of nonlinear trigonometric terms. All the variables are real ...
p8me's user avatar
  • 123
5 votes
1 answer
617 views

The existential theory of the reals

Some definitions of the existential theory of the reals (ETR) allow a real closed field and some definitions allow only rational numbers as coefficients of polynomials. Which one is correct? Will the ...
dschaehi's user avatar
2 votes
0 answers
103 views

Tools for "bound guessing"

I have a somewhat complicated symbolic expression of the form $\frac{J-a+\frac{q}{a}}{J(J-a)+q}$, where $J,a$ and $q$ are themselves affine functions of four other variables $d,r,c,s$, and I want to ...
Felix Goldberg's user avatar
2 votes
1 answer
399 views

Computer algebra system (CAS) with good re-presenting or transformation support

Such heavy-weight transformations as expanding or factoring are provided by most of CAS-es, but what about light-weight, but a useful transformations, like "reorder some terms to make expression more ...
Guest Here's user avatar
5 votes
0 answers
262 views

What became of PoSSo and FRISCO

I know PoSSo and FRISCO were pretty big projects involving many European universities. Interestingly, I couldn't find much information about these projects (the the top of the PoSSo homepage says "...
ssquidd's user avatar
  • 1,101
2 votes
2 answers
2k views

Solving a system of equations/inequalities that have trigonometric functions on the left-hand side

Is there any known (symbolic) method that solves a system of equations/inequalities that have trigonometric functions on the left-hand side of the system? Ex) Find $x,y,\theta \in \mathbb{R}$ that ...
SCL's user avatar
  • 23
5 votes
1 answer
800 views

Symbolic computations with differential operators (universal envelopings i.e. non-commutative variables) ?

Please give suggestions about soft to make symbolic computations with NON-commutative variables. Typical examples I am interesting - Capelli identities http://en.wikipedia.org/wiki/Capelli'...
Alexander Chervov's user avatar
1 vote
0 answers
1k views

symbolic diagonalization of a matrix

Hi, I am looking for algorithms that can perform a diagonalization, in a symbolic way, of a given matrix. I need to find a similarity transformation, if it exists. Desired features of the algorithms ...
Leslaw's user avatar
  • 11
3 votes
2 answers
541 views

What are some resources discussing mathematical notation?

I'm looking for resources discussing mathematical notation, the theory, the philosophy, the distinct advantages of various notations. Stuff about notation for computer algebra systems is interesting ...
user16513's user avatar
2 votes
2 answers
2k views

The easiest symbolic integration method to try implementing.

Hello! I wonder how hard is it to implement more or less general symbolic integration algorithm (number of lines in a certain language)? Maybe someone here did this or knows some good blog posts ...
Yrogirg's user avatar
  • 443
6 votes
0 answers
1k views

Why is mechanical differentiation so hard to get right?

This question is related to this question on differentiation/integration which asks why differentiation is mechanical but integration is an art. The answers given all make a huge assumption: that one ...
Jacques Carette's user avatar