Numeric computation usually uses floating point numbers. Symbolic computations use symbols, and can give exact answers, such as $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$. Mathematica, Maple, and Geometry Expressions all use symbolic computation, when desired. An online source is WolframAlpha.
Questions tagged [symbolic-computation]
281 questions
70
votes
4 answers
How much does symbolic integration mean to mathematics?
(Before reading, I apologize for my poor English ability.)
I have enjoyed calculating some symbolic integrals as a hobby, and this has been one of the main source of my interest towards the vast world of mathematics. For instance, the integral…
Sangchul Lee
- 181,930
33
votes
10 answers
Online tools for doing symbolic mathematics
This question is similar to this one, but specifically relates to resources available strictly as on-line web apps. Examples include:
Wolfram Alpha
Sage Notebook
hicalc
not sure what this is; the site is currently blocked from my location. I got…
Brian Vandenberg
- 962
25
votes
9 answers
symbolic computation program/software
What is your recommended symbolic computation program/software for free and commercial respectively?
What are its strength and weakness? For example, efficiency, comprehensiveness, etc
Thanks!
Tim
- 49,162
20
votes
1 answer
Algorithms for symbolic manipulation
If you take a look at WolframAlpha, or other computer algebraic system, you will find that it is able to do symbolic manipulation like real humans.
For example, if you type in an integral, it can show you step by step on how to solve the…
Graviton
- 2,364
17
votes
0 answers
Symbolic approximation through integration by parts
This is a slightly soft question. Suppose I have an integral $f(x) =\int_a^x g(t) dt $ which cannot be expressed in terms of elementary functions. One might still be able to integrate by parts to get something like:
$$ f(x) = h_1(t) |_{a}^{x} -…
msm
- 2,876
14
votes
0 answers
Techniques for doing "handwritten" math with current 2020 technology?
A bit of a soft question here. But what techniques are people using to efficiently do analytical math that would have typically been handwritten in the past?
For example, if you're working with a really complicated system, that involves many types…
Steven Sagona
- 326
- 2
- 20
12
votes
1 answer
Simplifying polynomials
Suppose I have a (multivariate) polynomial with coefficients in $\mathbb Z$ or $\mathbb Q$, given in fully expanded form. How can I simplify this to reduce the number of elementary operations (addition, subtraction, multiplication) used in its…
MvG
- 44,006
12
votes
2 answers
Computer algebra system to simplify huge rational functions (of order 100 Mbytes)
I have a huge rational function of three variables (which is of order ~100Mbytes if dumped to a text file) which I believe to be identically zero. Unfortunately, neither Mathematica nor Maple succeeded in simplifying the expression to zero.
I…
Yuji
- 322
11
votes
2 answers
What is the best way to factor arbitrary polynomials?
I am currently working on a Computer Algebra System and was wondering for suggestions on methods of finding roots/factors of polynomials. I am currently using the Numerical Durand-Kerner method but was wondering if there are any good non-numerical…
ternaryOperator
- 213
10
votes
0 answers
Is the Risch algorithm useful for calculating antiderivatives by hand?
In a German forum, a user asked how the "Feynman"-trick works. The example was $$f(x)=xe^x$$
Another user mentioned that the Risch algorithm should be taught. Therefore, I wonder whether the Risch algorithm is useful for calculating antiderivates by…
Peter
- 86,576
9
votes
4 answers
Easy ways to find partial fraction representation? (via a concrete example)
In a homework assignment (about generating functions) the students find themselves having to expand $\frac{3-7x+9x^{2}-3x^{3}}{\left(1-x\right)^{4}}$ intro partial fractions. Using some automated tool (e.g. Wolfram alpha) the expansion is found…
Gadi A
- 19,839
9
votes
0 answers
Composition of some linear differential operators $(D-A_n)...(D-A_1)$
Let $D=x\frac{d}{dx}$ and $A_i\in\mathbb{R}[[x]]$ for $i=1,...,n$. Let $B_i\in\mathbb{R}[[x]]$ for $i=1,...,n$ such that $$(D-A_n)...(D-A_1)=D^n+\sum_{i=1}^nB_iD^{n-i}.$$
Is there a well-known formula for $B_i$?
Also, suggestions for the resources…
Platonicsolids
- 390
8
votes
2 answers
Are there general methods to compare the largest real roots of two integer-coefficient polynomials using only their expressions?
Question
Are there general methods to compare the largest real roots of two integer-coefficient polynomials using only their expressions?
Example 1:
Consider:
$P_{1}\left ( x \right )= x^{5}- 20x^{4}+ 147x^{3}- 505x^{2}+ 784x- 432$
$P_{2}\left (…
Dang Dang
- 320
8
votes
3 answers
Integral of a rational function: Proof of $\sqrt{C}\,\int_{0}^{+\infty }{{{y^2}\over{y^2\,C+y^4-2\,y^2+1}}\;\mathrm dy}= {{\pi}\over{2}}$?
I suspect that
$$\sqrt{C}\,\int_{0}^{+\infty }{{{y^2}\over{y^2\,C+y^4-2\,y^2+1}}\;\mathrm dy}=
{{\pi}\over{2}}$$
for $C>0$.
I tried $C=1$, $C=2$, $C=42$, and $C=\frac{1}{1000}$ with Wolfram Alpha. But how to prove it?
slitvinov
- 319
8
votes
4 answers
Proof that "many differential equations cannot be solved using analysis"?
The wiki page on Numerical methods for ordinary differential equations states that "many differential equations cannot be solved using analysis." Is this literally true, or do they mean to say "it is not known how to solve many differential…
rhombidodecahedron
- 1,582