Questions tagged [symbolic-computation]
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.
262
questions
1
vote
1
answer
35
views
Software for symbolic matrix operations
Is there a software for symbolic manipulation that can treat matrices as whole variables?
For instance:
solve A = B + C * A for A
where ...
- 11
0
votes
0
answers
26
views
Maximization of a non-linear multi-variable symbolic piecewise function
I use math in building economic models and modeling/solving business problems.
I have a two-variable symbolic piece-wise function that I need to maximize i.e., fully and analytically characterize $(p^*...
- 101
0
votes
2
answers
86
views
CAS software similar to Nspire on macOS 10.13 [closed]
Last edit (2023-01-26):
This question (and answer) has been closed as off-topic, even though the What to ask here for this site says:
Software that mathematicians use (except Mathematica, which has ...
- 101
1
vote
0
answers
33
views
Expressing an equation in terms of its generators
I have a generating set of differential equations:
$$ I_1=u_t\\
I_2=-uu_{tt}\\
I_3=u^{-1}\\
I_4=uu_{xxx}-6u^2u_x $$
where $u=u(x,t)$. and the coordinate is $(x,t,u,u_x,u_t,u_{tt},u_{xxx})$. Infact we ...
1
vote
0
answers
57
views
Symbolic Evaluation of the Limit of an Undefined Function's Derivative in SymPy
Suppose that I have an unknown function $$f(x, y, z)$$ whose first order partial derivatives with respect to $(x, y, z$) are known to be everywhere smooth and well behaved (i.e., continuous, with no ...
2
votes
0
answers
26
views
How to get the conditions required to analyze the result? [closed]
I'm struggling to understand the method followed in the following analysis. Can someone please explain, how the author got the values of $Δ_1$ and $K_1$ and justify its analysis.
I have tried to ...
- 21
0
votes
0
answers
34
views
Finding eigenvalues of a 9 times 9 matrix with distinct symbolic values
I need to find the eigenvalues for a big symbolic matrix. Let
$$
K=\begin{pmatrix}
x_{11} & x_{12} & \ldots & x_{19} \\
x_{21} & x_{22} & \ldots & x_{29} \\
\vdots & \...
0
votes
0
answers
38
views
Understanding Richardson's Theorem
Richardson's Theorem states that (quoting from Wolfram MathWorld):
Let $\mathcal{R}$ be the class of expressions generated by
The rational numbers and the two real numbers $\pi$ and $\ln 2$,
The ...
- 181
0
votes
1
answer
117
views
Implementing a convolution integral symbolically using integration-by-parts
I'm working on implementing a convolution integral symbolically (not numerically), to be used in a mathematical modelling tool. This is more of a mathematical problem at this stage, NOT programming. ...
- 66
0
votes
0
answers
15
views
Solving Interesting Inverse Problems with Symbolic Neural Network
I've been messing around with Mathematica and found it isn't too difficult to encode and decode symbolic expressions into vectors (e.g., sin(x)+y can be represented by some vector {1,2,3,2...}).
I can ...
0
votes
0
answers
48
views
Is there a standard way of writing constructible numbers?
Every rational number can be represented by an irreducible fraction with a nonnegative denominator.
For expressions involving integers, the four basic operations of arithmetic and square roots, you ...
3
votes
1
answer
263
views
Software for solving symbolic matrix expressions
I've seen that there are multiple softwares to solve equations like sageMath, sympy, maxima... All of these seem to work great for scalars but I haven't found any software which can solve symbolic ...
- 303
1
vote
0
answers
54
views
Computing inverse elements of symbolic matrices with binary variables
I'm working with symmetric, symbolic matrices $A$ with real coefficients and linear binary variables like
$$
A = \begin{pmatrix} 0.5x_0 & 0.3x_1+0.002x_2 & 0 & 0 \\
0....
- 11
0
votes
0
answers
42
views
Norm of complex polynomials
I am working on the following problem:
For $f=a_nt^n+a_{n-1}t^{n-1}+\cdots+a_1t+a_0\in\mathbb{C}[t]$ we define the norm $\mid\mid f\mid\mid_2=(\sum_{k=0}^n\mid a_k^2\mid)^{\frac{1}{2}}$.
Now let $f\in\...
0
votes
0
answers
21
views
How to derive symbolic equations of a polynomial function fitting 3 points in 3d space?
Let's say I have three points in 3D space and I would like to fit a quadratic polynomial through these points.
$P_{1} = (10,2,3), P_{2} = (15,6,8)$ and $P_{3} = (22,1,12)$
I know that it is possible ...
0
votes
2
answers
76
views
Derivative of complex quadratic form-functions with respect to a vector
I have a quadratic form like this:
$Q_1=v_1^TC_1(x)v_1$
$Q_2=v_2^TC_2(y)v_2$
where
$x,y,z$ - $3 \times 1$-vectors
$J_1,J_2,C_1,C_2$ - $3 \times 3$-matrix
$v_1=x-(J_1(x)y-z),v_2=x-(J_2(y)y-z)$
...
- 681
1
vote
3
answers
83
views
Finding the symbolic expression for a particular matrix raised to the $n^{\text{th}}$ power
Consider the following matrix, with $a$ being a real number between $0$ and $1$.
\begin{equation}
\text{M}=\left[
\begin{array}{cc}
1-\frac{a^2}{3} & \frac{1}{3} \left(2 a-a^2\right) \\
\frac{2 ...
- 135
1
vote
0
answers
24
views
sympy: group given expressions within other expressions, e.g. $a + 2 x y + b x + 2 x z$
I would like to use sympy to locate and group instances of a given symbolic expression within a set of expressions, so that the given expression may be replaced by a variable. For instance, with ...
- 31
2
votes
1
answer
106
views
Computing an integral depending on two parameters
A few days ago I stumbled upon this integral
$$
f(y,c) :=\int_{-c-\sqrt{2}}^{-c+\sqrt{2}}\sqrt{2-(x+c)^2}\; \ln\left|\frac{x+y}{x-y}\right| \;\mathrm{d}x
$$
where $c>0$ and $y\in(-c-\sqrt{2},-c+\...
- 2,199
1
vote
1
answer
28
views
Are there any symbolic solvers that can use matrices and vectors directly?
I frequently deal with matrix/vector calculus problems and would like to get the answers in matrix/vector form. The number of elements in these matrices and vectors is not specified, and the desired ...
- 135
0
votes
1
answer
651
views
How to apply Risch Algorithm by hand to solve integrals?
In a German forum, a user asked how the "Feynman"-trick works. The example was
$$
f(x)=x e^{x}
$$
Another user mentioned that the Risch algorithm should be taught. Therefore, I wonder ...
- 179
9
votes
0
answers
199
views
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 ...
- 390
0
votes
0
answers
29
views
Identifying Hamiltonian dynamics in 2D
While studying a physical system I've encountered an two dimensional autonomous ODE defined on a torus. The vector field describing it is a messy expression involving rational functions of ...
- 1,704
1
vote
1
answer
53
views
Symbolic points in an elliptic curve over $\mathbb{Q}$ of the form $(u/e^{2},v/e^{3})$
Suppose I have an elliptic curve $E : y^{2} = x^{3} + D$ over $\mathbb{Q}$ , where $D$ is a symbolic constant (an integer). I want to define two points in the curve of the type $P = (u_{1}/e_{1}^{2}, ...
- 423
1
vote
1
answer
86
views
Symbolic point in an elliptic curve over $\mathbb{Q}$ in SageMath
I want to study the behavior of polynomials sitting in the co-ordinates of multiples of a rational point of an elliptic curve over $\mathbb{Q}, $ $E(\mathbb{Q}).$
Suppose if we take a point $P = (s,t) ...
- 423
1
vote
0
answers
420
views
How to use Octave with real numbers and syms at one time
I want to use GNU Octave to solve my equations. Equations contain syms variables and vectors with decimal numbers in them. How properly solve these equations without getting warnings that im doing ...
-1
votes
1
answer
113
views
Integral of Squared Sum of Sine [closed]
Let's say we have the following function
$$ \int \left( \sum_{i = 1}^\infty a_i \sin(b_ix)\right)^2$$
Assuming that this series converges (it is actually a Fourier Series of the Lagrangian of the Wave ...
- 121
1
vote
1
answer
177
views
Algorithm for expressing Gröbner basis in terms of ideal generators
Given a polynomial ring and an ideal
$$A \supset I = (f_1, ..., f_m)$$
there are plenty of implementations of an algorithm (e.g. Buchberger's) that produces a Gröbner basis
$$G = (g_1, ..., g_n)$$
and ...
- 1,014
0
votes
2
answers
74
views
How to prove a transcendental equation is never zero or sometimes zero?
I am an engineer by profession. Recently, while working on beams, I obtained the following transcendental equation: $$1-\cos(x)\cosh(x)+x^2\sin(x)\sinh(x)$$. I was wondering if this expression can ...
- 23
0
votes
1
answer
31
views
How can you show that $(p \rightarrow q) \land (q \rightarrow r) \leftrightarrow (p \rightarrow r)$?
Obviously, $(p \rightarrow q) \land (q \rightarrow r) \rightarrow (p \rightarrow r)$ due to Hypothetical Syllogism, but how about the converse?
2
votes
0
answers
53
views
Problem evaluating Wolfram Symbolic Integration Result
****** Note: In response to the comment below, I've modified the problem ******
I'm trying to evaluate the following indefinite integral
$$
\Large\int x^{-\frac{3}{2}} e^{-\frac{K}{x}\Big[(x-a)^2+(x-b)...
- 1,116
5
votes
2
answers
232
views
Handling elementary but messy computations in proofs
Often when working on a proof, I get to a computation which appears to be elementary (e.g. requiring only standard algebra and perhaps calculus) but messy. Solving this via pen and paper is tedious ...
- 2,161
2
votes
1
answer
64
views
Why does the function $\frac{1}{x^{2222}}$ seem to have *two* asymptotes at around $\pm0.744$?
When screwing around with the Dirac-delta function in Desmos i noticed that when i multiply it by a really low-degree rational function
$$f(x)=\frac{1}{x^i}$$
, the resulting function has the domain
$$...
- 908
0
votes
3
answers
40
views
How to write that the only integers that have a multiplicative inverse are $1$ and $-1$, in symbolic form
Could it be something like for $(1,-1) \in \mathbb{Z}$, there exists $y \in \mathbb{Z}$, such that $xy = 1$?
1
vote
0
answers
39
views
what is the meaning of this element $\varphi(f|_Y)?$
I have some confusion in symbolics .
My confusion is given below marked in red colour
what is the meaning of this element $\varphi(f|_Y)?$
My main confusion about the symbolics is $(f|_Y)$?
- 14k
1
vote
0
answers
52
views
What is this hybrid symbolic-numeric algorithm for solving non-linear ODEs called?
I'm an engineer so please forgive my lack of knowledge in exact mathematical terminology:
For linear differential equations, it is possible to use the principle of superposition or convolution ...
- 153
1
vote
1
answer
159
views
Factor SYMBOLIC third degree polynomial
I apology in advance for the long post...
Say I have a very general 3rd-degree polynomial
$$x^3 + a_1x^2 + a_2x+a_3$$
I would really like to factor that polynomial as a product of a 1st order and a ...
- 11
1
vote
2
answers
70
views
Finding $b$ such that $y=mx+c$ is tangential to $y=b^x$.
I was thinking about the following problem:
Find $b$ such that $y=mx+c$ is tangential to $y=b^x$.
My work so far:
To satisfy the condition we need to find $x$ such that $\frac d{dx}\left[b^x\right]=\...
- 2,948
2
votes
1
answer
96
views
How to find solve for second order pde with initial conditions using Wolfram Mathematica?
I have next task:
$$
\frac{\partial^2 u}{\partial x \partial y} = 0,~ u(x,x^2) = 0,~ \frac{\partial u}{\partial x}(x, x^2) = \sqrt{|x|},~|x| < 1
$$
I write this, but it don't work:
...
- 33
0
votes
1
answer
76
views
How to use the Wolfram Language or another tool to find a second order pde solution with initial conditions?
I want to find a solution the Cauchy problem using the Wolfram Language or some other tool.
I have next task:
$$
3\frac{\partial^2 u}{\partial x^2} + 8\frac{\partial^2 u}{\partial x \partial y} - 3\...
- 33
0
votes
0
answers
67
views
pipe (vertical bar) symbol in function
could you please explain me what pipe symbols | means in bellow's function?!
$$
force = -R_B * D(v_a^B|v_a^B|)*c_D
$$
where we have introduced a notation $ D(b) = diag(b_x; b_y; b_z) $for a vector $...
- 11
0
votes
1
answer
227
views
A very heavy computation
I'm doing calculations about elliptic curve, which involve the sum of two rational points and birational transformation. I need help regarding the symbolic substitution below, where $u,d,x,y$ are ...
- 1,817
1
vote
1
answer
49
views
Relation between symbolic computation and symbolic dynamics
From what I read about the fields symbolic computation and symbolic dynamics , I expect these two fields to be having vivid connections . I would like to know what are some of the prominent topics ...
- 53
1
vote
2
answers
85
views
What are $c$'s in antiderivative of double integral in Wolfram Alpha
I would like to solve the following double integral:
$$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} (x y)^{-\theta - 1} dx \hspace{1mm} dy$$
Using Wolfram Alpha for symbolic algebra, it gives a ...
- 1,434
1
vote
1
answer
177
views
What is parametric equation of a cylinder with an arbitrary norm in Cartesian system?
I just need to intersect a plane and a cylinder for an industry application and latter we would fit the intersection curve on to some sample data we have and that supposed to provide us with some ...
- 111
0
votes
2
answers
74
views
Pull `t` out of trig functions and solve it
I'm trying to come up with an analytical solution for inverse kinematics for my robot; but some of my trig has left my brain!
The robot is very simple; with one rotary and one prismatic joint. I'm ...
- 105
1
vote
0
answers
64
views
Symbolic sum similar to Zeta function or harmonic number
(I have no idea why this was closed, nor did I receive any information/email regarding what details or clarity were missing. I have rendered the Mathematica into mathematics.)
copied from mathematica....
- 111
0
votes
0
answers
167
views
How do I extend SymPy pretty printing for new structures in Jupyter notebook?
Note 1: I am asking this on Math Stack Exchange rather than Stack Overflow, where most SymPy questions are, because StackOverflow doesn't have MathJax, making it very difficult to pose the question.
...
- 112
1
vote
0
answers
49
views
Construction of the Steinhaus probability space
Q: Please let me know if the following sketch of the build-up of the Steinhaus probability space $(\Omega,{\cal A},P) = ((0,1), {\cal B}((0,1)), \lambda\vert_{[0,1]})$ is correct, as far as it goes:
...
- 112
2
votes
4
answers
676
views
Open source software for calculation of eigenvalues of symbolic matrix
I have following matrix
\begin{bmatrix}
-\alpha & 0 & \beta & \gamma\cdot\omega_m \\
0 & -\alpha & -\gamma\cdot\omega_m & \beta \\
R_r\frac{L_h}{L_r} & 0 & -\frac{R_r}{...
- 259