All Questions
1,732 questions
129
votes
2
answers
16k
views
What are the shapes of rational functions?
I would like to understand and compute the shapes of rational functions, that is, holomorphic maps of the Riemann sphere to itself, or equivalently, ratios of two polynomials, up to Moebius ...
- 25.1k
109
votes
19
answers
38k
views
Why were matrix determinants once such a big deal?
I have been told that the study of matrix determinants once comprised the bulk of linear algebra. Today, few textbooks spend more than a few pages to define it and use it to compute a matrix inverse. ...
64
votes
5
answers
18k
views
The unreasonable effectiveness of Padé approximation
I am trying to get an intuitive feel for why Padé approximation works so well. Given a truncated Taylor/Maclaurin series it "extrapolates" it beyond the radius of convergence.
But what I can'...
- 6,962
57
votes
2
answers
5k
views
Recent observation of gravitational waves
It was exciting to hear that LIGO detected the merging of two black
holes one billion light-years away. One of the black holes had 36
times the mass of the sun, and the other 29. After the merging the
...
- 50.8k
57
votes
8
answers
16k
views
There must be a good introductory numerical analysis course out there!
Background As a numerical analyst, I've frequently taught the 'Introductory Numerical Analysis' class. Such courses are found in many major universities; the audience typically consists of reluctant ...
46
votes
7
answers
13k
views
What is the time complexity of computing sin(x) to t bits of precision?
Short version of the question: Presumably, it's poly$(t)$. But what polynomial, and could you provide a reference?
Long version of the question:
I'm sort of surprised to be asking this, because ...
- 6,694
40
votes
2
answers
4k
views
Recent fundamental new directions in PDEs
My main interests are in modern geometry/topology, algebra and mathematical physics. I observe that there is a raising communication, language and social barrier between this community and the ...
- 5,232
37
votes
11
answers
8k
views
"Must read" papers in numerical analysis
In 1993, Prof. L.N. Trefethen published a NA-net posting with a list of thirteen paper he used for teaching the seminar Classic Papers in Numerical Analysis.
In Trefethen's words, ... this course ...
35
votes
9
answers
14k
views
What is... a grossone?
Y. Sergeyev developed a positional system for representing infinite numbers using a basic unit called a "grossone", as well as what he calls an "infinity computer". The ...
- 16.6k
35
votes
4
answers
5k
views
Why are optimization problems often called "programs"?
Why are optimization problems often called programs?
linear programming
geometric programming
convex programming
Integer programming
...
- 461
35
votes
5
answers
4k
views
Should computer code be included within publications that present numerical results?
Many research papers include numerical results obtained through computation. Most of the time such computations are performed using software that is used by many mathematicians, i.e., Maple, ...
34
votes
4
answers
2k
views
"Wild" solutions of the heat equation: how to graph them?
It has long been known that the Cauchy initial-value problem for the
classical heat equation on $\mathbb{R}$ (or $\mathbb{R}^n$) doesn't
have unique solutions, without additional assumptions. In ...
- 29.7k
34
votes
1
answer
5k
views
What are "variational crimes" and who coined the term?
I just caught sight on arXiv a paper by Holst and Stern titled Geometric Variational Crimes. Apparently a Variational Crime is an approach to solve problems using a finite element method (e.g. ...
- 39k
32
votes
4
answers
6k
views
How does Mathematica do symbolic integration?
I suppose there was at least once in our lifetime the point where we resorted to mathematica for help with an integral.-Unless you chose not to have the pleasure of using the continuum in your ...
- 536
29
votes
6
answers
8k
views
How to find a closest integer point to the intersection of two lines?
Here's a question that originates from StackOverflow.
Given are two lines on a plane, specified by equations ($a x + b y = c$) with integer coefficients. The lines aren't parallel and they don't ...
- 391
28
votes
4
answers
3k
views
Can Gröbner bases be used to compute solutions to large, real-world problems?
In particular, suppose I have an affine algebraic variety over $\mathbb{R}^n$ described by generators of a radical ideal $I$ and I want to find (perhaps not all of the) points in the variety. Several ...
- 3,059
27
votes
5
answers
2k
views
Is the matrix $\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}$ nonsingular?
Suppose we have a $(2m-1) \times (2m-1)$ matrix defined as follows:
$$\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}.$$
For example, if $m=3$, the matrix is
$$\begin{pmatrix}6 & 20 & 6& 0 ...
- 1,121
27
votes
6
answers
5k
views
Why not evaluate integrals using ODE-solvers?
Hello!
I have a question about the relationship between numerical integration and the solution of ordinary differential equations (ODE). Suppose I want to evaluate the integral
$I(x) = \int_{0}^{x} f(...
- 271
27
votes
3
answers
9k
views
Analytical formula for numerical derivative of the matrix pseudo-inverse?
Is there a simple numerical procedure for obtaining the derivative (with respect to $x$) of the pseudo-inverse of a matrix $A(x)$, without approximations (except for the usual floating-point ...
- 691
25
votes
2
answers
2k
views
An Interesting Optimization Problem
You are given n non-negative integers $a_1, a_2 ,, a_n$. In a single operation, you take any two integers out of these integers and replace them with a new integer having value equal to difference ...
- 363
25
votes
3
answers
2k
views
Is the Ford-Fulkerson algorithm a tropical rational function?
The Ford-Fulkerson algorithm
Let me recall the standard scenario of flow optimization (for integer flows at least):
Let $\mathbb{N} = \left\{0,1,2,\ldots\right\}$. Consider a digraph $D$ with vertex ...
- 33.8k
24
votes
1
answer
4k
views
Why is there a $\sqrt{5}$ in Hurwitz's Theorem?
Hurwitz's theorem is an extension of Minkowski's Theorem and deals with rational approximations to irrational numbers. The theorem states:
For every irrational number $\alpha$, there are infinitely ...
- 1,129
23
votes
4
answers
4k
views
Convergence of finite element method: counterexamples
There are many known results proving convergence of finite element method for elliptic problems under certain assumptions on underlying mesh [e.g., Braess,2007]. Which of these common assumptions are ...
- 1,488
22
votes
3
answers
1k
views
Distribution of the Error term in GH Hardy's "curious result" $\sum_{\nu \leq n } \{ \nu \theta \}^2 = \tfrac{1}{12} n + O(1)$
In an early paper, GH Hardy talks about the distribution of "curious" sum:
$$ \sum_{\nu \leq n } \{ \nu \theta \}^2 = \tfrac{1}{12} n + O(1)$$
where $\{x\}:=x-\left \lfloor x \right \rfloor -1/2$. ...
- 22.8k
22
votes
9
answers
17k
views
Fast evaluation of polynomials
Hello everybody !
I was reading a book on geometry which taught me that one could compute the volume of a simplex through the determinant of a matrix, and I thought (I'm becoming a worse computer ...
- 1,654
21
votes
9
answers
19k
views
What is the best algorithm to find the smallest nonzero Eigenvalue of a symmetric matrix?
see title.
An algorithm is 'good' if it is able to distinguish between zero Eigenvalues and nonzero Eigenvalues.
- 979
21
votes
3
answers
1k
views
Easy functions?
Let $f$ be an analytic function, and suppose that we want to compute
$f(x)$. The input consists of the digits of $x$ and the output of
a rational number approximating $f(x)$. A function $f$ is called ...
- 91.8k
20
votes
13
answers
9k
views
Finding all roots of a polynomial
Is it possible, for an arbitrary polynomial in one variable with integer coefficients, to determine the roots of the polynomial in the Complex Field to arbitrary accuracy? When I was looking into this,...
- 333
20
votes
4
answers
2k
views
show that $ \frac{\Gamma(\frac{1}{24})\Gamma(\frac{11}{24})}{\Gamma(\frac{5}{24})\Gamma(\frac{7}{24})} = \sqrt{3}\cdot \sqrt{2 + \sqrt{3}} $
Mathworld's discussion of the Gamma function has the pleasant formula:
$$ \frac{\Gamma(\frac{1}{24})\Gamma(\frac{11}{24})}{\Gamma(\frac{5}{24})\Gamma(\frac{7}{24})} = \sqrt{3}\cdot \sqrt{2 + \sqrt{3}...
- 22.8k
19
votes
3
answers
2k
views
Best known bounds on tensor rank of matrix multiplication of 3×3 matrices
Years ago I attended a conference where they taught us that matrix multiplication can be represented by a tensor. The rank of the tensor is important, because putting it into minimal rank form ...
- 3,359
19
votes
4
answers
1k
views
Applications of linear programming duality in combinatorics
So, I know that one can apply the strong LP duality theorem to specific instances of maximum flow problems to recover some nontrivial theorems in combinatorics, such as Hall's theorem, Koenig's ...
- 997
19
votes
2
answers
11k
views
Meaning of $\Subset$ notation
The symbol $\Subset$ (occurring in places where $\subseteq$ could occur syntactically) comes up frequently in a paper I'm reading. The paper lives at the intersection of a few areas of math, and I ...
- 1,601
19
votes
2
answers
781
views
"Fractally self-similar" numbers
This is another question about visualization of Ford circles, the previous one being Confusion with practically implementing rational approximations. Here is an output of zooming into Ford circles at $...
- 18.4k
19
votes
2
answers
2k
views
Is the tensor product of polyhedra a polyhedron?
Conventions: A polytope in a finite-dimensional $\mathbb R$-vector space $V$ is defined to be a convex hull of finitely many points in $V$. A polyhedron in a finite-dimensional $\mathbb R$-vector ...
- 33.8k
19
votes
2
answers
777
views
How can I distinguish a genuine solution of polynomial equations from a numerical near miss?
Cross-posted from MSE, where this question was asked over a year ago with no answers.
Suppose I have a large system of polynomial equations in a large number of real-valued variables.
\begin{align}
...
- 1,302
19
votes
0
answers
841
views
I found a (probably new) family of real analytic closed Bezier-like curves; is it publishable?
Given $n$ distinct points $\mathbf{x} = (\mathbf{x}_1, \ldots, \mathbf{x}_n)$ in the plane $\mathbb{R}^2$, I associate a real analytic map:
$f_{\mathbf{x}}: S^1 \to \mathbb{R}^2$
with the following ...
- 5,215
18
votes
3
answers
3k
views
Deciding membership in a convex hull
Given points $u, v_1, \dots,v_n \in \mathbb{R}^m$, decide if $u$ is contained in the convex hull of $v_1, \dots, v_n$.
This can be done efficiently by linear programming (time polynomial in $n,m$) in ...
- 667
18
votes
3
answers
3k
views
Counting roots: multidimensional Sturm's theorem
Sturm's theorem gives a way to compute the number of roots of a one-variable polynomial in an interval [a,b]. Is there a generalization to boxes in higher dimensions? Namely, let $P_1,\dotsc,P_n\in \...
- 7,836
18
votes
1
answer
1k
views
Who introduced the notion of "stability" in numerical analysis?
I am preparing a lecture course on the applications of operator theory where I intended to make some numerical analysis application. I was wondering about this question while browsing the literature I ...
- 4,729
17
votes
3
answers
6k
views
The cone of positive semidefinite matrices is self-dual? (reference needed)
I'm seeking a reference for the following fact.
The cone of positive semidefinite matrices is self-dual (a.k.a. self-polar).
This result is relatively easy to prove, has been known for a long time,...
- 1,513
17
votes
3
answers
2k
views
The minimum of a sum of absolute values of inner products in $\mathbb{R}^d$
Consider a collection of unit vectors $v_1, \ldots, v_n$ in $\mathbb{R}^d$ (we think of $n$ being much larger than $d$). I would like to minimize the sum:
$$\sum_{i\neq j}|\langle v_i,v_j\rangle|.$$
...
- 2,288
17
votes
2
answers
1k
views
The Chow & Robbins game ≈ 0.79295350640: improvements could come from simple statistics, or from a continuous version of the game
This question seeks help with improving a numerical estimate of the value of the Chow and Robbins game. Much about this game is unknown, such as whether its value is rational, but there are two routes ...
- 199
16
votes
3
answers
1k
views
Can a convex polytope with $f$ facets have more than $f$ facets when projected into $\mathbb{R}^2$?
Let $P$ be a convex polytope in $\mathbb{R}^d$ with $n$ vertices and $f$ facets.
Let $\text{Proj}(P)$ denote the projection of $P$ into $\mathbb{R}^2$.
Can $\text{Proj}(P)$ have more than $f$ facets?
...
- 163
16
votes
3
answers
3k
views
Current Research in Numeric Mathematics
To me, as an non-expert in the field, it seems as if numeric mathematics should have lost its importance because nowadays symbolic calculations or calculations with unlimited precision are generally ...
16
votes
3
answers
2k
views
Are there any known quantum algorithms that clearly fall outside a few narrow classes?
I'm trying to refresh myself on quantum algorithms and have been skimming Childs and van Dam's 2008 RMP paper among other things. From my preliminary surfing it looks like the known quantum algorithms ...
- 15.4k
16
votes
2
answers
819
views
Numerical integration using interval arithmetic, nowadays
This is an update to my question Rigorous numerical integration from three years ago.
Is there now a package for rigorous numerical integration that uses interval arithmetic and has access to a well-...
- 20.2k
16
votes
7
answers
6k
views
Numerical integration over 2D disk
I have a real-valued function $f$ on the unit disk $D$ that is fairly well behaved (real-analytic everywhere) and would like to find the integral $\int_D f(x,y)dxdy$ numerically. After much searching, ...
- 8,271
15
votes
9
answers
9k
views
Exponential of large matrices
I want to make a diffusion kernel, which involves $e^{\beta A}$, where A is a large matrix (25k by 25k). It is an adjacency matrix, so it's symmetric and very sparse.
Does anyone have a ...
- 151
15
votes
3
answers
2k
views
How should I approximate real numbers by algebraic ones?
Given a high precision real number, how should I go about guessing an algebraic integer that it's close to?
Of course, this is extremely poorly defined -- every real number is close to a rational ...
- 7,800
15
votes
2
answers
1k
views
Minimal polynomial with a given maximum in the unit interval
Find the lowest degree polynomial that satisfies the following constraints:
i) $F(0)=0$
ii) $F(1)=0$
iii)The maximum of $F$ on the interval $(0,1)$ occurs at point $c$
iv) $F(x)$ is positive ...
- 457