All Questions
109 questions
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, ...
12
votes
1
answer
5k
views
Closest 3D rotation matrix in the Frobenius norm sense
Given a 3 by 3 matrix $M$ I would like to find the rotation matrix $R$ minimizing the Frobenius norm:
\begin{equation}
\|R-M\|_F
\end{equation}
Is there a closed form solution for $R$, or is it ...
- 561
11
votes
3
answers
6k
views
Random Sampling a linearly constrained region in n-dimensions...
Hi,
So here is my problem:
Given a nonlinear, discontinous, cost function $f(x_1,x_2,..,x_N)$ along with linear constraints $x_n \ge 0, \forall n$
$x_n \le c_n$
and $\sum_{n=1}^N x_n = 1$ find an ...
- 113
6
votes
3
answers
490
views
Non-polynomial splines, a non-linear problem
I'm looking for references on how to construct spline-like functions from a basis that does not include piecewise polynomials.
To be specific, given a class of functions such as "decaying ...
- 3,065
6
votes
6
answers
3k
views
Circumference of Convex Shapes
Here is a puzzle I found in Mitteilungen der DMV (roughly, "Letters of the German Society of Mathematicians"), issue 19/2011. It was posed by Alfred Schreiber in "Wie man Hasen fangt" (How to catch ...
3
votes
1
answer
328
views
LP Constraints for Connected Subgraphs of Fixed Size
Question:
how can the connectedness-constraint for a subgraph, that is induced by a proper subset $W\subset V$ of the vertices of $G(V,E),\ |V|=n,\ |W|=m$, be formulated in a $LP$ or $ILP$?
...
- 13.2k
2
votes
1
answer
165
views
Boundary condition for elliptic problems and domain decomposition
This question is motivated by one that has been previously asked on this website: Elliptic problem on a domain split in two subdomains
Consider an open domain $U$ split in two non-overlapping ...
user60665
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
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
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
$$\begin{pmatrix}6 & 20 & 6& 0 ...
- 1,121
19
votes
2
answers
2k
views
Is the tensor product of polyhedra a polyhedron?
Conventions: A polytope in a finite-dimensional $\mathbb R$-vector space $V$ is defined to be a convex hull of finitely many points in $V$. A polyhedron in a finite-dimensional $\mathbb R$-vector ...
- 33.8k
17
votes
3
answers
2k
views
The minimum of a sum of absolute values of inner products in $\mathbb{R}^d$
Consider a collection of unit vectors $v_1, \ldots, v_n$ in $\mathbb{R}^d$ (we think of $n$ being much larger than $d$). I would like to minimize the sum:
$$\sum_{i\neq j}|\langle v_i,v_j\rangle|.$$
...
- 2,288
14
votes
2
answers
2k
views
Rigorous numerical integration
I need to evaluate some (one-variable) integrals that neither SAGE nor Mathematica can do symbolically. As far as I can tell, I have two options:
(a) Use GSL (via SAGE), Maxima or Mathematica to do ...
- 20.2k
8
votes
3
answers
859
views
Books and resources on PDEs that use Mathematica and Matlab
Can you recommend some reference books that use software like MATLAB and Mathematica to deal with the basic topics in
analysis of PDE (the ones you can find in Strauss' book Partial Differential ...
user60665
7
votes
2
answers
7k
views
Computational complexity of calculating the nth root of a real number
Several sources state that the computational or time complexity of square rooting is the same as that of multiplication (or division). See for example:
Jean-Michel Muller, "Elementary Functions: ...
- 385
5
votes
3
answers
1k
views
Algorithm for the intersection of a vector subspace with a cone of non-negative vectors
Hi,
I would like to know whether there is some more effective way of how to compute an intersection of a vector subspace of $\mathbb{R}^{n}$ with a cone of vectors with non-negative entries than the ...
4
votes
1
answer
3k
views
optimization of inverse matrix with constraint on matrix elements
everyone! I have this optimization problem with constraint.
$D$ and $T$ are symmetric matrices, where T is known and D is the unknown parameter.
$x$ and $v$ are two known p-dimensional vectors.
The ...
- 49
4
votes
1
answer
8k
views
Detection of Redundant Constraints
Suppose I pose the following query to a constraint logic programming
system:
?- Y <= 6 - X, Y <= (- 4) + 4 * X, Y <= 4 + X / 3.
Are there systems that would recognize the last inequality as
...
Countably Infinite
3
votes
2
answers
2k
views
Is there an example where the error of Gauss-Laguerre quadrature does not vanish?
The $n$th Gauss-Laguerre quadrature aims to approximate integral $$\int_{\mathbb{R}_+} f(x) \exp(-x)$$ by the sum
$$\sum_{i=1}^n f(x_i) w_i$$
where $x_1,...,x_n$ are the roots of the $n$th Laguerre ...
- 1,839
2
votes
0
answers
105
views
Optimization over a convex cone generated by a set is equal to optimization over the set
Within my research I found an important doubt and that prevents me from advancing, the context of my doubt is as follows:
We considerer the following optimization problem
$$
\left\{\begin{array}{cl} \...
- 159
1
vote
0
answers
443
views
Multivariate solution to Lambert W / product-log function
Consider solving the following system for $x$
\begin{align*}
a - b e^{\theta x} - cx = 0
\end{align*}
According to your favorite computer algebra program, one possible (and the simplest) is
\begin{...
- 229
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
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 ...
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
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
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
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
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
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
14
votes
1
answer
900
views
“Taylor series” is to “Volterra series” as “Padé approximant” is to _________?
Padé approximants are often better than Taylor series at representing a function. Given a Taylor series, one can use Wynn's epsilon algorithm to easily produce the Padé approximants to it.
Volterra ...
- 4,897
14
votes
2
answers
2k
views
Computing Gauss Legendre quadrature for large $N$
I've been scanning across the web, and haven't found a good method to compute the Gauss Legendre abscissas and weights $\{ x_j, w^j \} _{j=1}^N$ for large $N\in\mathbb{N}$. My question is how to do it,...
- 3,574
13
votes
2
answers
946
views
Computing a large permanent
Is there a practical way to compute the permanent of a large ($91 \times 91$) $(0,1)$ matrix?
I have tried to use the matlab function written by Luke Winslow which works great for smaller matrices ...
- 6,962
12
votes
2
answers
5k
views
Why Householder reflection is better than Givens rotation in dense linear algebra?
It’s obvious that Givens rotation works better with sparse matrices. But I don’t know why Householder reflection is better for dense matrices. Does it require less computations? Or it’s numerically ...
- 253
11
votes
1
answer
3k
views
Best way to find a closest vector in a lattice
Let $v_1,\dotsc,v_n$ be linearly independent vectors in $\mathbb{R}^n$, and let $\Lambda=\bigoplus_{i=1}^n \mathbb{Z}v_i$. The question is, given a vector $w$ in $\mathbb R^n$, find the element $v$ ...
- 351
10
votes
1
answer
449
views
How to compute $\sum_{x \in \mathbb{Z}^n} e^{-x^TMx}$ efficiently
Let $M$ be a real symmetric integer valued positive definite matrix with $\det(M) \geq 1$. I would like write code to compute
$$S_M= \sum_{x \in \mathbb{Z}^n} e^{-x^TMx}.$$
One option is to simply ...
- 3,377
10
votes
1
answer
297
views
Confusion with practically implementing rational approximations
Writing a program visualizing Ford circles I've encountered a seemingly purely programmatic puzzle but then gradually realized there are some mathematical aspects of it which I don't understand. Let ...
- 18.4k
10
votes
2
answers
1k
views
Machin-like formulas for logarithms
I found this math puzzle blog post
http://fredrikj.net/blog/2013/03/machin-like-formulas-for-logarithms/
which I'm reposting here with permission. I'm setting this to community wiki to minimize the ...
10
votes
2
answers
3k
views
How do you tell if a system of linear inequalities has a solution?
A naive solution would be to optimize a dummy variable via linear programming and see if a result is returned. I imagine there must be a more direct way.
- 693
10
votes
1
answer
411
views
Network flows with capacities on pairs of edges
Take a standard network flow problem: a directed graph with nonnegative capacities on each edge, a source $s$, a sink $t$. We all know how to find the maximum flow from $s$ to $t$.
Now add edge-pair ...
- 37.7k
9
votes
1
answer
295
views
Definition of packing property
Definition 1:
A clutter $C$ is said to have the packing property if $C$ and all of its minors satisfy the König property.
where,
vertex cover of $C$ is a set of vertices that have non-empty ...
- 381
9
votes
3
answers
657
views
Degree necessary of a polynomial?
Given $-1<a<b<0$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[a,b]$ at every $x\in[b^2,a^2]$ and $f(0)=0$. What is minimum degree that is needed and maximum degree that ...
- 13.9k
9
votes
2
answers
775
views
Is there a systematic theory for Gibbs measures (better if on Hilbert spaces)?
During these first months in my PhD, I realized how my computational problems can be drastically reduced to one single problem:
Find an efficient way to sample from a Gibbs measure.
Let me ...
- 211
9
votes
1
answer
2k
views
Uniform sampling from general simplex with a twist
This is part of a question I had asked elsewhere, and then some of the links redirected me to CS stack exchange.
Given $0\leq a_1\leq\dots\leq a_D\leq1$ (all strictly positive), I want to draw points ...
- 221
9
votes
2
answers
818
views
Approximation theory on the disc
Chebyshev theory provides a very effective method for approximating continuous real valued functions on the unit interval. Is there something similar for continuous real valued functions on the ...
- 56.9k
8
votes
5
answers
15k
views
Eigenvalues of A+B where A is symmetric positive definite and B is diagonal
If I have a symmetric positive definite matrix A and a diagonal matrix B, and I know the eigenvalues of both A and B (by iterative numerical computation in A's case and trivially for B), is there any ...
- 135
8
votes
2
answers
2k
views
error estimates for multi-dimensional Riemann sums
Suppose that $f$ is a continuous function of bounded variation from $R^2$ to $R$ that's negative outside of some bounded set, and let $F=\max(f,0)$. Let $S_n$ be the Riemann sum for the integral of $F$...
- 19.7k
7
votes
1
answer
247
views
Add a multiple of $I$ to a matrix to minimize its operator norm
Given $A\in\mathbb{C}^{n\times n}$, what is $s_* = \arg\min \|A-sI\|$?
Here $\|A\|$ is the operator norm, $\|A\|=\rho(A^*A)^{1/2}$, and $I$ is the identity.
The corresponding problem for the ...
- 20.2k
7
votes
1
answer
505
views
Numerical linear algebra: how to compute $b^TA^{-1}b$ efficiently
What is the most efficient way to compute $b^TA^{-1}b$ for a given $A$ and $b$?
Do we have to calculate $A^{-1}b$, or is this not necessary?
edit: I forgot to mention that A is symmetric and ...
- 493