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Weyl inequalities for largest eigenvalue of matrix sum

The $k^{\rm th}$ largest eigenvalue (arranged in decreasing order) of the sum of two $N \times N$ Hermitian (real symmetric) matrices $\bf{A}$ and $\bf{B}$ can be stated using the Weyl inequalities as ...
aslan's user avatar
  • 385
6 votes
1 answer
609 views

Attempt at applying linear programming to the partial sums of the Möbius inverse of the Harmonic numbers

Let $a(n)$ be the Dirichlet inverse of the Euler totient function: $$a(n) = \sum\limits_{d|n} d \cdot \mu(d) \tag{1}$$ and let the matrix $T(n,k)$ be: $$T(n,k)=a(\gcd(n,k)) \tag{2}$$ It has been ...
Mats Granvik's user avatar
  • 1,183
6 votes
1 answer
861 views

Is Binary Integer Linear Programming solvable in polynomial time?

The paper Solving the Binary Linear Programming Model in Polynomial Time claims that Binary Integer Linear Programming is in P. However, it seems that no subsequent literature in the mainstream has ...
aroyc's user avatar
  • 221
6 votes
1 answer
877 views

Is $O(10^{-6})$ an acceptable notation in numerical analysis? [closed]

The following question has been on math.SE for several days. Without having a satisfying answer, I'd like to ask the experts here. In mathematics, the big $O$ notation is used to describe the ...
user avatar
6 votes
3 answers
11k views

Maximum flow with negative capacities?

I'm trying to compute an (s-t) maximum flow through a network which includes a number of arc pairs ((u,v), (v,u)) that have equal, negative capacities (weights). I'm not aware of any efficient ...
Fumiyo Eda's user avatar
6 votes
1 answer
239 views

Numeric equality testing?

Suppose we have two closed-form expressions with $k$ unknowns which are hard to test for equality but easy to evaluate numerically over $\mathbb{R}^k$. One could then approach the problem of equality ...
Yaroslav Bulatov's user avatar
6 votes
1 answer
234 views

Stopping criteria for damped Newton iterations with backtracking line search

Are there better criteria than the Armijo criterion for damped Newton iteration with backtracking line search, when the objective is standard self-concordant? (See Boyd and Vandenberghe.) Let $F(x)$ ...
Sébastien Loisel's user avatar
6 votes
1 answer
779 views

If $\ell_0$ regularization can be done via the proximal operator, why are people still using LASSO?

I have just learned that a general framework in constrained optimization is called "proximal gradient optimization". It is interesting that the $\ell_0$ "norm" is also associated with a proximal ...
ArtificiallyIntelligent's user avatar
6 votes
4 answers
1k views

Applications of Hodge-de Rham Laplacian on p-forms ($p\neq 0,n$) in physics or engineering

The Hodge-de Rham Laplacian $L=(d+d^*)^2$, where $d$ is the boundary operator of the de Rham complex, is well-known in the math community. Recently, I tried very hard to search for examples of its use ...
Lizao Li's user avatar
  • 473
6 votes
3 answers
854 views

Error in Polynomial Root Finding Algorithm with Synthetic Division

I have written a program which finds the roots of polynomial using Newton's Method. After finding the first root to within a tolerance (note that this also finds complex roots), I use synthetic ...
Geekgirl's user avatar
  • 163
6 votes
1 answer
737 views

Rank of the absolute-value matrix $|M|$ vs. rank of $M$

Let $M$ be a real matrix of rank $r$ (and let us set $M=UV^T$, with $U,V^T\in\mathbb{R}^{n\times r}$, to fix the notation). Let $|M|$ be the matrix obtained by taking the absolute value of each entry ...
Federico Poloni's user avatar
6 votes
4 answers
2k views

Quantitative de Moivre–Laplace theorem (reference request)

The classical de Moivre-Laplace theorem states that we can approximate the normal distribution by discrete binomial distribution: $${n \choose k} p^k q^{n-k} \simeq \frac{1}{\sqrt{2 \pi npq}}e^{-(k-np)...
András Bátkai's user avatar
6 votes
1 answer
222 views

Computing $(AA\otimes BB + AB \otimes BA)^{-1}$

Can anyone suggest a way to numerically compute the following matrix vector product? $$u=A^{-1}b=(AA\otimes BB + AB \otimes BA)^{-1}\operatorname{vec}(C)$$ Here $AA,BB,AB,BA$ and $C$ are $d\times d$ ...
Yaroslav Bulatov's user avatar
6 votes
2 answers
1k views

Is the matrix positive definite given the Gauss-Seidel method converges?

I know that the Gauss-Seidel method converges given that the matrix you want to solve is symmetric positive definite. However, I'm wondering if the "converse" of the statement is true. Namely, if $A$ ...
bernard's user avatar
  • 205
6 votes
1 answer
797 views

Celestial mechanics and Runge Kutta methods

I am working on an example here to simulate the orbit of Earth for one year. As you can see in the notebook, RK45 doesn't conserve energy, and after one simulated year it has spiraled in substantially....
AllenDowney's user avatar
6 votes
1 answer
415 views

Difference stencils approximating Laplacian

Let $\Delta$ be the Laplace operator on the interval $[0,1]\subset \mathbb{R}$. Divide $[0,1]$ into small intervals of size $h$ to get an equidistant grid. One can approminate $-\Delta$ on this grid ...
Kira G.'s user avatar
  • 161
6 votes
4 answers
633 views

Expected value of a function over random sets

I am doing an analysis on the complexity of some set-related algorithm where the input is a random set. One of my setbacks can be formulated as follows: Pick $k$ distinct numbers out of numbers $[1,n]...
Matjaž Krnc's user avatar
6 votes
2 answers
973 views

Divergence of the Lagrange interpolation on the Chebyshev nodes

Faber theorem states that for every $\lbrace x_k^{(n)} \rbrace$ there exists a continuous function $f$ such that $\| f - L_n \|_{\infty} \not\rightarrow 0$, where $L_n$ is an interpolation polynomial ...
Benjamin's user avatar
  • 109
6 votes
1 answer
1k views

Arnoldi method to compute the dominant eigenvector

Hi, everyone! I have a problem of computing the dominant eigenvector. When I want to approximate the dominant eigenvector of a large sparse matrix via the famous Arnoldi method, I am wondering how to ...
jane's user avatar
  • 61
6 votes
2 answers
993 views

An orthogonal companion matrix

Let $P\in{\mathbb R}[X]$ be a monic polynomial with roots on the unit circle. For the problem below, we may assume wlog that the roots are simple and distinct from $\pm1$. It can be shown that there ...
Denis Serre's user avatar
  • 52.3k
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 ...
gmvh's user avatar
  • 3,065
6 votes
3 answers
615 views

What is the current fastest method to calculate Lerch's Phi transcendent?

Lerch's Phi transcendent is $$ \Phi(z,s,a) = \sum_{k=0}^{\infty} \frac{z^k}{(k+a)^s} $$ I am trying to compute this for the following parameters: $z$ is complex, $|z| \approx 1$ and $|z|$ < 1 (...
Mike Battaglia's user avatar
6 votes
1 answer
218 views

Any convergence rule for ${\mathbf X}_k={\mathbf A}{\mathbf X}_{k-1}{\mathbf B}$?

We know iteration ${\mathbf X}_k=\mathbf{A}{\mathbf X}_{k-1}$ converges if the spectral radius of $\mathbf A$ is smaller than 1 (see here). Is there any known rule for iteration ${\mathbf X}_k={\...
Tony's user avatar
  • 272
6 votes
2 answers
834 views

Symmetric matrix formula for Gauss-Legendre quadrature

While searching the web, I came across the following algorithm for the Gauss-Legendre quadrature. I wasn't able to find a reference or a proof of my own as to why it works. I'll present it, and the ...
Amir Sagiv's user avatar
  • 3,574
6 votes
1 answer
378 views

Compensated compactness for system of conservation laws?

As far as I knew, the method of compensated compactness can be used only for one-dimensional scalar and $2\times 2$ systems of conservation laws, i.e. $u_t+f(u)_x=0$. But if I understood correctly ...
Hamed's user avatar
  • 105
6 votes
1 answer
217 views

numerically track spectrum curves of a parameter dependent linear operator

Hi, I am interested in how to numerically track spectrum curves of a parameter dependent linear operator. Given a linear operator in square matrix form $M(t)$, where the matrix is smooth dependent of ...
bobye's user avatar
  • 135
6 votes
1 answer
155 views

Adding constraints as penalty with $\| \cdot \|_0$ norm

In the paper Image Denoising Via Sparse and Redundant Representations Over Learned Dictionaries (page 2), the authors rewrite the minimization problem \begin{align} \min_{\alpha \in \mathbb R^k} \| \...
TheWaveLad's user avatar
6 votes
1 answer
263 views

Algorithm that solves every Mixed Integer Linear Program (to optimality)?

Given a Mixed Integer Linear Program with rational coefficients (both for the objective functions and all constraints), is it always possible to solve it algorithmically? I know that you usually ...
J Fabian Meier's user avatar
6 votes
1 answer
216 views

Estimates of Hausdorff dimension (and its derivatives)

For example, the cookie cutter maps, say $T:I_1 \cup I_2 \subset [0,1] \to [0,1] $ is a $C^2$ map such that $|T'|>1$ and provided $I_1$ and $I_2$ are disjoint closed intervals and $T(I_i)=[0,1]$. ...
Nyima Kao's user avatar
  • 165
6 votes
1 answer
218 views

Approximating an iteratively defined function

Let $f_0,f_1,\ldots$ be a sequence of functions $f_n : [0,1] \rightarrow R$ defined as follows: $$f_0(x) =1+2x$$ $$f_{n}(x) := \left\{\frac{5+t}{2} : \text{ where t solves } f_{n-1}\left(\frac{x}{t}...
Mark Lewko's user avatar
6 votes
2 answers
3k views

Computational complexity of integration in two dimensions

I have an algorithm for solving a certain problem that requires that I compute two-dimensional integrals as a subroutine, and I'd like to make some kind of statement about its running time. Suppose $...
Joel Ford's user avatar
6 votes
2 answers
2k views

Computation of a Drazin inverse

I need to compute the Drazin inverse $A^D$ of a singular M-matrix $A$, i.e., a matrix in the form $A=\lambda I -P$, where $P$ has nonnegative entries and $\lambda$ is the spectral radius (Perron value)...
Federico Poloni's user avatar
6 votes
1 answer
1k views

Frozen coefficient method (von Neumann stability analysis)

Earlier it was considered that frozen coefficients method for Neumann stability analysis for finite difference scheme is more heuristic than rigorous. But I have read some information in a book by ...
cool's user avatar
  • 161
6 votes
2 answers
2k views

Numerical solution to diffusion-like equation with negative diffusion coefficient region?

I am trying to numerically solve the initial value problem (see later discussion for ICs) $$ x \frac{\partial f}{\partial t} = \frac{\partial}{\partial x} (1-x^2) \frac{\partial f}{\partial x} - f$$ ...
vojta havlíček's user avatar
6 votes
1 answer
443 views

Algorithm for numerically approximating the Prokhorov metric?

Question: What is known about algorithms for numerically computing/approximating the Prokhorov distance between two measures? Recall that the Prokhorov distance metrizes the topology of weak(-*) ...
Nick Alger's user avatar
  • 1,160
6 votes
1 answer
1k views

Condition number for a symmetric positive definite matrix

I am trying to get an estimate for the induced 2-norm condition number $\kappa_{2}(M)$ of this matrix $M$: $$M_{ij} = \frac{1}{(n-i)!(n-j)!(2n-i-j+1)} = \displaystyle\int_{0}^{1}\frac{x^{n-i}}{(n-i)!} ...
Abhishek Halder's user avatar
6 votes
1 answer
1k views

Speed up Linear programming

I have a linear programming problem like this: minimize $c^t X$ under the constraint that $AX \ge b$. I will need to solve this linear programming problem online many times. I need it to be as fast ...
Robert's user avatar
  • 83
6 votes
1 answer
1k views

Efficient computation of Markov chain transition probability matrix

Consider a continuous Markov chain $X = (X_t)$ on a finite state space and let $Q$ be the (given) transition rate matrix. This matrix is very sparse, with non-zero values on 3 diagonals only (so from ...
Johannes's user avatar
6 votes
2 answers
660 views

analytic approximation of a non-negative matrix by a sequence of positive matrices

Let $L \in \{0,1\}^{n \times n}$ be a non-negative matrix whose row sum is 1. ($L$ stands for "limit"). It is known that there exists a unique $r \times r$ principal minor of $L$ that is a permutation ...
Ngoc Mai Tran's user avatar
6 votes
0 answers
233 views

Newton type method for finite fields?

I have a polynomial $p(x)$ in $\mathbb{Z}/q\mathbb{Z}$ that is easy to compute for any $x$ but has an absurdly large degree $d > 2^{256}$. I know for a fact that it has a zero and I would like to ...
mtheorylord's user avatar
6 votes
0 answers
137 views

Why wavelet methods are not popular anymore in nonparametric statistics?

Back in my master years, I took a nonparametric statistics class. In this class, a few nonparametric methods were presented, but I remember spending a lot of times on methods based on wavelet ...
BabaUtah's user avatar
6 votes
0 answers
197 views

Where to cut off a double sum?

I have to compute a double infinite sum to within a given accuracy $\epsilon$. Let us say the sum is of the form $$\sum_{m\geq 1} \sum_{n\geq 1} \frac{a_{m,n}}{m^2 n^2 \max(m,n)},$$ where $|a_{m,n}|\...
H A Helfgott's user avatar
  • 20.2k
6 votes
0 answers
394 views

Numerical analysis with p-adic numbers

How should one go about doing numerical analysis with $p$-adic numbers? By that I mean, how should one go about implementing numerical integration (using analogues of Newton-Cotes or perhaps Gaussian ...
gmvh's user avatar
  • 3,065
6 votes
0 answers
320 views

Can we approximate any eigenvalue of an infinite matrix via eigenvalues of some sequence of submatrices which approximates the matrix?

Let $T:\ell^2\to\ell^2$ be a compact linear operator. Let $[T]=(a_{i,j})_{i,j=1}^{\infty}$ be the representing infinite matrix of $T$ with respect to the canonical base. Let $T_n$ be the finite rank ...
Chilote's user avatar
  • 596
6 votes
0 answers
206 views

Degree of Chebyshev polynomial necessary

In general, given $0<a<1$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[1-\frac{a}2,1+\frac{a}2]$ at every $x\in[1-a,1+a]$ and $f(0)=0$. What is minimum degree that is ...
Turbo's user avatar
  • 13.9k
6 votes
0 answers
97 views

Finding the optimal mixture of two convex functions

I am trying to find an efficient way to solve the problem $$\min_{p,x_1,x_2} p\cdot f(x_1)+ (1-p) \cdot f(x_2)~~~~~ s.t.\\p\cdot g_1(x_1) + (1-p)\cdot g_2(x_2)\leq 1 \\ 0\leq p \leq 1$$ where $x_1,x_2\...
Robert Lowell's user avatar
6 votes
0 answers
379 views

Numerical Methods for stochastic PDE, from rough paths to backward equations

this question is about some literary references regarding the state of the art in terms of numerical methods for SPDE's. In particular, Have the numerical implications, if any, of the results in ...
Sergio Almada's user avatar
6 votes
0 answers
317 views

Variant of orthogonal Procrustes problem

The orthogonal Procrustes problem seeks a matrix $M$ that minimizes $||AM-B||_F$ subject to $M^TM=I$, where $M$ is $d\times d$ and both $A$ and $B$ are $n\times d$. Geometrically, $M$ rotates a set of ...
Matt's user avatar
  • 61
6 votes
0 answers
565 views

What are the eigenvectors of the Lagrange interpolation matrix?

Let $F$ be a field. Let $x_1,\ldots,x_k,y_1,\ldots,y_k\in F$ be distinct elements in the field. Consider the $k\times k$ matrix that in position $i$, $j$ has the element $\frac{\prod_{l\neq i}(y_i - ...
user17119's user avatar
  • 179
5 votes
4 answers
2k views

Determining a recurrence relation

I would like to solve the general problem of determining a linear recurrence relation that fits a given integer sequence of length $n$, or stating that none exists (with fewer than $n/2-k$ ...
Charles's user avatar
  • 9,114

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