Numerical algorithms for problems in analysis and algebra, scientific computation

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6
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3answers
673 views

Square Root Algorithm

I would like an efficient algorithm for square root of a positive integer. Is there a reference that compares various square root algorithms?
7
votes
3answers
366 views

accelerating convergence of a class of sequences

Do any of the standard methods of acceleration convergence of series, when applied to the series $1 - 1 + 1/2 - 1/2 + 1/3 - 1/3 + ...$, give convergence to 0 with error $o(1/n)$? I tried applying ...
2
votes
2answers
318 views

What is the definition of an antilimit?

I've seen some references to antilimits in the numerical analysis literature, but no definition of the term. The impression I get is that in specific contexts where every sequence $x_0,x_1,x_2,\dots$ ...
1
vote
1answer
154 views

Understanding the rationale behind “batch means” estimation

Hello all, I am implementing an MCMC algorithm for my work, and I've come upon something in the literature which I just can't understand. Specifically, I am attempting to estimate the amount of ...
4
votes
0answers
136 views

Are there some numerical test to check if a map is a contraction?

Let's say I have a multivariate function $$ f:D \to D, D \subset \mathbb R ^n, D \text{ compact}, $$ for which there is no closed form. That is the only way to evaluate the function is to do it ...
4
votes
1answer
425 views

Numerical multivariate definite integration

I need to compute a set of multivariate definite integrals with infinite integration domain $$\displaystyle \int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} f(x_1,x_2, \ldots , x_n)\;\;dx_1 ...
2
votes
0answers
200 views

minimize a cost function with matrix traces

Hi, I have a cost function of the form $$F(X) = \operatorname{tr}(X'AX)+\operatorname{tr}(X'B),\quad\textrm{ s.t. }X'X=I.$$ $X$ is a $m\times n$ matrix, ($m>n$), with orthonormal columns. $A$ is ...
2
votes
1answer
88 views

non convex quadratic optimization

Hi I would like to optimize the following system: $$\min_{q,\|q\|=1} \sum_i^n |q^T M_i q|$$ More details: the size of the unknown vector q is $4\times 1$, M_i is a matrix of size $4\times 4$. It is ...
4
votes
3answers
333 views

Lower bound for sum of square root of the degrees of a connected graph

By Cauchy-Schwartz and the handshake lemma, it is easy to see that $\left( \sum_{i=1}^n \sqrt d_i \right)^2 \leq n \sum_{i=1}^n d_i =2mn$, with equality iff the graph is regular (constant degree). ...
1
vote
1answer
154 views

Discretizing a cosine function?

I'd like to start by noting that for some fixed natural $N$ basis functions for my system will be generated by $f(k,x)$ as defined and explained here or in numerous other sources: $$f(k,x) = \sqrt2 ...
0
votes
1answer
129 views

How to handle a scalar product in an integral?

I am having a problem with a certain inequality I try to understand. I think it's just a basic idia (/trick) I'm missing, but I can't seem to find it. Here's a simplification of the problem: $ ...
0
votes
0answers
70 views

Approximate closed-form solution for a recurrence

Find an (approximate) closed-form solution for $S(m, b)$. $$S(m,b)=\sum_{i=0}^{\lfloor (e-1)/2\rfloor}{e \choose i}S(m-1, b-i) \quad + \sum_{i=\lfloor (e-1)/2\rfloor+1}^{\min(b,e)}{e\choose ...
4
votes
0answers
196 views

Inadmissibility of Simpson's rule

(An earlier version of this at stackexchange got no answers.) Bayesianism says that all uncertainties, or at least all uncertainties about the truth or falsity of propositions, can be expressed by ...
3
votes
1answer
72 views

Conjugate gradient algorithm where first search direction is not equal to residual

In usual formulation of conjugate gradient algorithm initial search direction is taken to be the residual (so residual and search direction spans Krylov subspace). However, in cases where inexact ...
32
votes
2answers
1k 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 ...
7
votes
1answer
163 views

Problems where Conjugate gradient works much better than GMRES

I am interested in cases where Conjugate gradient works much better than GMRES method. In general, CG is preferable choice in many cases of SPD because it requires less storage and theoretical bound ...
11
votes
2answers
412 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 ...
3
votes
1answer
125 views

The discrete theory of compressible fluids dynamics

I am working on the discrete theory of compressible fluids dynamics, i.e., numerically solving and simulating the compressible fluids , we are interested in the way using discrete exterior calculus, ...
7
votes
1answer
143 views

Who first observed that Conjugate Gradient for Symmetric Positive Definite linear systems is a Krylov method?

Conjugate gradient was originally presented in the 50's before the modern understanding of Krylov subspaces (and the resulting iterative methods) was fully realized. As such, the method was derived ...
3
votes
1answer
495 views

Stability in algebraic geometry

Suppose I have a collection of polynomials with multiple variables (more polynomials than variables, say), and I'm given noisy versions the values of these polynomials at a certain unknown point. I ...
10
votes
0answers
171 views

What are the difficulties in proving almost-everywhere stability of Gaussian elimination?

It is well known that Gaussian elimination without pivoting is numerically unstable, and in practice Gaussian elimination is done with row pivoting (partial pivoting). A theorem of Wilkinson states ...
5
votes
1answer
731 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 ...
7
votes
0answers
504 views

American put option pricing by “binomial trees”

Dear MO World, I'm teaching a financial mathematics course and have found a fascinating (to me) numerical phenomenon and wonder if anyone has studied it, or knows anything similar. I'll try and give ...
10
votes
3answers
480 views

Rapid evaluation of multivariate normal integral

I'm implementing a model that requires me to numerically evaluate a multivariate normal integral of the following form $$\int_{-\infty}^\infty \phi(z)\displaystyle\prod_{i=1}^N \Phi(a_iz+b_i) \, ...
3
votes
1answer
145 views

Using Fourier Transform to speed up calculation of forces following an inverse square law

Suppose I have $n$ electric point charges in, say, two dimensions. Is there any algorithm (and I have a hunch that it might be related to the Fourier transform) to compute the net forces that act on ...
15
votes
1answer
481 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 ...
1
vote
1answer
167 views

Dragilev method

What are the advantages and disadvantages of the Dragilev ( http://www.mathnet.ru/php/person.phtml?&personid=32359&option_lang=eng and ...
8
votes
1answer
365 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 ...
2
votes
0answers
76 views

Application and relevance of Sobolev gradients

The Sobolev gradient concept has been developed in the 1970s, with a first publication in 1985, and an introduction can be found at: Ranka I would like to learn how strong the impact of Sobolev ...
2
votes
0answers
65 views

Approximating solutions to minima of the discrete Lagrangian

I have been stuck on this problem for a week and I'm not sure whether or not it is hard or I'm just missing something obvious. General gist of the problem I have a variational problem on a ...
1
vote
1answer
497 views

Software to numerically solve partial differential equation

When we use software to numerically solve differential equation, for example, using finite difference, finite element or finite volume methods, etc., is it possible that people input differential ...
23
votes
3answers
983 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 ...
2
votes
2answers
277 views

Computing hypergeometric function of matrix argument

In the context of the Bingham probability distribution the ${ }_1F_1$ hypergeometric function of matrix argument naturally arises as a normalization constant of the probability distribution function. ...
6
votes
2answers
270 views

Rigorous numerics for maxima and minima (one variable)

Let $f:\mathbb{R}_0^+\to \mathbb{R}$ be defined by some combination of the four basic operations and square roots. (The argument of square-roots is assumed is to be non-negative, and the value of ...
1
vote
1answer
183 views

integral basis of orthogonal complement

Suppose there are $r$ linearly independent vectors $v_1,\dots,v_r\in \mathbb{R}^n$, all of them have integer-valued entries and $\|v_i\|_\infty\leq m$ for some integer $m$. My goal is to find an ...
2
votes
0answers
226 views

eigenvalues of the sum of a stochastic matrix and a diagonal matrix

Let $D$ be a real diagonal matrix $D=diag(a_1,a_2,\ldots,a_n)$ with $a_1\le a_2\le\ldots\le a_n$. Assume that at least one of the $a_i$ is positive. Let $P$ be an irreducible, real, row-stochastic ...
3
votes
2answers
278 views

Questions on Discrete Exterior Calculus in numerial computing

I have several questions about the Discrete Exterior Calculus (DEC) in the numerical method for solving partial differential equation in physics: (Discrete Exterious Calculus is the newly developed ...
1
vote
1answer
126 views

For what values of the parameter does this function have an elementary anti-derivative?

I am a grad student working on some independent research trying to derive some exact formulas for a particular class of power series. During my study I came across the following integral which would ...
2
votes
1answer
168 views

Reducing the error of Algorithms by assigning variables formulas instead of values

Let me first give the intuition for my question: Suppose that you want to use a ruler to mark $n$ points in a line on a page, with 1 cm distance between neighbor points. There are two ways: 1- Mark ...
4
votes
1answer
457 views

Hilbert Matrix and Approximation Theory

I was reading about the Hilbert matrix and Cauchy determinants: \[ \det \left[ \frac{1}{i+j-1} \right]_{i,j} \] By guessing where this determinant is $0$ or $\infty$ we can guess the right formula. ...
5
votes
0answers
253 views

Approximation by polynomials

The following is a well-known theorem (see e.g. The Chebyshev Polynomial by Rivlin): If $p(x) = x^n + a_{n_1} x^{n-1} + \ldots + a_0$, then $\max_{-1\leq x \leq 1} |p(x)| \geq 2^{1-n}$ for $n \geq 1$ ...
2
votes
1answer
134 views

Optimization problem

I'm trying to solve a very practical optimization problem and I think I hit a dead-end. There are $N$ products ($N \sim 50$). Each product can have a price $p_i$ in range between 1 and 40 dollars. ...
0
votes
1answer
88 views

Avoiding epsilon in mixed integer linear and quadratically constrained programs

I would like to represent the following constraint as MILP constraint where $x \in [a, b]$ with fixed $a, b \in \mathbb{R}$ and $y \in \lbrace 0, 1 \rbrace$. $(x = 0 \wedge y = 1) \vee (x \neq 0 ...
1
vote
0answers
76 views

Orthogonal projection of discontinuous piecewise polynomial space in energy scalar product

Let $I = [0,1]$ be the unit interval Let $I$ be partioned into $n$ closed subintervals $(I_j)_J$, each of length $1/n$. Let $X_{DC} = \{ v \in L^2[0,1] | 1 \leq j \leq n : v_{|I_j} \in \mathcal P_1( ...
9
votes
2answers
672 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 ...
0
votes
2answers
204 views

Proof that polynomial evaluated at roots of unity is DFT [closed]

Hello All, I hope I am not abusing the forum here. I am just trying to understand the efficient implementations of the fast fourier transform. My reading and searching has led me to understand that ...
14
votes
1answer
2k views

The unreasonable effectiveness of Pade approximation

I am trying to get an intuitive feel for why the Pade approximation works so well. Given a truncated Taylor/Maclaurin series it "extrapolates" it beyond the radius of convergence. But what I can't ...
4
votes
1answer
258 views

best rank r approximation for non-Frobenius norm

The best rank $r$ approximation to a given matrix $M$ in Frobenius norm, according to Eckart-Young theorem, is truncated SVD - just keep $r$ largest singular values. What if I need to construct best ...
2
votes
2answers
1k views

Finding the formula for Bezier curve ratios (hull/point : point/baseline)

Given a cubic Bezier curve defined by points p₁, p₂, p₃, and p₄, a point B on that curve at some t value (where 0 ≤ t ≤ 1), a point A on the line (p₂ — p₃) at distance ratio t from p₂, and a point C ...
1
vote
1answer
127 views

Recommendations for binomial system solver

I am interested in solving binomial systems of the form $$ \begin{cases} a_1 x_1^{d_{11}} x_2^{d_{12}} \cdots x_n^{d_{1n}} + b_1 x_1^{d_{11}} x_2^{d_{12}} \cdots x_n^{d_{1n}} &= 0 \\\\ ...