Numerical algorithms for problems in analysis and algebra, scientific computation

**33**

votes

**2**answers

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

**1**answer

171 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

**2**answers

452 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

**1**answer

131 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

**1**answer

153 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

**1**answer

513 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

**0**answers

186 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

**1**answer

809 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

**0**answers

573 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

**3**answers

588 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

**1**answer

150 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

**1**answer

517 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

**1**answer

182 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

**1**answer

393 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

**0**answers

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

**0**answers

73 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

**1**answer

681 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

**3**answers

1k 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

**2**answers

348 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

**2**answers

283 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

**1**answer

210 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

**0**answers

241 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

**2**answers

315 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

**1**answer

127 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

**1**answer

184 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

**1**answer

491 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

**0**answers

260 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

**1**answer

145 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

**1**answer

91 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

**0**answers

82 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

**2**answers

699 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

**2**answers

228 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 ...

**16**

votes

**1**answer

3k 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

**1**answer

290 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

**2**answers

2k 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

**1**answer

137 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 \\\\
...

**3**

votes

**1**answer

127 views

### approximation methods in integral equations

Recently I was reading about integral equations and I am a beginner in it. There was a constant reference to the non-availability of methods to find the exact solutions and hence lot of approximation ...

**1**

vote

**1**answer

202 views

### How are real-analytic functions encoded in computer algebra?

I would like to know how are encoded the real-analytic functions on the interval by the computers. When I think in a real-analytic function I just think in a composition of the ''typical'' analytic ...

**4**

votes

**1**answer

585 views

### Matrix perturbation theory

I am having matrix $M_0$ with coresponding eigenvectors and 4 eigenvalues {0,0,a,-a}. Eigenvalue $\lambda=0$ is double degenerated. Now I am appliing small perturbation $\epsilon M_1$ and want to get ...

**2**

votes

**0**answers

199 views

### Convergence rate of iterated nonlinear equations?

For $i=1, \dots, n$ ($n$ could be large) we have variables $x_i$ and $y_i$ relating to probability bounds s.t. $x_i, y_i \geq 0, x_i+y_i \leq 1 \; \forall i$. Each $i$ has a constant $\theta_i$, and ...

**2**

votes

**0**answers

217 views

### Heat transfer coefficients - correlation equations for Nusselt number - vertical pipe

I need a heat transfer coefficient applicable to free convection of water INSIDE a vertical pipe. I have the correlation equation from Churchill and Chu, but it is only valid for D/L>=35/Gr^(1/4), ...

**3**

votes

**1**answer

115 views

### Kronecker-structured matrix kernel

Let $A,B\in\mathbb{C}^{n\times 3n}$ be two matrices, and denote the Kronecker matrix product by $\otimes$. The matrix
$$
M=
\begin{bmatrix} A \otimes I_n \\\\ I_n \otimes B\end{bmatrix}
$$
has size ...

**2**

votes

**1**answer

229 views

### Approximation theory under $L_1$-error

Is there a reference for results in approximation theory of bounded functions of one (and multiple) variables under $L_1$-error?
Formal definitions for functions of one variable are below.
Let $C$ ...

**2**

votes

**0**answers

138 views

### Radius of convergence to be proved more precisely (differential equation)

There is a differential equation in polar coordinates:
$r'^2+r^2=(kt)^2$, $r(t=0)=0$, k- Const.
It is possible to get a solution which is a power series (see below). However, I am looking for an ...

**3**

votes

**1**answer

351 views

### Lebesgue constant as condition number of polynomial interpolation

Let $T = \{ x_0,\ldots,x_n \}$ be a set of $n+1$ different points in the real interval $[a,b]$. Let $X_T$ be the associated interpolation operator on $C[a,b]$: it takes a function $f \in C[a,b]$ into ...

**4**

votes

**1**answer

1k views

### Smoothing L1 norm, Huber vs Conjugate

I'm trying to minimize a convex (not necessarily strictly convex) function involving an L1 norm (similar to lasso), which makes it non-differentiable at some points. So I'd like to smooth it and treat ...

**4**

votes

**0**answers

51 views

### Recovering Shared Eigenvector Set

Suppose we are given a set of $M$ pairs $\{(\vec{x}^{(i)},\vec{y}^{(i)})\}$, with
$\vec{x}^{(i)}\in\mathbb{R}^N$,
$\vec{y}^{(i)}\in\mathbb{R}^N$,
$M\gg N$ such that
$\vec{y}^{(i)} = Q^{(i)} ...

**6**

votes

**2**answers

886 views

### Approximating erf by tanh

It appears to be well-known that $\tanh(x)\le \mathrm{erf}(x)$ on $[0,\infty)$. It's off-handedly mentioned here, for example. Where can I find a formal proof? On the one hand, it's hard to imagine ...

**2**

votes

**0**answers

188 views

### integrate square of bessel 1st and 2nd kind

Has anyone ever solved the integral of square of Bessel and Neumann functions? This is for standing wave analysis on a cylindrical object.
I need to integrate $(AJ(kr) + BY(kr))^2$ from $r=a$ to ...

**1**

vote

**0**answers

88 views

### Integrating B-Spline composed with log

If $f$ is a real B-Spline and $a, b$ are real numbers, then is there a numerically stable way to evaluate the following expression?
$\int_a^b f (\log x) \mathrm{d}x$