Numerical algorithms for problems in analysis and algebra, scientific computation

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4
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1answer
586 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 ...
8
votes
0answers
404 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 ...
9
votes
3answers
424 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
129 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
397 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
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0answers
61 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
307 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
75 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
60 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
412 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 ...
21
votes
3answers
876 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
222 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
260 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
148 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
192 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
230 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
158 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
372 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
232 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
125 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
76 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 ...
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vote
0answers
62 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
618 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
190 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 ...
12
votes
1answer
1k 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
210 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
942 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
111 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
1answer
115 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
1answer
195 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
1answer
490 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
0answers
179 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
0answers
155 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
1answer
89 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
1answer
212 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
0answers
127 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
1answer
258 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
1answer
694 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
0answers
45 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
2answers
649 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
0answers
114 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
0answers
75 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? $\int_a^b f (\log x) \mathrm{d}x$
16
votes
3answers
831 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 ...
5
votes
0answers
131 views

Numerical linear algebra: how to compute $B^TC^{−1}B$ efficiently

Hi, my question is similar to this one. I have to compute $B^TC^{−1}B$, where $C$ is a strictly positive definite $n\times n$ matrix and $B$ is $n\times m$. The matrix $C$ is huge ($n$ up to a ...
13
votes
2answers
287 views

Condition number of matrix after partial orthogonalization

I'm wondering about which bounds one can put on the condition number of a $n\times n$ square matrix which is obtained from another $n\times n$ square matrix by orthogonalizing the first $m < n$ ...
3
votes
0answers
132 views

Upper bound on integrals of Legendre polynomials

Hi, If $P_n(x) $ is unnormalized shifted Legendre polynomial, and $g_{n,m}(x) = \int_0^x P_n(x_1)x_1^m dx_1, n>m $ then what is the upper bound $ |g_{n,m} (x)|_{max} , x\in (0,1) $ as a function ...
5
votes
0answers
415 views

Parabolic cylinder functions - explicit estimates?

I need estimates for the parabolic cylinder functions $U(a,z)$ (first studied by Whittaker). Most work in the literature focuses on $a$ real. As it happens, I am interested in $U(a,z)$ on a strip in ...
7
votes
2answers
285 views

How to solve a system of linear equations without storing the matrix?

I have a procedurally defined Hermitian matrix $M$, i.e. I can get any matrix element by calling a black box function (e.g. a library function), and a vector $Y$. And I have to solve a system of ...
4
votes
4answers
723 views

When we use Bernstein polynomials in application

When it is preferable to use Bernstein polynomials to approximate a continuous function instead of using the only following preliminary Numerical Analysis methods: "Lagrange Polynomials", "Simple ...