Numerical algorithms for problems in analysis and algebra, scientific computation

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10
votes
1answer
216 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 ...
0
votes
0answers
26 views

Monotonicity per dimension of multivariate scattered data

For my thesis, I am working on interpolation using the RBF method (Radial Basis Functions). Before interpolating, I want some a priori insight into the data, for example check in which dimensions it ...
5
votes
1answer
76 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 ...
1
vote
1answer
133 views

Iterative Root Finding

Consider a function $f(x)=g(x,h(x))$, which we know has a unique root $x^*$. The functions $f$, $g$ and $h$ are all continuous in $x$ and behave nicely. Iteratively solving $g(x_{i+1}, h(x_i))=0$ with ...
4
votes
1answer
146 views

How to calculate log or exp of a value in GF(2^n) using log/exp table of GF((2^k)^m) where n=k*m?

Consider Galois fields $\mathbb{F}_{2^n}$ and $\mathbb{F}_{2^k}$, where $n=km$, and $\mathbb{F}_{2^k}$ is a ground field of $\mathbb{F}_{2^n}$. I’d appreciate pointers to papers or suggestions on: ...
2
votes
1answer
108 views

Does Gaussian Quadrature actually refer to Gauss-Legendre Quadrature?

When the term Gaussian Quadrature appears in most Literatures, does it actually refer to Gauss-Legendre Quadrature. In other words, do they implicitly admit that they use the Legendre orthogonal ...
2
votes
0answers
127 views

An alternative to the Euler--Maclaurin formula: Approximating sums by integrals only

The Euler--MacLaurin summation formula can be written as $$ \sum_{i=0}^{n-1} f(k)\approx \int^{n-1}_0f(x)\,dx + \frac{f(n-1) + f(0)}2 + \sum_{j=1}^m\frac{B_{2j}}{(2j)!}[f^{(2j - 1)}(n-1)...
0
votes
0answers
34 views

Numerical integration over a cube with non-product weight

Numerical integration over an interval with (well-behaved) weight functions is a research area that has received considerable attention in the past centuries. Any cubature formula over a interval ...
0
votes
1answer
64 views

The classical two phase Stefan problems

What is the most commonly used treatment method of the moving interface in the classical two phase Stefan problems with the finite element method. Here I mean the water-ice two phase problem under ...
1
vote
0answers
142 views

Is there a brute force method for determining irreducible representations?

Suppose I have some groups $G_1$, $G_2$, $G_3$, etc... Then the direct product is given by $G = G_1 \times G_2 \times G_3 \ldots$ I know that the sub-representations of a reducible representation ...
0
votes
0answers
35 views

On the computation of generalized eigenvalues of a low-rank approximation using SVD

I have trouble deriving an expression of the generalized eigenvalues of a matrix pair, found in http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4618700 . The setup is the following and can ...
2
votes
0answers
65 views

Natural neighbor interpolation

Recently I am interested in Natural neighbor interpolation, that is : Given a function $P(x)$ and some interpolation points $\{x_i,P(x_i)\}_{i=1}^N$, we have the interpolation function $$P^*(x)=\sum_{...
20
votes
2answers
656 views

Distribution of the Error term in GH Hardy's “curious result” $\sum_{\nu \leq n } \{ \nu \theta \}^2 = \tfrac{1}{12} n + O(1)$

In an early paper, GH Hardy talks about the distribution of "curious" sum: $$ \sum_{\nu \leq n } \{ \nu \theta \}^2 = \tfrac{1}{12} n + O(1)$$ where $\{x\}:=x-\left \lfloor x \right \rfloor -1/2$. ...
0
votes
0answers
177 views

Boundary conditions in the Finite Element Method

I just want to solve a Sturm-Liouville problem in 1D, i.e., \begin{align} (p(x)u'(x))'+q(x)u(x) = f(x) \end{align} with boundary conditions \begin{align} u(0)=a \hspace{1cm} u'(0)=b \end{align} How do ...
3
votes
1answer
187 views

Evaluating elliptic integrals

I am interested in evaluating some elliptic integrals, and I have not been able to secure a reference to do exactly what I need. Most of the references I've found seem to focus on reducing more ...
1
vote
1answer
175 views

Stone-Weierstrass Theorem, polynomial interpolation, divided difference in complex plane

Setting: Let $\Gamma$ be a simple smooth($C^\infty$) curve in $\mathbb{C}$ parametrized by the injective map $\gamma:[0,1] \to \mathbb{C}$. Assume $f$ is a function defined on $\Gamma$ s.t. $f$ is ...
1
vote
1answer
161 views

Values of the completed Riemann $\xi(1+it)$ for small t?

I'm editing this question heavily for clarity: I am looking for methods to compute $\zeta(1+it)$, or the (partially) completed Riemann zeta function $$\pi^{-s/2}\Gamma(s/2)\zeta(s)$$ along the line ...
1
vote
0answers
105 views

Finding an error estimation for the De Moivre–Laplace theorem with Stirling's formula

Context for my question: For one part of my thesis I try to find an upper bound for the error in the normal approximation of the binomial distribution following the standard proof of the De Moivre–...
5
votes
0answers
101 views

On existence of rapid Arithmetic geometric procedure?

We know that $\pi$ can be computed by Arithmetic Geometric mean using Gauss-Legendre procedure which does provide fastest convergence rate as well with a guarantee of $2^n$ bits of $\pi$ at $n$th ...
2
votes
0answers
81 views

Computing the density of a set of multiples

Erdős and his coauthors often wrote about problems relating to the densities of sets of multiples. I have a computational question about the same topic. I have a finite* set $A=a_1<\cdots<a_r$ ...
14
votes
3answers
1k views

Current Research in Numeric Mathematics

To me, as an non-expert in the field, it seems as if numeric mathematics should have lost its importance because nowadays symbolic calculations or calculations with unlimited precision are generally ...
-2
votes
1answer
272 views

Calculating a sum including large numbers [closed]

Let $\theta(x)=\begin{cases} 0 & \text{ if } x<0 \\ 1 & \text{ if } x\ge 0 \end{cases}$ Do you know any way to calculate this number: $$\sum_{r=493701}^{506199}\sum_{k=0}^{100}(-1)^k\...
2
votes
0answers
48 views

Find Moment condition for generalized method of moments

Consider a scalar system with 2K outputs and K+2 unknowns $y_{k,1}=x_ka_1+n_{k,1} \quad y_{k,2}=x_ka_2+n_{k,1}$. The variables $n_{k,\ell}$ are zero mean noise variables. To estimate $a_1$ and $a_2$, ...
0
votes
0answers
39 views

How to implement conjugate gradient method to minimize this nonlinear action?

Given a 2D stochastic differential equation: \begin{align} \dot{x}_{i}=f_{i}(\textbf{x})+g_{ij}\xi_{j}(t), \end{align} where $i=2$, $g_{ij}g_{jk}=2\epsilon\delta_{ik}$ and $f(\textbf{x})=-\nabla\phi(\...
1
vote
1answer
115 views

Solving Schroedinger Equation for the electronic energies of the Molecular Ion Hydrogen H2+ in the Elliptic coordinate system [closed]

Electronic Energies of Molecular Ion Hydrogen $H_2^{+}$ $r_1$ is the distance between the proton $1$ and the electron. $r_2$ is the distance between the proton $2$ and the electron. $R$ is the ...
0
votes
1answer
130 views

Numerical methods for solving a hyperbolic nonlinear PDE

What type of numercial methods are there to solve PDE of the sorts of: $$f(x,t,u(x,t))u_{xx} - g(x,t,u(x,t))u_{tt} = F(x,t,u(x,t))$$ $$u(x,0)=G_1(x) , \frac{\partial u(x,0)}{\partial t}=H_1(x) ,u(0,t)...
1
vote
0answers
93 views

Closed form answer to a naive integral [closed]

Let a and b be positive real numbers. How to find a closed form answer to the integral $$\int_0^t \left(-a t + \big(1+ \dfrac{2bt}{3}\big)^{-3/2}\right)^{5/3} dt$$ If it is not possible to find a ...
12
votes
1answer
164 views

Interval arithmetic with different definitions of intervals

Interval arithmetic normally deals with intervals defined as $[a,b]$ with rules like $$[a,b]+[c,d]=[a+c,b+d]$$ I am interested in interval arithmetic with different interval definitions such as $$\{a\}...
4
votes
0answers
112 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 ...
0
votes
2answers
130 views

The condition number of a scaled Vandermonde matrix

Let $V(x_1,..,x_n)$ be the Vandermonde matrix induced by $x_1,..,x_n$, and let $\tilde{V} := V(\frac{x_1}{h},...,\frac{x_n}{h})$. My intuition says that the condition number should be invariant under ...
0
votes
1answer
87 views

Fit a system of linear ODEs from several experiments

Assume we are given several initial vectors $x^{(1)},\ldots,x^{(r)} \in \mathbb{R}^n$, where the dimension $n=6$ (in any event a number below 10) , and the number of initial vectors $r$ is in the ...
1
vote
0answers
65 views

Level Set Advection with Extension Velocity

We're studying the following system of PDEs for a scalar function $F(x, t)$ with $x \in \mathbb{R}^3$ and $t \in \mathbb{R}$. The function $F(\cdot, t)$ is a level set function for a time-dependent ...
-2
votes
1answer
134 views

Solving a nonlinear PDE numerically

I want to solve numerically the following PDE: $$ u_x + u_t - (u_{xt})^2 = u(x,t) $$ The boundary conditions are no concern of mine, pick the ones that work. So which numerical method will work for ...
4
votes
1answer
135 views

Solving over-determined system of polynomials

I am trying to solve the following over determined system of polynomials \begin{align} & p_1(x_1,x_2,\ldots,x_n)=0, \\ & p_2(x_1,x_2,\ldots,x_n)=0, \\ & \vdots \\ & p_m(x_1,x_2, \...
2
votes
0answers
127 views

What am I missing in this highly oscillatory integral? [closed]

I want to numerically integrate this equation (in python): $\int_{0}^{\infty}{\rm d}k f(k) J_v(r k)J_v(s k) $, where f(k) is a non-smooth function, and $J_v$ are the Bessel function of the fist kind....
3
votes
1answer
219 views

Are piecewise linear functions dense in $W^{1,\infty}$?

Are piecewise linear functions dense in $W^{1,\infty}$ ?
0
votes
0answers
46 views

Time-stepping numerical scheme for the advection dispersion equation

I am facing a simple (at first glance) problem. I need to implement a numerical scheme for the solution of the first order wave propagation equation with chromatic dispersion included. My original ...
3
votes
1answer
497 views

What is the rate of convergence? [closed]

How quickly does the series defined by $$x_0 = 0, \ x_{n+1} = \frac{x_n^2+1}{2}$$ converge to $1$?
1
vote
1answer
65 views

Finding t vlaue in Bezier curve [closed]

According to this question, I'm looking for some method to find the t value in Quadratic bezier curve equation: $$ B(t)=P_0+t(1-t)P_1+t^2P_2 \space \space where \space 0 ≤ t ≤ 1 $$ In this ...
1
vote
0answers
54 views

Monotonic convergence of Newton's method for boundary value problems

I’m interested in solving nonlinear elliptic boundary value problems of the type $$ -a\Delta u + f(u) = 0, $$ $$ u|_\Gamma = u_0 $$ by Newton’s method when its convergence is global and monotonic. ...
16
votes
1answer
1k views

Why is there a $\sqrt{5}$ in Hurwitz's Theorem?

Hurwitz's theorem is an extension of Minkowski's Theorem and deals with rational approximations to irrational numbers. The theorem states: For every irrational number $\alpha$, there are infinitely ...
1
vote
0answers
29 views

Open volumetric time series data set

Does anyone know where I can find a good open volumetric time series data set? I had a look at some of Stanford's open data sets (https://graphics.stanford.edu/data/voldata/ ) But these do not seem ...
2
votes
2answers
97 views

Fixed point iteration on symmetric biconvex function

Suppose $X\subseteq\mathbb{R}^n$ is a convex set and that a function $g(x,y):X\times X\rightarrow\mathbb{R}_+$ is smooth, "strictly biconvex" (strictly convex in $x$ and $y$ independently but not ...
0
votes
0answers
33 views

Weighted Perturbation Bound for Polar Decomposition

Setup: Let $X\in\mathbb{R}^{n\times r}$ be a matrix with orthogonal columns, with $\Sigma = X^TX$, and assume that $\Sigma$ is invertible (note, $\Sigma$ is not necessarily the identity). Suppose we ...
3
votes
0answers
107 views

Determining Nullspace Basis such that only one column is deleted or added as row is added or deleted, and remaining columns of basis stay the same

I would like to compute, in MATLAB, the basis Z for the nullspace of an m by n matrix A, such that if one row of A is added (resulting in A_a), the basis for A_a is n-m-1 of the n-m columns of Z, i.e.,...
7
votes
1answer
288 views

Can I find the gap between the two least eigenvalues of this special matrix A(t)?‎

I am interested in finding the gap between the two least eigenvalues of $A(t)$, a Hermitian $N\times N$ sparse ‎matrix whose diagonal elements are $a_it+b_i\,(1\leq i\leq N)$, and all off-diagonal non-...
1
vote
1answer
88 views

Convergence of sequence of polynomials defined by boundary conditions

I'm sorry if my question sounds trivial, but analysis is not my field. Consider the interval $[a,b]\subset \mathbb{R}$. On $[a,b]$, for every $n\in\mathbb{N}$, $n\ge 3$, I define the polynomials $P_n:...
2
votes
1answer
90 views

Numerical solution of singular ODE

Consider the singular ODE $y''+\frac{y'}{r}+p(r)y=0 \ \ with \ \ y(0)=1 \ \ and \ \ y'(0)=0$. Theoretically such solution exists and is unique if $p$ is nice. Is there a method to numerically ...
3
votes
2answers
531 views

Who is currently researching topics concerning applying algebraic topology and/or differential geometry to numerical methods? [closed]

I am interested in pursuing a PhD in mathematics from a top ranked university with a faculty member researching something akin to the following description: applications of algebraic topology and/or ...
0
votes
2answers
72 views

Solving sparse linear least squares or a positive definite 5-band matrix system fast

I want to quickly solve linear least squares problem for $x \in \mathbb{R}^n$ $$min_x \left\| A x - b \right\|_2^2$$ with a special sparse structure where each row in $A$ has only up to 4 ...