Numerical algorithms for problems in analysis and algebra, scientific computation

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33 views

What's the advantage of majorization-minimization (MM) algorithm [on hold]

The majorization-minimization (MM) algorithm is a framework for convex and nonconvex optimization. When applied to nonconvex optimization, the MM algorithm solves a sequence of convex problems to ...
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0answers
89 views

Evaluate a Function to Full Machine Precision [closed]

If we want to evaluate $$f(x)=\frac{e^x-1-x}{x^2}$$ then we have to observe its large relative error as $x\to 0$. My question is that how can we find a method so that we can compute $f(x)$ to full ...
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25 views

state-of-art numerical contour (complex) integration method when contour is square and available values are evenly spaced

What is current state-of-art for numerical contour integration method (for $f(z)$ with $z$ being complex number and $f$ complex-valued) when contour is square on complex plane, and one only has ...
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0answers
22 views

numerical differentiation of sum of one-dimensional sinusoids with angular frequency close to Nyquist one

Suppose that $f(t) = \sum_i C_i e^{i\omega_i t}$, and $f$ is sampled at certain sampling angular frequency $\omega_s$. All $\omega_i$s are very close to $\omega_s/2$, and thus standard finite ...
3
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1answer
48 views

Injectivity of vector functions: Numerical Verification

Problem Setup Let $f:A\rightarrow B$, be a continuous function, $A\subset\Re^{n}$,$B\subset\Re^{m}$, $m\geq n$ and $A, B$ compact. The function $f(\cdot)$ can only be evaluated numerically. ...
2
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1answer
103 views

Reference Request: Variational Problem

I want to solve approximately the following variational problem: Given a function $c:[-1,1]^2\rightarrow [0,1]$, constants $p_1...p_n\in \mathbb{R}^+$, $\alpha_1...\alpha_n\in \mathbb{R}$, and ...
2
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1answer
108 views

Solving a simple Schrödinger equation with Fast Fourier Transforms

While trying to solve a stochastic Gross-Pitaevskii equation I have found a problem that can be tracked down to something buggy occurring in the simplest Schrodinger equation possible: $$\partial_t ...
5
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2answers
174 views

Alternating binomial Dirichlet series

I have come across the following deceptively simple expression: $$ H_n^s=\sum_{j=1}^n(-1)^{j-1}\left(\begin{array}{c}n\\j\end{array}\right)j^{-s} $$ We have (using eg mathematica, though probably ...
10
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2answers
246 views

Why Householder reflection is better than Givens rotation in dense linear algebra?

It’s obvious that Givens rotation works better with sparse matrices. But I don’t know why Householder reflection is better for dense matrices. Does it require less computations? Or it’s numerically ...
3
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2answers
138 views

Roots of the Chebyshev polynomials of the second kind

It is known that the roots of Chebyshev polynomials of the second kind, denote it by $U_n(x)$, are in the interval $(-1,1)$. I have noticed that, by looking at the low values of $n$, the roots of ...
3
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1answer
565 views

Claimed Quadrature Results seem Impossible

We've been preparing a preprint that shows that the convergence bounds proved for tanh-sinh quadrature for numerical integration, cannot possibly hold, and an error must exist - since they imply a P ...
3
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39 views

numerical stability of root identification via Newton-Raphson iteration of Stieltjes residue sums

I have asked several questions on math.SE in order to compute numerically the poles of high-degree Padé approximations for $e^{-x}$, because a computation directly from the polynomial ...
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1answer
42 views

Is spectral properties a general term for condition number?

I am reading an article about solving large sparse linear systems, in this paper it’s said that most of the iterative methods to solve $Ax = b$ are very much influenced by the spectral properties of ...
21
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9answers
2k views

What is… A Grossone?

Y. Sergeyev developed a positional system for representing infinite numbers using a basic unit called a "grossone", as well as what he calls an "infinity computer". The mathematical value of this ...
3
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72 views

Numerical inversion involved confluent hypergeometric (1F1) (or Kummer function)

Edit: The question is solved !! The code is actually correct. There is not error in the codes. I miss-used it. Thank you for your attention : ) This problem arises when I tried to compute the valua ...
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0answers
25 views

Numerical techniques for nonlinear, coupled integro-differential equations

The gist of the problem I have is I want to be able to find a numerical solution to these three coupled, rather unpleasant looking integro-differential equations (1): $$ \frac{d^2 x(t)}{dt^2} = ...
17
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2answers
399 views

“Fractally self-similar” numbers

This is another question about visualization of Ford circles, the previous one being Confusion with practically implementing rational approximations. Here is an output of zooming into Ford circles at ...
4
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1answer
188 views

Existence of nonlinear equation

How can we prove that equation (1) has solutions for every $p$. I mean, is there an analytic method that can be used to show that there exist solutions for every $p$ for this nonlinear equation: ...
10
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1answer
203 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 ...
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0answers
19 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
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1answer
68 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 ...
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1answer
130 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 ...
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1answer
129 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
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1answer
99 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
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0answers
109 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 - ...
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22 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 ...
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1answer
60 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 ...
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0answers
131 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 ...
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0answers
29 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
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0answers
60 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 ...
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2answers
595 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 $\{x\}:=x-\left \lfloor x \right \rfloor -1/2$. ...
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0answers
145 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 ...
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1answer
143 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 ...
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1answer
150 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 ...
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1answer
158 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 ...
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77 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 ...
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0answers
95 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 ...
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0answers
80 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 $A=a_1<\cdots<a_r$ ...
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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 ...
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1answer
266 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: ...
2
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0answers
43 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 $a_2$, ...
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0answers
37 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 ...
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1answer
112 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 ...
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1answer
115 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) ...
1
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0answers
88 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
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0answers
110 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 ...
4
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0answers
105 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 ...
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2answers
125 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
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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 ...
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0answers
55 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 ...