Numerical algorithms for problems in analysis and algebra, scientific computation

**10**

votes

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**3**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

219 views

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

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

**0**

votes

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**2**answers

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 ...