Numerical algorithms for problems in analysis and algebra, scientific computation

**8**

votes

**1**answer

126 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 ...

**9**

votes

**1**answer

151 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 ...

**3**

votes

**1**answer

142 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**

votes

**1**answer

4k views

### Meaning of $\Subset$ notation

The symbol $\Subset$ (occurring in places where $\subseteq$ could occur syntactically) comes up frequently in a paper I'm reading. The paper lives at the intersection of a few areas of math, and I ...

**0**

votes

**0**answers

11 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

56 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 ...

**0**

votes

**2**answers

120 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 ...

**1**

vote

**1**answer

128 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 ...

**2**

votes

**2**answers

162 views

### Numerical solution of SDEs with colored noise

I am trying to numerically solve an SDE with both white and colored noise that models a non-linear circuit:
$$
dX_t = f(X_t) dt + \sigma_w dW + \sigma_c dC
$$
where $W$ is a standard Brownian motion ...

**2**

votes

**1**answer

90 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 ...

**4**

votes

**1**answer

112 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

**0**answers

98 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 - ...

**0**

votes

**0**answers

68 views

### Generalization of asymptotic notations like big O or little o notation [migrated]

Context and question
For a thesis I need to prove all the arithmetic rules for asymptotic notations – such as $O(\cdot)$ or $\omega(\cdot)$ – I have used (which are a lot). Currently I look for a ...

**2**

votes

**1**answer

103 views

### numerical methods for discontinuous ODEs

Greetings,
what are state of art methods for numerical solution of ODEs with discontinuous right side?
I'm mostly interested piecewise-smooth right side functions, e.g. sign.

**0**

votes

**0**answers

17 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

54 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 ...

**5**

votes

**1**answer

257 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 ...

**1**

vote

**0**answers

124 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 ...

**2**

votes

**0**answers

58 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 ...

**0**

votes

**0**answers

25 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 ...

**1**

vote

**1**answer

125 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 ...

**35**

votes

**6**answers

5k views

### There must be a good introductory numerical analysis course out there!

Background As a numerical analyst, I've frequently taught the 'Introductory Numerical Analysis' class. Such courses are found in many major universities; the audience typically consists of reluctant ...

**18**

votes

**2**answers

558 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

126 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 ...

**0**

votes

**0**answers

167 views

### Solving a parabolic PDE with boundary conditions given over ranges

How can one solve a Parabolic PDE (like the wave or diffusion equations) if the boundary conditions were given over ranges?
Here is an example: How to solve the equation ...

**3**

votes

**1**answer

116 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

149 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 ...

**0**

votes

**0**answers

51 views

### Exact diagonalization of tridiagonal centrosymmetric matrices

It is said that one can diagonalize a tridiagonal matrix using the analytical Lanczos method http://arxiv.org/abs/cond-mat/9712283v1. In some references in it, they always say that the starting point ...

**0**

votes

**0**answers

32 views

### Non-coherent estimation problem

I have the following signals
$$\left[\begin{array}{c} y_{mn} \\ y_{nm}\end{array}\right] =\left[\begin{array}{c} x_{n} \\ x_{m}\end{array}\right]h_{nm} +\left[\begin{array}{c} e_{mn} \\ ...

**12**

votes

**7**answers

4k views

### Numerical integration over 2D disk

I have a real-valued function $f$ on the unit disk $D$ that is fairly well behaved (real-analytic everywhere) and would like to find the integral $\int_D f(x,y)dxdy$ numerically. After much searching, ...

**3**

votes

**0**answers

32 views

### Solving a Certain Constrained Isoperimetric Approximation-Problem

This question is related to my question Differential Geometric Aspects of Rubber Bands, where I asked for a mathematical model of contracting rubber bands.
In contrast to my former question, the ...

**0**

votes

**0**answers

31 views

### How to treat non-identifiable states in Kalman filtering/dynamic linear models?

Let $x_t = G_tx_{t-1}+\omega_t$ with $\omega_t \sim \mathrm{N}(\mathbf{0}, \mathbf{W}_t)$ be a state equation, and let $y_t = F_tx_t+\nu_t$ with $\nu_t \sim \mathrm{N}(\mathbf{0}, \mathbf{V}_t)$ be a ...

**1**

vote

**0**answers

22 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 ...

**11**

votes

**2**answers

508 views

### Who is Petrov of the Petrov-Galerkin method?

I was not able to find the origin of the name Petrov in the Petrov-Galerkin method for the numerical approximation of PDEs.
Wikipedia refers to a certain Alexander G. Petrov, but it is still not ...

**1**

vote

**0**answers

62 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 ...

**0**

votes

**1**answer

256 views

### Comparing iterative methods for linear systems

For a tridiagonal matrix of the from
\begin{bmatrix}
a & -b & \newline
-b & a & -b \newline
& \ddots & \ddots & \ddots \newline
& & & & ...

**5**

votes

**0**answers

91 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 ...

**11**

votes

**0**answers

106 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 ...

**2**

votes

**0**answers

75 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 $A=a_1<\cdots<a_r$ ...

**-1**

votes

**1**answer

85 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 ...

**-2**

votes

**1**answer

261 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:
...

**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 ...

**5**

votes

**1**answer

589 views

### Solution of Helmholtz-Equation where Phase is restricted by additional PDE

Hello!
Let's say I have
$(\partial_x^2 + \partial_y^2 + a)f(x,y)=0$
with $f(x,y) \in \mathbb{C}$, ($\lim_{x,y \to \infty} f(x,y)=0$).
Now separate the Amplitude and Phase of the solution:
...

**1**

vote

**0**answers

36 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

32 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 ...

**1**

vote

**1**answer

108 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

**2**answers

53 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 ...

**0**

votes

**1**answer

101 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**

vote

**0**answers

83 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 ...

**1**

vote

**2**answers

304 views

### Are there some algorithms to solve the diagonal matrix $X$ to the following matrix equation?

Suppose $X$ is an unknown $m \times m$ diagonal matrix. Given a scalar $0<c<1$, and a matrix $A$ of $m \times m$ size whose entries $0<A_{i,j}<1$. Are there some algorithms to find the ...