All Questions
Tagged with cv.complex-variables na.numerical-analysis
31 questions
129
votes
2
answers
16k
views
What are the shapes of rational functions?
I would like to understand and compute the shapes of rational functions, that is, holomorphic maps of the Riemann sphere to itself, or equivalently, ratios of two polynomials, up to Moebius ...
- 25.1k
19
votes
2
answers
11k
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 ...
- 1,601
11
votes
3
answers
726
views
Can computers find zeros of order $2$?
We assume we are given an entire function $f: \mathbb C \to \mathbb C$ with $f(0)=1$ and $f'(0)=0$ and $f$ is real on the real axis.
We assume (as a fact about $f$, that we want to demonstrate ...
- 124
7
votes
2
answers
386
views
Solution to at least one ODE in a family of ODE's
In my research I have stumbled across the following 1st order complex differential equation for smooth functions $\eta:\mathbb{R}/2\pi\mathbb{Z}\to\mathbb{C}-\lbrace0\rbrace$ defined on the circle,
$$...
- 17.5k
6
votes
2
answers
660
views
analytic approximation of a non-negative matrix by a sequence of positive matrices
Let $L \in \{0,1\}^{n \times n}$ be a non-negative matrix whose row sum is 1. ($L$ stands for "limit"). It is known that there exists a unique $r \times r$ principal minor of $L$ that is a permutation ...
- 1,010
5
votes
2
answers
4k
views
sparsity of QR decomposition
Hi, everyone!
I have a sparse $n \times n$ matrix $A$ with $nnz(A)$ denoting the number of non-zero entries in $A$. Now I use QR factorization to decompose $A$ into an orthogonal matrix $Q$ and ...
- 51
5
votes
1
answer
481
views
Stability of root-finding near the unit circle
It is stated in several sources on numerical analysis that the general problem of polynomial root-finding is ill-conditioned, but that it is well-conditioned if the roots are near the unit circle. (e....
- 173
5
votes
1
answer
471
views
Padé multipoint approximants of the exponential function
One says that a pair of polynomials $(P_m,Q_n)$ over $\mathbb C[z]$, with
$\text{deg }P_m=m$, $\text{deg }Q_n=n$, is a "multipoint Padé approximant of the exponential function" if $P_m(z)e^z-Q_n(z)$ ...
- 3,998
5
votes
1
answer
169
views
Efficients method for finding a zero of a multilinear complex polynomial in an specified region
Let P be a given multilinear polynomial in $\mathbb{C}[z_1,\dots,z_n]$ and $D\subset \mathbb{C}$ be a given disc in the complex plane. Does there exist an efficient method for checking that $P$ has a ...
- 4,484
5
votes
0
answers
211
views
Numerical analytic continuation/asymptotics
I posted this question, quite a while ago, on math.stackexchange.com, here. I received an interesting answer but not sufficiently accurate for my purposes, so I'm trying here.
I have a class of ...
- 516
5
votes
0
answers
225
views
Validity of $\ln z=\frac{\pi}{\operatorname{AGP}(\theta_2^2(1/z),\theta_3^2(1/z))}$
Definitions
For the definitions of $\operatorname{AGP}$ and $\operatorname{AGO}$, see here or here. $\theta_2(z)$ and $\theta_3(z)$ are defined as follows:
$$\theta_2(z)=\sum_{n=-\infty}^\infty z^{(n+...
- 83
4
votes
3
answers
682
views
Approximation for complex variables
Approximation theory, which aims to provide the optimal polynomial function approximating the target function in a given domain such as $x\in[-1,1]$, has been well-developed for real variables. In ...
- 149
4
votes
3
answers
643
views
Traceless GUE : Four Centered Fermions
The proof of the Wigner Semicircle Law comes from studying the GUE Kernel
$$ K_N(\mu, \nu)=e^{-\frac{1}{2}(\mu^2+\nu^2)} \cdot \frac{1}{\sqrt{\pi}} \sum_{j=0}^{N-1}\frac{H_j(\lambda)H_j(\mu)}{2^j j!} ...
- 22.8k
4
votes
1
answer
211
views
Rate of convergence of Padé approximants
Let $f$ be an entire function of order $1$. Two questions:
1) Can one assert that the diagonal Padé approximants converge to $f$ (pointwise or uniformly over compacts of $\mathbb C$)?
2) if yes, can ...
- 3,998
4
votes
1
answer
173
views
Numerical equality testing
I am working on developing an online homework system.
One thing I would like to have is something which compares a student's answer (like $2\sin(x)\cos(x)$) with the intended answer (maybe $\sin(2x)$)...
- 12.1k
3
votes
1
answer
711
views
Residues and values of Riemann Zeta function at some points
I need the following computational results for proving something.
Let $1/2 + i\gamma_0$, be the first nontrivial zero of Riemann zeta function, $\zeta(s)$,
i.e. $\gamma_0\sim 14.134...$.
1) what is ...
- 897
3
votes
1
answer
326
views
Polynomial and rational approximation of continuous functions in $\mathbb{C}$
I am wondering what the state of the art is for polynomial and rational approximations to continuous/holomorphic functions in $\mathbb{C}$. The particular domains of interest are the closed unit ball $...
- 465
3
votes
1
answer
187
views
Variation of steepest descent/Laplace methods for non-exponential integrands
I was wondering if versions of the Laplace/steepest descent methods exists for integrals of the type
$$\int_C f(z) M(\lambda g(z)) dz$$
for $\lambda >>0$ functions $f(z), g(z): \mathbb C \...
- 135
3
votes
0
answers
170
views
Asymptotic expansion of an integral, related to Maass forms
I am trying to compute the asymptotic expansion of the integral
$I(t) = \int_{C} e^{\sqrt{1+u}(\frac{1}{t}+\frac{t^2}{\sqrt{u}})}\frac{u^\eta}{\sqrt{1+u}}du$
as $t$ is real and $t\rightarrow +\infty$, ...
- 31
2
votes
2
answers
178
views
Roots of modified polynomials
Consider the following two polynomials:
$$
g=x^3 - x^2 - (c + 2)x + c
$$
and
$$
h=x^3 - x^2 - cx + c
$$
The roots of $h$ are $1$ and $\pm \sqrt{c}$. I am interested in obtaining the roots of $g$, ...
- 6,962
2
votes
1
answer
5k
views
Pade approximant to exponential function
Suppose:
a) $p(z)$ is an even degree polynomial (of degree $k = 2j$) with real coefficients;
b) $p(0) = 1$;
c) $p(z)$ and $p(-z)$ have no roots in common anywhere in the complex plane;
d) $f(z) = ...
- 21
2
votes
1
answer
110
views
Proving that $\lim_{j \to i} Z_{ij} = [\ln(\frac{\Delta s_i}{2})-1]\Delta s_i$
If I have the following integral equation $$\phi(\vec{x})=\frac{1}{\pi}\int [\phi\frac{\partial (\ln r)}{\partial n} -\ln(r) \frac{\partial \phi}{\partial n}] ds$$
An approximate solution of $\phi$ ...
- 111
2
votes
0
answers
48
views
Convergence of finite-difference method for Cauchy-Riemann equations
Let $I\subseteq \mathbb{R}$ an open interval. Let $f:I\rightarrow \mathbb{C}$ real analytic. Suppose we want to numerically compute an analytic extension of $f$.
We will assume the following: we are ...
- 312
2
votes
0
answers
89
views
Finding a branch cut or a branch point [closed]
Is there a way to find a branch cut or a branch point, through which a curve over a complex function goes, or in general in some region of complex function, say $\ln(f(z))$, using numerical methods or ...
- 67
2
votes
0
answers
406
views
Gaussian type integral with inverse square root
Hi,
I have encountered an integral of the following type in an engineering application:
$\int_{-\infty}^\infty dx \frac{1}{\sqrt{x^2+a^2}}\exp(-x^2/2+i x b)$,
where $a$ and $b$ are real ($a$ could ...
- 51
1
vote
1
answer
534
views
Stone-Weierstrass Theorem, polynomial interpolation, divided difference in complex plane
Setting:
Let $\Gamma$ be a simple smooth($C^\infty$) curve in $\mathbb{C}$ parametrized by the injective map $\gamma:[0,1] \to \mathbb{C}$.
Assume $f$ is a function defined on $\Gamma$ s.t. $f$ is ...
- 398
1
vote
1
answer
182
views
Analyze a function defined in terms of an integral
Here is a question that really has puzzled me for quite a while. I happened to see this function defined in terms of an integral
$$f(x):=\int_0^{\pi/2}\frac{2e^{x+e^x\cos y}}{1+\left(e^{e^x\cos y}\...
- 1,350
1
vote
0
answers
443
views
Multivariate solution to Lambert W / product-log function
Consider solving the following system for $x$
\begin{align*}
a - b e^{\theta x} - cx = 0
\end{align*}
According to your favorite computer algebra program, one possible (and the simplest) is
\begin{...
- 229
0
votes
0
answers
57
views
Complexity of evaluation of analytic functions
Given an analytic function $f(x)$ (say as combination of elementary functions and operators), is it possible to compute $n$ first bits of the value of the function on the whole interval $[a, b]$ ...
- 67
0
votes
0
answers
80
views
Non-triviality of the sum of simple rational functions
Recently, in the study of unicity problems in complex analysis, I met a problem that can be stated in the following way,
Let $\{m_i\}_{i=0}^{3}$ and $\{n_i\}_{i=0}^{3}$ be eight integers in $\mathbb{Z}...
- 1,706
-1
votes
1
answer
214
views
Best approximation of the modulus function
While there is extensive study regarding the best approximation of function with polynomial functions in the real domain, the study of approximation of complex variables becomes much sparse. See this ...
- 149