Questions tagged [na.numerical-analysis]
Numerical algorithms for problems in analysis and algebra, scientific computation
1,224
questions
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Partitioned Runge-Kutta (Lobatto IIIAB)
I am wondering, if anybody knows some paper, that study convergence and stability of Partitioned Rung-Kutta Methods (especially Lobatto IIIAB) applied on separable Hamiltonian system.
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How to numerically compute the operator norm of an operator acting on a matrix algebra?
Let $M_n(C)$ denote the $n\times n$ matrices with complex entries acting on the Hilbert space $C^n$. As norm on $M_n(C)$ we take the operator norm, i.e. the largest eigenvalue of its absolute value. ...
3
votes
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answers
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Rate of uniform approximation by piecewise constant functions
Definitions and Notation:
Fix a positive constant $M>0$ with positive integers $m,n$ and the standard orthonormal basis $e_1,\dots,e_n$ of $\mathbb{R}^n$.
For every positive integer $N$, define the ...
10
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2
answers
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Parameter estimation for stochastic differential equation from discrete observations
Suppose we have a time-series $x(t_i)$ at discrete times $t_i$ and we want to estimate the parameters of an underlying SDE corresponding to this time-series:
$$dx_t = f(x_t,\theta)dt + \sigma(x_t,\...
3
votes
0
answers
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Chebyshev-like polynomials [closed]
In some approximation problems I'm working on, the errors turned out to be polynomials of various degrees whose graphs on the interval $[-1,1]$ look like this:
As you can see, these things look a bit ...
5
votes
1
answer
262
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Unbounded solution but bounded Euler discretization
Is there an ordinary differential equation in $\mathbb{R}^d$ induced by a gradient vector field with unbounded solutions, for which the difference equations obtained by using the forward Euler method ...
6
votes
1
answer
884
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Resultant of linear combinations of Chebyshev polynomials of the second kind
The Chebyshev polynomial $U_n(x)$ of the second kind is characterized by
$$
U_n(\cos\theta)=\frac{\sin(n+1)\theta}{\sin(\theta)}.
$$
It seems that
$$\operatorname*{Res}_x \left( U_n(x)+tU_{n-1}(x),\...
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answers
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Direct (first-order ?) algorithm to minimize $u(x) := \|x-a\|_C + r\|x\|_p$
Fix $a \in \mathbb R^n$, $r \ge 0$, $p \in \{1,2\}$, and a positive-definite matrix $C$ of order $n$. Define $u:\mathbb R^n \to \mathbb R$ by $u(x) := \|x-a\|_C + r\|x\|_p$, where $\|z\|_C := \sqrt{z^\...
0
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answers
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Numerically expanding a function in a rational-power "basis"
I have some scientific code which interfaces with a library which accepts real functions specified as any number of additive terms with exponential powers. For instance, it is capable of accepting ...
11
votes
3
answers
711
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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 ...
64
votes
5
answers
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The unreasonable effectiveness of Padé approximation
I am trying to get an intuitive feel for why Padé approximation works so well. Given a truncated Taylor/Maclaurin series it "extrapolates" it beyond the radius of convergence.
But what I can'...
1
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1
answer
460
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Calculating the eigenvalues of the Laplacian numerically
I am trying to find the eigenvalues of the Laplacian operator, or in other words, solve the Helmholtz equation
$\nabla^2f=\lambda f$
on a compact 2D domain (comes from a quantum mechanics particle-in-...
6
votes
3
answers
431
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Non-polynomial splines, a non-linear problem
I'm looking for references on how to construct spline-like functions from a basis that does not include piecewise polynomials.
To be specific, given a class of functions such as "decaying ...
1
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0
answers
54
views
Functional approximation with derivatives
I am trying to solve a functional approximation problem.
Consider a set of measurements of a d-dimensional state $\mathrm x \in \mathbb{R}^d$, together with velocities $\dot{\mathrm x}$ and ...
0
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0
answers
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Solving a nonlinear equation maybe with Lambert W function
Can you please help me solve the following nonlinear equation?
\begin{equation}
\boldsymbol{z} \odot\left(\boldsymbol{\Gamma}^{\top} \boldsymbol{y}\right)=(\beta)^{\frac{1}{m-1}}\left(\frac{m-1}{...
0
votes
1
answer
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A simple procedure to simulate multifractional Brownian motion paths
In a paper by Peltier and Vehel the multifractional Brownian motion (mBm) was defined for the first time, and they also give a procedure to simulate mBm sample paths. Briefly, mBm generalizes the ...
3
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0
answers
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how to efficiently find level sets (using a modification of a root-finding algorithm)?
I'm trying to find a set of points $\{ a_i | f(a_i) = c_i \}_{i=1}^k$ where $f$ and $\{ c_i \}_{i=1}^k$ are given in sorted order. All $c_i > 0$, $f$ is continuous and monotonically increasing, $f(...
3
votes
1
answer
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Is there a classical textbook/reference on numerical discretization schemes?
I found that it is relatively easy to find a book that discusses Euler discretization or Runge-Kutta discretization, but I am not aware of one that is well-known and/or common knowledge (i.e., field-...
2
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1
answer
148
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Obtain 3D function from 2D slices [closed]
I am given a graph of a motor's torque vs RPM values at two different current draws, 730A and 300A (see graph). I need to obtain the 3D function to find current as a function of torque and RPM by ...
3
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0
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Preconditioners for $Ax=y$ that rely on hierarchical statistical modeling
Solving $Ax=y$ exactly can be done as:
fit a linear autoregressive model by treating rows of $A$ as data
apply this model to $A^T y$
Imperfect predictive model corresponds to an approximate inverse ...
1
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0
answers
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How is the quasipotential in Freidlin-Wentzell theory of large deviations affected by $C^1$ transformations?
I have a 3D differential equation I'm interested in studying the potential landscape using quasipotentials, described in depth in this paper. I need to calculate the potential landscape several times, ...
10
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2
answers
950
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Machin-like formulas for logarithms
I found this math puzzle blog post
http://fredrikj.net/blog/2013/03/machin-like-formulas-for-logarithms/
which I'm reposting here with permission. I'm setting this to community wiki to minimize the ...
1
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0
answers
128
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Implementation of Mellin transform of exponential decay
I'm trying to understand this paper: 10.1016/j.jmr.2010.05.015. It is about using a Mellin transform of curves that contain multiple exponential decays of varying contributions (CPMG data from Nuclear ...
3
votes
1
answer
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Deconvolution using the discrete Fourier transform
Summary: From discrete convolution theorem, it is understandable that we need 2N-1 point DFT of both sequences in order to avoid circular convolution. If we need to do deconvolution of a given ...
5
votes
1
answer
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Is there a general method for determing the domain of dependence of (higher-order) PDEs?
Is there a general method for determining the domain of dependence of (higher-order) PDEs? I would be plenty happy with a reference to a paper or textbook I could look at; despite much reading around,...
16
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7
answers
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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, ...
0
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0
answers
108
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Degeneracies in linear combination of tensor product of Pauli matrices
Let $P_i \in \{I,X,Y,Z\}^{\otimes n} $, that is $P_i = \bigotimes_{i =1 }^n \sigma_i$ with $\sigma_i \in \{I,X,Y,Z\}$, where
$$
I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \hspace{1cm} X =...
1
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0
answers
62
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Newton-Raphson and bisection for solving for a system of nonlinear equations?
We know that for a single equation root-finding, we can use the Newton's method, or a combination of Newton with bisection to guarantee convergence. Can we use Newton+bisection for a system of ...
10
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1
answer
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Area of filled Julia sets
The recent question Area of the boundary of the Mandelbrot set ? prompted me to ask this question.
There has been some work on estimates for the area of the Mandelbrot set, e.g., a paper by John H. ...
13
votes
1
answer
792
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“Taylor series” is to “Volterra series” as “Padé approximant” is to _________?
Padé approximants are often better than Taylor series at representing a function. Given a Taylor series, one can use Wynn's epsilon algorithm to easily produce the Padé approximants to it.
Volterra ...
6
votes
1
answer
624
views
How can I efficiently find the "simplest" rational in an interval?
For a hobby software project I am working with exact rational arithmetic, as it happens this produces numbers $\frac{n}{k}$ of huge size even after reducing them, I am searching for an efficient ...
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0
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Flux that can be represented by low and high resolution schemes
In the wiki page of Flux limiter, it writes:
If these edge fluxes can be represented by low and high resolution schemes, then a flux limiter can switch between these schemes depending upon the ...
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3
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On mathematical studies of the Mpemba effect
Since the days of Aristotle and Descartes, it has been known that under certain circumstances warm water freezes faster than cold water. This effect is now commonly known as the Mpemba effect, named ...
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Convex maximization over the boundary of a convex set
Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be the objective function that is strictly convex. We would to like maximize $f$ over a convex compact set $S \subseteq \mathbb{R}^n$. Assume that $f$ has ...
2
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Derivation of the Cahn-Hilliard PDE from the point of view of finite difference methods
Consider the Cahn-Hilliard equation
$$\frac{\partial c}{\partial t} = \nabla^2(f(c)-\varepsilon^2 \nabla^2 c)$$
defined on your favorite domain. I'm looking for a literature reference that formally ...
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0
answers
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Function uniquely determined by its values at integer arguments
A smooth enough, slow growing real-valued function $f(x)$, is uniquely determined by its values at integer arguments. I don't remember the name of the theorem and the conditions for this to be true. ...
14
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3
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Accelerating convergence for some double sums
I am interested in the following general double sums, for integers $a\geq 1$ and $b\geq 2$,
$$Z(a,b) = \sum_{k,\ell \geq 0} \frac{2k+3}{\binom{k+2}{2}^a} \frac{2\ell+3}{(\binom{k+2}{2}+\binom{\ell+2}{...
2
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0
answers
237
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A general question about spectral methods vs finite element methods
According to this Wikipedia article:
Spectral methods can be computationally less expensive and easier to implement than finite element methods; they shine best when high accuracy is sought in simple ...
0
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1
answer
105
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FEM based solution to parabolic problem
Consider the problem
$$
\begin{cases}
u_t - \Delta u = 0 &\text{ on } \Omega\times (0,T)\\u=0 &\text{ on } \partial \Omega\times (0,T) \\ u(x,0)=g(x) &\text{ on } \Omega
\end{cases}
$$
...
3
votes
0
answers
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Tight bounds on the iterates of the Clenshaw algorithm for Chebyshev polynomials
I'm trying to bound the iterates of the Clenshaw algorithm when applied to the Chebyshev series, related to a question I'm running into related to the stability of this algorithm. Recall that, for $p(...
5
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3
answers
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Square root algorithm
I would like an efficient algorithm for square root of a positive integer. Is there a reference that compares various square root algorithms?
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0
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Is there a more efficient computer algebra system to solve the system of nonlinear equations in N-R method or other numerical methods?
Consider the system of infinite series
\begin{align}
&F=x+\frac{y^{3^2}}{3}+\frac{x^{3^5}}{3^2}+\frac{y^{3^7}}{3^3}+\frac{x^{3^{10}}}{3^4}+\frac{y^{3^{12}}}{3^5}+\cdots=0
\\
&G=y+\frac{x^{3^3}}...
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votes
1
answer
226
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Correct way to conduct equilibrium scaling of linear/integer/MIP program
I would like to scale my linear/integer program and also mixed-integer program using the equilibrium scaling method. I have worked on two research papers and one research book. However, they did the ...
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0
answers
181
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Complexity of singular value decomposition using matrix multiplication oracles
Suppose I have an $n\times m$ real matrix $A$, $n\ll m$ with full row rank $(\mathrm{rank}(A) = n)$. I have an oracle that can compute $Ax$ or $A^T y$ for any $x\in \mathbb{R}^m, y\in \mathbb{R}^n$. ...
0
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1
answer
180
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Numerical methods for evaluating singular integrals
The Helmholtz decomposition for a vector field B contains both volume integrals and two boundary integrals (https://en.wikipedia.org/wiki/Helmholtz_decomposition). For brevity I show just one of the ...
1
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0
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143
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Specific L1 piece-wise linear approximation of the convex function of one variable
Consider a convex function $f(x)$ of one variable $x$ on some interval $[a,b]$.
Question:
What is known about the L1 approximation of $f(x)$ by piecewise linear functions? How to construct ...
7
votes
1
answer
183
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Reporting inconclusive experimental searches
In many areas of mathematics it is informative to conduct numerical experiments.
But, it not uncommon that the searches do not lead to the examples or data one was hoping for. Since the numerical ...
10
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5
answers
8k
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Shifted QR algorithm—why does the shift help?
I read that a way to speed up the convergence rate of the QR algorithm is to shift the target
matrix. It is not so clear to me why this helps. The convergence rate depends on the
minimum gap between ...
2
votes
1
answer
395
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Given a function $f(t): \mathbb{R}\to\mathbb{R}^n$, can 2D, or $n$D discrete Fourier transforms be used on $f(t)$ to perform frequency analysis?
$\DeclareMathOperator{\R}{\mathbb{R}}$Frequency analysis is often performed on wave forms (1D DFT = discrete Fourier transform), and images (2D DFT), where the function in question often takes the ...
0
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0
answers
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3D interpolation function
I've got a 3D figure created using around 30k points and has different regions colored in an specific way according to some unrelated variables that come from a project I'm creating. Taking in ...