Questions tagged [na.numerical-analysis]
Numerical algorithms for problems in analysis and algebra, scientific computation
1,049
questions
2
votes
1answer
46 views
Cubic spline interpolation without a constant term
Two main questions:
I am wondering if it is possible to construct a cubic spline that interpolates data WITHOUT a constant term $a$. That is, the polynomial takes the form $f(t) = bt + ct^2 + dt^3$, ...
5
votes
1answer
104 views
Can you use Oseledet's theorem to numerically approximate the Lyapunov spectra?
Let's say you have a set of first order differential equations with known Jacobian $J$. Let $x_0, x_1, ..., x_n$ be sampled points on the trajectory near the attractor.
Let $T_n = J(x_{n-1})J(x_{n-2})....
2
votes
1answer
57 views
Spline Interpolation error of higher degree
It is well-known that the interpolation error of a cubic spline has at best order $O(h^4)$, which results from polynomials of degree $3$.
Can I assume that, if one uses polynomials of degree $p$ and ...
0
votes
1answer
79 views
Is it possible to numericaly solve functional equation
Given a functional equation of form $f(f(x))=T(x)$ is there any good ways to solve it numerically? If not then at least approximate in some small region $x\in(-a;a)$.
E.g. with the equation $f(f(x))=x+...
1
vote
1answer
58 views
Probability finite precision random matrix has distinct eigenvalues
copied from math stack exchange
There is a theorem which says the probability/size of a random matrix having repeated eigenvalues is 0 and this result is used in many fields. What I am wondering is, ...
30
votes
4answers
4k views
How does Mathematica do symbolic integration?
I suppose there was at least once in our lifetime the point where we resorted to mathematica for help with an integral.-Unless you chose not to have the pleasure of using the continuum in your ...
5
votes
0answers
194 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:
$$\theta_2(z)=\sum_{n=-\infty}^\infty z^{(n+...
1
vote
0answers
38 views
Implementation of Mellin transform of exponential decay
I'm trying to understand this paper: 10.1016/j.jmr.2010.05.015. It is about using a Mellin transform of curves that contain multiple exponential decays of varying contributions (CPMG data from Nuclear ...
3
votes
0answers
89 views
Best approximation of piecewise constant function by Lipschitz functions
Let $f=\sum_{n=1}^N k_n I_{E_n}$ where $E_n$ are Borel subsets of $\mathbb{R}^n$ and $k_n\in \mathbb{R}^m$ with non-negative entries, and let $\mu$ be a finite Borel measure on $\mathbb{R}^n$. What ...
2
votes
1answer
79 views
Error in Gauss-Laguerre numerical quadrature scheme
The $n$-th Gauss-Laguerre quadrature scheme aims to approximate integral of exponentially decreassing function over $[0 ; \infty[$ by a finite sum, according to:
$$ \int _0
^{+ \infty}
...
0
votes
1answer
156 views
Hamilton equations-Symplectic scheme [closed]
We know that $\dot{q} = \frac{\partial H}{\partial p}$ and $\dot{p} = -\frac{\partial H}{\partial q}$, and we also know the values $Q$ and $P$ respectively of $q$ and $p$ at a later time step $\Delta ...
1
vote
0answers
34 views
Is it desirable to establish a CAD-like geometry medium for four dimensional space-time topologies and FEM? [closed]
Finite element and numerical methods over 4d space-time is a topic of interest for elastodynamics right now. Do you think it is desirable to establish a geometry/CAD system for space-time involving ...
1
vote
0answers
46 views
Convergence of numerical method [closed]
I would like to prove that the following sequence :
$S^{n+1} =1 - I +\frac{β}{α}\ln(S^{n})$,
where $\alpha,\,\beta,\,I$ are constants and $S^{0} = 1$
converges as long as $\alpha\cdot S<\beta.$
1
vote
0answers
64 views
symplectic Runge-Kutta for matrix differential equation
I would like to solve, for $t>0$ the following matrix differential equation:
$$U'(t)=H(t)U(t)$$
with initial condition $U(t=0)=U_0$ ($2N\times2N$, symplectic and unitary matrix) and $H(t=0)=H_0$ ($...
1
vote
1answer
92 views
How many Fourier coefficients of a sparse signal $f=\sum_{n=1}^Nc_n\delta_{t_n}$ are needed to determine $f$ uniquely?
Let $N \in \mathbb N$ and $c_n \in \mathbb C$, $t_n \in \mathbb R$ for $n=1, \dots, N$. Suppose that $f$ is a linear combination of dirac-deltas with locations $t_n$ and coefficients $c_n$, i.e.
$$
f=\...
0
votes
0answers
30 views
Numerically finding matrix approximation by lower-dimensional “pseudo-similar” matrix
Consider an $N\times N$ (real or complex) matrix $A$, and some $n<N$. Is there a good numerical algorithm that finds the set consisting of an $n\times n$ matrix $B$, an $n\times N$ matrix $I$, and ...
3
votes
0answers
45 views
What is the purpose of converting a level-set function into a signed distance function?
In the paper Electrical impedance tomography using level set representation and total variational regularization, the authors tried to implement an iterative algorithm to find the interface of two ...
3
votes
0answers
64 views
Smoothly connecting PDEs with finite differences
A PDE with non-smooth inhomogeneity
Let $\mathcal{L}$ be a second-order, linear, elliptic differential operator acting on $\mathcal{C}^2([0,2]^2)$.
I'm numerically solving the inhomogeneous PDE
\begin{...
1
vote
1answer
87 views
Example of a differentiable function optimization where derivative free methods are used
While preparing a workshop on the derivative free methods, and fminsearch in MATLAB, I found an example function where fminsearch converges better and in less iterations than fmincon with calculated ...
0
votes
1answer
58 views
Finding numerical solution for nonlinear Poisson-like equation using finite difference method
I am trying to use finite difference method to solve for $u(x,t)$ in the equation:
\begin{align}
\frac{\partial^2u}{\partial x^2} = \frac{au}{1+bu},
\end{align}
which is actually part of a system of ...
1
vote
0answers
72 views
Best approximation of a Lipschitz function with a piecewise polynomial Lipschitz function
Let $g : [-1, 1] \to R$ be a $1$-Lipschitz function and $f_{k,d} : [-1, 1] \to R$ a $1$-Lipschitz function whose restriction to any subinterval $[h_i, h_{i+1}] \subset [-1, 1]$, $i = 0 ... (k-1)$ with ...
0
votes
1answer
89 views
Gaussian quadrature, with no exact result over polynomial, but on inverse functions
Generally, a Gaussian quadrature of degree $n$ over an interval $I$ is defined so that it integrates exactly polynomials up to degree $2n - 1$. The main tool are the orthogonal polynomials.
When $I$ ...
2
votes
1answer
315 views
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 ...
3
votes
0answers
137 views
Lower bound of the modulus $|\eta(s)|$ of the Dirichlet Eta function if $0.6 < \Re(s) < 0.9$
Let $s=\sigma + it$, with $0.6 < \sigma < 1$ and $\sigma=\Re(s)$. I am trying to get good enough approximations for $\eta(s)$, hoping something useful might come out of it. I stumbled upon a ...
0
votes
1answer
75 views
Solution of complex linear system
In Brubeck, Nakatsukasa, and Trefethen - Vandermonde with Arnoldi (example 3) they solve the following linear system:
$$\operatorname{Re}\left(\begin{array}{ccc}1 & \cdots & z_{1}^{n} \\ 1 &...
1
vote
1answer
73 views
Quadrature methods for high-dimensional Gaussian integration
Suppose that $f$ is the density of a high(-$d$)-dimensional Gaussian measure with mean $\mu$ and non-singular covariance matrix $\Sigma$. Let $g:\mathbb{R}^d\rightarrow \mathbb{R}$ be a continuous ...
3
votes
1answer
83 views
Assignment problem with priorities and scores
I have run into a real problem that is actually a sort of assignment problem. I am describing it here because I am interested in knowing whether this problem already has a name (and whether there is ...
10
votes
1answer
2k views
Did human computers use floating-point arithmetics?
Before the proliferation of computers in the 1950s, did human computers use floating-point formats for their computations?
Floating-point calculation was reportedly implemented already in the 1910s (...
0
votes
0answers
37 views
A parametrized saddle point problem with linear constraints
I am struggling to find any potential algorithm for solving a saddle point problem.
More precisely let $\mathcal{P}=\{ \mathbf{x}\in \mathbb{R}^{d}; \mathbf{A}\mathbf{x}=\mathbf{b}, \mathbf{x} \geq 0\}...
1
vote
1answer
87 views
Norm of a matrix with clustered eigenvalues
On page 271 of Trefethen and Bau's Numerical Linear Algebra, it is constructed a matrix
$$A=2I_{m\times m}+0.5\cdot\frac{\text{rand}(m)}{\sqrt{m}}$$
for $m=200$, where rand(m) is an array with $m\...
1
vote
0answers
92 views
Long-term behavior of asynchronous, stochastic, numerical solution to a dynamical system
I am simulating the behavior of a dynamical system, say $$\dot{x} = f(Ax; \lambda), $$
with an Euler update, where $x\in \mathbb{R}^n$ and $\lambda$ are some parameters. In my scenario, $A\in \mathbb{...
1
vote
0answers
44 views
Existence of efficiently computable integrals for “spiky” functions
$\DeclareMathOperator\spikify{spikify}$Apologies if I'm misusing the word spiky, I mean it only as a visual description of a function, not in any technical mathematical sense!
We define the function
$\...
2
votes
2answers
165 views
Numerically differentiated values and their corresponding x-coordinates
If we numerically differentiate a given time series data consisting of N points by finite forward difference method, we will have N-1 points corresponding to first derivative. If it is a second ...
1
vote
0answers
55 views
Recursive formula for integral of Chebyshev-type integral
Define
$$
I_{m,n}(x,y,r) = \int_a^b T_m(x + r \sin(\gamma)) T_n(y-r \cos(\gamma)) d\gamma
$$
where $T_m(x)$ are the Chebyshev polynomials of the first kind, and $a$ and $b$ are constants. Assume that ...
2
votes
0answers
48 views
How to constrain the integral of the control function to a fixed value?
In the following, I am referring to the general "Hamiltonian control theory" using the conventions defined here.
I am working on a very simple S($x_1$)I($x_2$)R($x_3$) model for infectious ...
0
votes
0answers
59 views
Can we improve the error bounds for spline interpolation if the interpolated function is smooth?
Let me first state the original problem I want to solve:
Given a closed curve $C:[a,b]\to\mathbb R^2$ that is smooth ($C^\infty$), a partition in the parameter space $a=t_0<t_1<\cdots<t_n=b$,...
1
vote
2answers
114 views
Robust estimation of $Ax=b$
Problem setting :
$ \underset{x}{\text{min}} \|Ax-b\|$, where $A \in \mathcal{R}^{m \times n}, m\gg n $, full rank.
L1 loss is used for robust estimation using IRLS. The corresponding equation to ...
2
votes
1answer
97 views
Newton-Raphson with multiple root [closed]
To approximate the root of a function, which also happens to be of multiplicity greater than 1, how do I choose the starting point of the algorithm? For example, I am trying to approximate the root $0$...
0
votes
1answer
91 views
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 ...
2
votes
1answer
118 views
Quadrature for numerical integration over infinite intervals
I am looking for book recommendations or hints on numerical integration over infinite intervals. I am particularly interested in integrals of the form
$\int\limits_{-\infty}^{+\infty} g(x) \exp(p_d(x))...
1
vote
1answer
108 views
Complexity of solving $\sum_i A_i X B_i = C$
Is anything known about computational complexity of finding $X$ which satisfies the following matrix equation?
$$\sum_i^n A_i X B_i = C$$
With $A_i,B_i,C$ dense $d\times d$ matrices. Any literature ...
0
votes
1answer
147 views
Estimate for computing the $L^2$-norm of a function from its data
Let $f:\mathbb{T}^m \to \mathbb{R}$ is a function of bounded variation(BV). Let $D=\{\boldsymbol{p}_i,i=1,2,3\ldots\}$ be a countable dense subset of $(0,1)^m$. Let $E_n, n = 1,2,3\ldots$ be a ...
0
votes
0answers
36 views
Solving nonlinear equations involving expectations
Let $X$ be a random variable and $g(x,y)$ be a function of two variables. Consider the equation
$$
\mathbb{E}_Xg(X,y) = 0
$$
Are there any specialized techniques for solving such equations (...
3
votes
0answers
45 views
Computing the stabilizer of a specific vector in a Lie group representation
Let $x$ be a fixed vector in the carrier vector space of an irrep $\rho$ of a compact Lie group $G$, and let $G_x$ be the stabilizer subgroup of $G$ with respect to $x$. Assume that $x$ is not ...
6
votes
1answer
191 views
Computing $(AA\otimes BB + AB \otimes BA)^{-1}$
Can anyone suggest a way to numerically compute the following matrix vector product?
$$u=A^{-1}b=(AA\otimes BB + AB \otimes BA)^{-1}\operatorname{vec}(C)$$
Here $AA,BB,AB,BA$ and $C$ are $d\times d$ ...
0
votes
0answers
46 views
Optimal approximation of circles with sum of logarithms
By playing around, I found that
$$\left\|\frac{\log{(a\cdot(x+1)+1)}+\log{(1+(1-x)a)}-\log{(2a+1)}}{2\log{(a+1)}-\log{(2a+1)}}- \sqrt{1-x^2}\right\|_\infty\lt 0.12$$
indicating that the fraction quite ...
1
vote
1answer
59 views
Solving equation for higher degree of composition
Given this function $f(x) = x - 1/x$, the equation $f(f(x)) = x$ has two solutions: $\frac{1}{\sqrt{2}}$, $\frac{-1}{\sqrt{2}}$. But how about solving this equation for a higher degree of composition, ...
0
votes
0answers
104 views
A method for extracting a condition to check whether a feature is related to an object
Let we have the object $\bf S$. This object has some properties such as length, temperature and other features. Assume that for the object $\bf S$ we selected $n$ features.For example the vector
${\bf ...
1
vote
0answers
60 views
Randomly determining the maximum of a continuous function
Take a continuous function $f:[-1,1]\to\mathbb{R}$ and a sequence of independent random variables $X_1,X_2,\ldots$ uniformly distributed in $[-1,1]$.
Define $Y_n=\max\{f(X_1),f(X_2),\ldots,f(X_n)\}$. ...
6
votes
1answer
334 views
Condition number for a symmetric positive definite matrix
I am trying to get an estimate for the induced 2-norm condition number $\kappa_{2}(M)$ of this matrix $M$:
$$M_{ij} = \frac{1}{(n-i)!(n-j)!(2n-i-j+1)} = \displaystyle\int_{0}^{1}\frac{x^{n-i}}{(n-i)!} ...
