Questions tagged [na.numerical-analysis]

Numerical algorithms for problems in analysis and algebra, scientific computation

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2
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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
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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
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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
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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
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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
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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+...
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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
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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
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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
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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 ...
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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 ...
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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 $\alpha\cdot S<\beta.$
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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
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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=\...
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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 (...
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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)!} ...

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