Questions tagged [na.numerical-analysis]

Numerical algorithms for problems in analysis and algebra, scientific computation

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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}{...
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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 ...
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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 ...
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Approximation of inverse trigonometric functions [closed]

I would like to implement algorithms from scratch for inverse trigonometric functions (inverse of sine, cosine, and tangent). I found on Wikipedia that those functions are approximated by the Taylor ...
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0 votes
1 answer
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FEM based solution to parabolic problem

Consider the problem $$ \begin{cases} u_t - \Delta u = 0 &\text{ on } \Omega\times (0,T)\\u=0 &\text{ on } \partial \Omega\times (0,T) \\ u(x,0)=g(x) &\text{ on } \Omega \end{cases} $$ ...
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1 vote
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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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What are the convergence requirements for Inverse Power Method?

I'm struggling to find the convergence requirements for the Inverse Power Method. I implemented this method in MATLAB as shown below. ...
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Fourier spectral methods for an elliptic equation

I would like to study a linear elliptic problem on the torus $\mathbb{T}^n$ (i.e. periodic boundary conditions) which is of the following form: $$ -\Delta u + b^i \partial_i u + c u = f $$ where $b^i$,...
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3 votes
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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(...
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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 $x\in \mathbb{R}^m, y\in \mathbb{R}^n$. ...
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1 answer
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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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6 votes
1 answer
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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 ...
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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 ...
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29 views

Finding the maximum of a certain trigonometric series

Let $a_0,_1,a_2,\dots,a_n$ be a finite sequnce of $n $ real numbers and consider the function $$f(t)=\sum_{j,k=0}^{n}a_j a_k \cos\left((j-k)t\right)$$ for $t \in [0,2 \pi]$ .If we want to maximise ...
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1 vote
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Generating Hermite polynomial with coefficient recurrance relation algorithm

I am writing a math paper for my numerical analysis class about orthogonal Hermite polynomials. I want to implement the algorithm for generating the "probabilist's Hermite polynomials": $$ \...
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1 vote
0 answers
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p-adic taylor polynomial [closed]

This might be an easy question but i am sorry for asking this. Let $f(x)\in\mathbb{Z}_p[x].$ Is it always true that $$f(x+y)=f(x)+f'(x)y+f''(x)\frac{y^2}{2}+zy^3$$ for some $z\in\mathbb{Z}_p.$ if it ...
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2 votes
0 answers
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Iterative method of finding root

I don't know much about numerical analysis. I need the following for help with my research in number theory. Is there a simple(not multistep) iterative method of finding the root of a real-valued ...
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0 votes
0 answers
57 views

Sampling a distribution related to factoring

Consider these two problems. Given two numbers $a$ and $N$, find the smallest $r$ such that $a^r= 1 \pmod N$. Given three numbers $a$, $r$, and $N_{max}$, find $N\le N_{max}$ such that $a^r= 1 \pmod ...
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6 votes
3 answers
350 views

How to numerically compute $x \ln x$ and related functions near $0$?

I was recently trying to find a numerical solution to a thermodynamics problem and the expression $x\ln x$ appeared in one of the computations. I did not have to find its value very near $0$, so the ...
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1 vote
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Minimizer for a certain variational problem

In the process of estimating certain hitting probabilities for Brownian motion I've run into the following variational problem. Let $\mathbb{D}\subseteq{\mathbb{C}}$ be the unit disk in the plane, ...
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0 votes
1 answer
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Maximizing a skew-symmetric 4D cross product

How do I find two orthonormal 4D vectors, $(x_0,x_1,x_2,x_3)$ and $(y_0,y_1,y_2,y_3)$, which maximize this function: $-19x_1y_0 - 33x_2y_0 + 11x_3y_0 + 19x_0y_1 - 21x_2y_1 - 33x_3y_1 + 33x_0y_2 + ...
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  • 1,425
4 votes
1 answer
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Singular value decomposition of truncated discrete Fourier transform matrix

Let $\mathbf{F}$ be a discrete Fourier transform (DFT) matrix such that \begin{align} F_{m,n}=e^{-j2\pi(m-1)(n-1)/N},\quad m,n=1,\ldots,N. \end{align} What we can say about the singular value ...
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  • 129
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References for discrete calculus of variations

What would be good references for discrete calculus of variations? For applications such as minimizing a functional not on a $[0,1]\times[0,1]\to \mathbb{R}$ but on a bitmap image that approximates ...
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Convergence conjugate gradient method

For Conjugate Gradient (to solve a linear system $Ax=b$) there is a theorem: "if $m$ is the number of distinct eigenvalues of matrix $A$, then the Conjugate Gradient method converges at the ...
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4 votes
0 answers
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How can I numerically solve the Laplace equation with cohomological data?

Consider the problem of solving for $u$ where $-\Delta u = f$, $[u] = [g]$ where $[\cdot]$ denotes cohomology class and $u, f, g$ are $p$-forms on a Riemannian manifold $M$. If $g$ instead was ...
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5 votes
3 answers
181 views

closest equidistant point to N points in M dimensions

Is there a formula/algorithm/etc. to find the closest equidistant point (assuming it exists) to a set of points, allowing that the number of dimensions of the space is independent of the number of ...
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  • 425
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0 answers
38 views

Padé–Hermite approximants of the exponential of type II

I found the explicit expression of the Padé–Hermite approximants of the exponential function, of type I by complex integrals (see for instance Khémira - Approximants de Hermite–Padé, déterminants d’...
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3 votes
1 answer
173 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 $...
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0 answers
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Fitting a model

I have a function expressed as the ratio of two exponential series with certain parameters $$\frac{\sum\limits_{j=1}^{i-1} \frac{e^{-ar_jt}}{\prod_{l=1\\l \ne j}^{i-1} (b^j-b^l)}}{\sum\limits_{j=1}^{i}...
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2 votes
0 answers
44 views

Harmonic function over a square with linear Neumann boundary conditions

For a rectangle with height 1 and length 2, here is the unique numerical solution (showing contours of the equipotential from 0, defined by the bottom, to 0.54, the numerically-calculated maximum) to ...
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  • 1,425
0 votes
1 answer
108 views

Are root finding algorithms stable for bounded polynomials? [closed]

Suppose that we have a bounded polynomial defined on $[0,1]$. I think because it is just polynomial, root finding algorithms would easily and without any instability find all its roots. Am I right? ...
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  • 113
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0 answers
62 views

Translating between interpolation vs approximation

tl;dr: Is there a way to translate interpolation error estimates into approximation error estimates for non-interpolating polynomials? Given a function $f\in L^2(\Omega)$ and a basis $\{\phi_n\}$ for $...
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0 answers
42 views

Error estimates for orthogonal polynomial approximation

tl;dr: Are there explicit bounds for the approximation error by orthogonal polynomials? There are various ways to formulate this question more precisely, so want I emphasize up front that this is a ...
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6 votes
0 answers
187 views

Where to cut off a double sum?

I have to compute a double infinite sum to within a given accuracy $\epsilon$. Let us say the sum is of the form $$\sum_{m\geq 1} \sum_{n\geq 1} \frac{a_{m,n}}{m^2 n^2 \max(m,n)},$$ where $|a_{m,n}|\...
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2 votes
1 answer
106 views

Is it possible to obtain orthogonal (but not normalized) vectors after QR factorization?

After QR decomposition of a matrix, $M$, the columns of Q are orthonormal. Is it possible after obtaining Q, we recover unnormalized column vectors from $Q$? For example, the matrix M has the ...
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1 vote
0 answers
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Largest nuclear norm of $n \times n$ symmetric matrices whose entries are between -1 and 1

Let $\cal M$ be the set of real symmetric $n \times n$ matrices whose entries are all in the interval $[-1, 1]$. I'm interested in understanding the largest possible nuclear norm of these matrices as ...
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3 votes
1 answer
97 views

Proof of Levinson-Durbin algorithm

Is there any article or reference book with a full proof of the Levinson-Durbin algorithm used for solving linear system with a Toeplitz matrix ?
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1 vote
0 answers
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P1-finite element as convolution of P0-finite element

For a vector $u\in\mathbf{R}^N$ let's denote $\pi_N(u)$ the unique piecwise linear and $1$-periodic function matching the components of $u$ on the discretization $x_k = \frac{k}{N}$ of the unit ...
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  • 2,058
1 vote
1 answer
143 views

Computing global maximum

For $\lambda\in\mathbb{R}$, I want to find the expression of $f(\lambda)$: $$f(\lambda)=\max_{E\in\mathbb{C}}arccosh{\frac{|E^2+i\lambda E|+|E^2+i\lambda E-4|}{4}}-arccosh{\frac{|E^2-i\lambda E|+|E^2-...
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  • 91
1 vote
0 answers
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Numerical calculation of a double integral from the slowly-decaying oscillating function

Let us consider the following integral $$ I = \int\limits_{0}^{+\infty}dx\int\limits_{-\infty}^{+\infty}dy \left[f(x,y) + g(x,y) \right]. $$ We know several properties of these functions. There are ...
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2 votes
0 answers
56 views

What are desirable properties that data should satisfy to reasonably use the dynamic mode decomposition?

In the dynamic mode decomposition, we consider a sequence of data vectors $\{z_0, \dots, z_m\}$ where $z_k \in \mathbb{R}^n$ for all $n$. We assume that the data satisfies the linear relationship $z_{...
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1 vote
3 answers
247 views

Determining polynomial approximations of piecewise constant functions

Let $t_1 < t_2 < \cdots <t_m$ be real, and $X = \cup_{i=1}^{m-1} (t_i, t_{i+1})$ be a union of real open intervals. Let $f:X \rightarrow \{-1, 1\}$ be any piecewise constant function of form ...
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  • 823
1 vote
0 answers
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How does a computer program recognize shocks given data of a solution to a conservation law?

Conservation laws are PDEs of the form $u_t +j_x=0.$ A discontinuous solution (for $u$ and $j$) to an equation like this can be easily found. Let's suppose that we are working with a piecewise ...
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2 votes
0 answers
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Is there a generalisation of this perturbation result about rank-one modifications of diagonal matrices?

In Theorem 1 of [1] we have the following result: Let $D$ be a real $n \times n$ diagonal matrix and consider the rank-one modification $C = D + \rho z z^T$, where $\rho > 0$ is a real scalar and $...
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0 votes
0 answers
71 views

Is there an efficient algorithm to project a vector onto the eigenbasis of a symmetric matrix?

Let $H$ be a symmetric matrix over $\mathbb R^n$. Given some vector $u$, I would like to express $u$ in the eigenbasis for $H$. Can this be done efficiently, perhaps using some kind of iterative ...
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  • 611
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Coordinate gradient descent in small time-step limit

I'm looking for a reference, proof or intuitive explanation that the $n$-dimensional coordinate-wise gradient descent sequence with step size $h>0$ converges to the $n$-dimensional gradient flow. ...
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  • 143
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0 answers
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Polynomial Lagrange splines

Question: if $x_0 \lt x_1\lt\cdots\lt x_{N-1}\lt x_N$, how can the spline-analogues $\mathscr{L}_i(x)$ of Lagrange polynomials be calculated when they are defined via $\mathscr{L}_i^n(x)\in C^{n-1}$ $...
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1 vote
1 answer
80 views

Typo in error a-priori estimate in a discontinuous Galerkin paper?

I'm looking at this famous paper which is available in the link below: Franco Brezzi, LD Marini, Endre Süli, Discontinuous Galerkin methods for first-order hyperbolic problems, Mathematical Models ...
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0 answers
64 views

Why does explicit Runge-Kutta 4 allow positive eigenvalues?

When solving a system of ODEs: $$\dot{x} = A x,$$ we call the system unstable when the eigenvalues of A have a positive real part. However, the stability region of the explicit Runge-Kutta 4 method ...
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  • 17
5 votes
1 answer
142 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 ...
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