Questions tagged [na.numerical-analysis]

Numerical algorithms for problems in analysis and algebra, scientific computation

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2 votes
1 answer
229 views

How to solve this set of equations as efficiently as possible (with "efficiently" measured in FLOPS)?

The system of equations is the following: $$ \Gamma_i^{\ -1} = \sum_{i=1}^nA_{ij}\Gamma_j, $$ where $\Gamma = (\Gamma_i)$ is a vector of size $n$ and $A$ is a matrix of size $n\times n$, with $n \gt ...
3 votes
1 answer
105 views

Converting an algebraic equation into a ODE

I'm working on a method to solve algebraic equations by converting them into ordinary differential equations (ODEs) and then integrating these ODEs over time. Given an algebraic equation $f(x(t), t) = ...
-1 votes
0 answers
27 views

Numerical solution 2-D Poisson equation [closed]

I have some kind of Poisson equation: $-au_{xx}-bu_{yy} = f(x,y)$, $\ a,b>0$ and trying to solve it using FDM and Iteration methods (no matter which one, each method has the same problem). $$-a\...
2 votes
1 answer
129 views

Are the coefficients in the stationary phase approximation computed explicitly somewhere

In Stein's "Harmonic analysis" book, page 334, one can find the asymptotic expansion An instructive proof is given for the case $k=2$. It is clear enough to generalize to the cases $k\geq ...
0 votes
0 answers
71 views

Interpolation polynomials with constraints

Lets consider a collection of $n$ points $\{Z_i\}_{i=1}^n\subseteq \mathbb{D}^k(1)=\{(z_1,\ldots,z_k)\in\mathbb{C}^k:\forall j\leq k, |z_j|\leq 1\}$. Let $h: \{Z_i\}_{i=1}^n \to \mathbb{D}^1(1)$ be a ...
0 votes
0 answers
72 views

What the scientists in numerical solutions of PDE are concerned recently? [closed]

I'm a new one in the numerical solution of PDE, mainly interest in NS equations, Convection-Diffusion equations and I know a lot of people are solving equations by ML, but I really don't like this ...
1 vote
1 answer
274 views

Numerical estimation of partial derivatives of convolved functions when closed forms do not exist

Summary: Some peak functions are convolutions which may not have a closed form solution. A classical example can that of a Voigt which is a convolution of a Lorentzian and a Gaussian, followed by ...
1 vote
0 answers
43 views

Galerkin’s Method for hyperbolic PDEs: proving convergence without using compactness

Lawrence Evan's PDE book prove the existence of solution to the following problem where $L$ is an elliptic operator: $$ \begin{cases} u_{tt} = -Lu+f,\\ u|_{t=0} = u_0,\\ u|_{\partial U} = 0 \end{cases}...
10 votes
5 answers
14k views

Numerical differentiation. What is the best method?

What is the best method for 1D numeric differentiation? Something as glorious as Gaussian quadrature for numeric integration. Maybe differential quadrature is such a method? What is its accuracy? I'...
4 votes
1 answer
383 views

Linear convergence rate of proximal point algorithm

For $T : R^n \to P({R^n})$ maximally monotone, the proximal point algorithm (step size $c>0$) $$ x^{k+1} = (I + c T)^{-1} x^k, $$ converges linearly with rate $\kappa = \frac{1}{1 + c \sigma}$ if $...
2 votes
2 answers
426 views

Approximating a subclass of $L^2(\mathbb{R})$ by Schwartz functions within similar subclass

It is well-known that real valued Schwartz functions on $\mathbb{R}_+$ $\mathcal{S}(\mathbb{R}_+)$ are dense in the set of square integrable functions on $\mathbb{R}_+$ $L^2(\mathbb{R}_+)$. We can ...
2 votes
2 answers
287 views

Reference request on computational schemes for $\inf_{x\in\Omega^n}\sup_{y\in\mathbb R^n}F(x,y)$

Let $\Omega\subset \mathbb R^d$ be compact, $\rho$ be a density function on $\Omega$ and $p_1,\ldots, p_n\in (0,1)$ be weights satisfying $\int_{\Omega}\rho(z)dz=1=\sum_{k=1}^n p_k$. We consider the ...
0 votes
0 answers
47 views

Concentration of bilinear forms

This is a bit vague so I'll begin by indicating the motivation. I am looking for ways to [do something interesting or useful] with the self-attention in transformer models. Ultimately the self-...
8 votes
2 answers
3k views

Condition number related to root finding problems

Suppose we want to find the root of the equation $f(x)=\phi(x) - d = 0$, where d is a real constant and $f$ is continuously differentiable function. The problem is well posed if the inverse $\phi^{-...
0 votes
1 answer
54 views

Integration algorithm and analytic property

This question is the continuation of the previous one. In the article about the integration of analytical polynomial - time computable function $f(x)$ with the Taylor series $$f(x) = \sum_{n=0}^{\...
1 vote
0 answers
112 views

Integration in polynomial time

The work of Friedman and Ko and Müller guarantee the polynomial time computability of the integrals of analytic functions inside the circle of convergence. But do algorithms have practical value? Is ...
1 vote
0 answers
30 views

Error bounds for a Romberg-style improvement of a non-linear approximation

I have a (possibly non-linear) functional $F$, which I want to numerically approximate by a (typically non-linear) $\widehat{F}_h$. For a suitable class of functions, I have asymptotic error behaviour ...
1 vote
0 answers
42 views

Computing smallest singular value of a matrix with explicit error control?

Many good algorithms are out there computing truncated SVD: What is the time complexity of truncated SVD?. I am trying to implement some codes to find the smallest singular value of a big matrix $A$. ...
14 votes
3 answers
2k views

Guaranteed correct digits of elementary expressions

Let $E$ be the set of elementary expressions, defined as the smallest set of mathematical expressions such that $1 \in E$, and if $a,b \in E$ then $a + b$, $a - b$, $ab$, $a/b$, $\exp(a)$, $\log(a)$ ...
0 votes
0 answers
30 views

Ensuring symmetry in mixed derivatives using RBF-FD method

I'm working on a numerical problem where I have the first-order partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ of a bivariate function $f(x, y)$ at a set of ...
-1 votes
1 answer
162 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 ...
3 votes
2 answers
475 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 ...
5 votes
2 answers
331 views

Locating the maximum point $x_n$ of $f_n(x):=e^{-1/x}\Bigl(1+\frac{1}{n^2 x^n} \Bigr)$ in $(0,1)$

I am trying to observe the behavior of $x_n \in (0,1)$ defined such that the function \begin{equation} f_n(x):=e^{-1/x}\Bigl(1+\frac{1}{n^2 x^n} \Bigr) \end{equation} attains its maximum inside the ...
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 $g$, ...
2 votes
1 answer
233 views

Computing 3-term connection coefficients for wavelets

I am trying to calculate the three-term connection coefficients $$ Λ_{l,m}^{d_1,d_2,d_3} = \int_{-\infty}^\infty \varphi^{(d_1)}(x) \varphi^{(d_2)}_l(x) \varphi^{(d_3)}_m(x) dx $$ for Daubechies ...
1 vote
2 answers
184 views

Numerical integration method that doesn't involve derivative in the error bound

Consider the integral $\int_a^b f(t)dt$. There are many numerical integration methods, like trapezoidal rule, Simpson's rule, Gaussian quadrature, but all they involve derivative in the error bound. ...
0 votes
0 answers
31 views

Convergence of numerical scheme for HJB equation

Convergence of numerical scheme for HJB equation has been widely studied, the key paper is the Barles's one. Essentially, the convergence is guarenteed if the scheme is: Consistent Stable Monotony ...
0 votes
0 answers
41 views

Godunov splitting convergence research

The approximation of Godunov splitting on certain differential equations is known to be first order accurate. In 2011, a paper has also shown that it is first order accurate for nonlinear ordinary ...
1 vote
1 answer
271 views

Best approximation with tensors of rank $\ge2$

Let $k\in\mathbb N$, $H_i$ be a (finite-dimensional, if necessary) $\mathbb R$-Hilbert space for $i\in I:=\{1,\ldots,k\}$, $H:=\bigotimes_{i\in I}H_i$ denote the tensor product$^1$ of $(H_i)_{i\in I}$ ...
0 votes
1 answer
474 views

How do I get an analytical solution to this nonlinear equation?

I posted this question over on Math Stack Exchange (link), but have not received a response. I'm wondering if it's too complicated for that audience, so I'm posting it here in the hopes that someone ...
1 vote
5 answers
581 views

Avoiding overfitting by averaging polynomials fit to part of the data?

I was thinking about the problem of overfitting data. Suppose you have a hundred data points sampled from an unknown function (call this the training set). You could try fitting a hundred-...
0 votes
0 answers
49 views

How to solve with FEM a semilinear elliptic equation?

I searched in many books regarding FEM how to solve semilinear elliptic equation, but I did not find too many things. They mostly treat linear and simple problems. For example in P.Ciarlet-The finite ...
1 vote
0 answers
174 views

Connection of eigenspace of finite Hilbert matrix and its continuous operator counterpart

I am trying to understand the connection between the eigenspace of the continuous operator $$ H(x,y) = \frac{1}{x+y} $$ which is nothing but the square of the Laplace operator, and its discrete ...
1 vote
0 answers
47 views

Computing geodesic length of Euclidean lines in the manifold of positive definite matrices

I am working with the manifold of positive definite matrices $PD(n)$ equipped with the affine-invariant Riemannian metric (AIRM) $g_P(V,W):=tr(P^{-1}VP^{-1}W)$, where $P \in PD(n)$ and $V,W \in T_P PD(...
7 votes
1 answer
7k views

Upper bound on largest eigenvalue of a real symmetric $n \times n$ matrix with all main diagonal entries positive, everywhere else nonpositive

Is there a good analytic upper bound on the largest eigenvalue of a real symmetric n*n matrix with all main diagonal entries strictly positive, all other entries <=0 with typically many of them ...
4 votes
0 answers
192 views

Computational complexity of zeros of an analytic function

The work of Friedman and Ko, page 342, Corollary 4.3.1 states that all zeros of analytic polynomial time computable function are polynomial time computable, but for me that is not clear how it could ...
3 votes
1 answer
267 views

Root finding algorithm for an analytic function

Given an analytic function $f(x)$. What is the best algorithm to find roots on the interval $[a,b]$ inside the radius of convergence> What is its complexity with respect to the length of input of ...
3 votes
1 answer
156 views

Numerical scheme for convex optimization

Given $(e_n)_{-N\le n\le N}\in\mathbb R^{2N+1}$ and $-1<x<1$, solve \begin{eqnarray} &&\max_{(q_n)_{-N\le n\le N}\in\mathbb R^{2N+1}_+}~ \sum_{n=-N}^N (e_n-\log(q_n))q_n \\ \mbox{s.t.} &...
0 votes
1 answer
137 views

Where can I find the paper by Tappert and Hardin on split-step Fourier transform method?

The split-step method is a numerical method that can be used to solve a nonlinear PDE (https://en.wikipedia.org/wiki/Split-step_method). Even Wikipedia does not refer to the original authors (F.D. ...
0 votes
1 answer
94 views

Explicit expression of Padé–Hermite approximant of type I

It is well known that the Padé approximants $(P,Q)$ of an analytic function in the neighborhood of $0$ can be expressed as a quotient of Hankel determinants built on the coefficients of the function $...
1 vote
1 answer
540 views

Strong convergence of differential quotient in $L^2(0,T;V^*)$

I have got a problem regarding the weak differentiability of Bochner-integrable functions. Let $(V,H,V^*)$ be a Gelfand-triple and \begin{align*} w \in W(0,T) := \{w\in L^2(0,T;V) ~\vert~ \exists w' \...
1 vote
1 answer
191 views

Solving (or approximating) a certain delay differential equation

I'm interested in finding the (unique?) solution to the set of delay differential equations $$f_w(w,x) = xf(w,w^2x)+w^3x^2f(w,w^4x), $$ $$f_x(w,x) = wf(w,w^2x)$$ With the initial condition $f(1,x) = e^...
4 votes
2 answers
626 views

Difference between Chebyshev first and second degree iterative methods

Consider linear equation $Au = f$. We want to solve it with iterative method (assuming $A$ is good). First order iterative method is: $$ u^{k+1} = u^k - \alpha_{k+1}(Au^k - f), $$ The second degree ...
2 votes
0 answers
64 views

Are there spectral Galerkin methods for PDE of the form $\partial_tu=\nabla\cdot f(\nabla u)\nabla u$?

Question is in the title. The nonlinearity due to the term $f(\nabla u)$ makes it difficult to directly apply the spectral Galerkin method as it can be done for PDE of the form $\partial_tu=\nabla\...
0 votes
0 answers
70 views

Error bound for stochastic gradient descent method

To solve an optimization problem $\min_x G(x)$ using standard stochastic gradient descent method, we let $x_0$ be the initial point and $x_k$ be the $k$-th point such that \begin{equation} x_k = x_{k-...
1 vote
1 answer
173 views

Average distance between points of lower dimensional simplices in $\mathbb R^n$

Notation: By a simplex, we mean the convex hull of a finite set of distinct points in $\mathbb R^n$, which are called the vertices of the simplex. $\mathcal H^n$ will denote the $n$-dimensional ...
0 votes
1 answer
32 views

How to handle the evaluation of functions on staggered ghost nodes?

I have a convection-diffusion-reaction steady state PDE in the form $$ \frac{\partial C}{\partial x} = \frac{1}{u_0(x)}\left(\frac{\partial}{\partial z} \left( \mathcal{D}(z) \frac{\partial C}{\...
3 votes
0 answers
123 views

Series acceleration for $\sum_{k=0}^\infty\left(\frac{H^k}{k!}\right)^\beta$, $\beta\ll 1$

The probability mass of the Conway-Maxwell-Poisson variable $K$ is given by $$ \mathsf P(K=k)=\frac{1}{Z(H,\beta)}\left(\frac{H^k}{k!}\right)^\beta $$ where $$ Z(H,\beta)=\sum_{k=0}^\infty\left(\frac{...
0 votes
1 answer
124 views

Matrices and vectors of intervals

I'm working on a project and think that matrices and vectors of intervals will be useful. I'm aware about interval arithmetic, but there is little information on the internet, regarding matrices and ...
3 votes
2 answers
1k views

About Newton's forward and backward interpolation

I am new to Math Overflow, so I am not really sure whether this question fits the community standards. But, I posted this question in Stack Exchange and recieved no answers. Moreover, nothing even ...

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