Questions tagged [na.numerical-analysis]

Numerical algorithms for problems in analysis and algebra, scientific computation

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10 views

Projection estimate for finite element basis?

Let $ \mathcal{T}_n $ be equidistant mesh of $ (0,1) $ with $ n+ 1 $ nodes, $ X_n $ be corresponding finite element space, $ P_n $ be the orthogonal projection mapping $ L^2(0,1) $ onto $ X_n $. We ...
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39 views

Gradient of the function in the BFGS quasi-Newton algorithm

I have a function $f=\sum\limits_{k=1}^{K}|R_{k}^{dl}- R_{k}^{ul}|$ that I want to minimize using the BFGS Quasi-Newton algorithm. If $R_{k}^{dl} = y_k \times A \times B \times C$. $y_k$ and $R_k^{...
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30 views

Finite element method reference, from the perspective of the finite elements themselves

I found the finite element chapters in The Finite Element Method of Elliptic Problems especially enlightening and would like to learn more about the theory behind the base components of a general ...
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1answer
109 views

More important or relevant progress in discretizing hard problems in physics in last decade

This is a reference request, and soft question as companion. I'm curious to ask, from an informative point of view, what about the more important progress in the goal to discretize hard problems in ...
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2answers
97 views

Is the matrix positive definite given the Gauss-Seidel method converges?

I know that the Gauss-Seidel method converges given that the matrix you want to solve is symmetric positive definite. However, I'm wondering if the "converse" of the statement is true. Namely, if $A$ ...
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1answer
42 views

How can I find minimum and maximum eigenvalue of non-positive define matrix [closed]

There is a power iteration method, but it only returns the greatest(in absolute value) eigenvalue of matrix. So when we have negative eigenvalues it'll give wrong results. Is there any method, which ...
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2answers
75 views

Finding the nearest quadratic Bézier curve

Given a set of three-dimensional quadratic Bézier curves. I'm looking for some analytical solution to find the nearest curve to an arbitrary point in space. Example I already have a brute force ...
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66 views

How to solve a non-local self-consistent equation

I have been struggling lately with solving numerically an equation of the form: $$ g(x\pm x_{0}) = F[ g(x) ] $$ where $g(x)$ is a matrix satisfying the condition $g(x\to\pm\infty)=0$. My question is ...
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36 views

Fastest way to calculate the eigenvalues of a product of two Toeplitz matrices

I have the following problem: I need to find the fastest way to calculate the eigenvalues of a matrix that is the product of two Toeplitz matrices. $B = A U$. The first is a regular Toeplitz matrix $A$...
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66 views

The numerical evaluations of special values of $L$-functions associated to modular forms

LMFDB has a database of lots of classical modular forms. Suppose we have a modular form $f$ listed in LMFDB, which usually gives the first 100 terms of the $q$-expansion. Are there any packages/...
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120 views

A question about rationality, irrationality or transcendence of definite integral [closed]

Forgive me for the following fundamental question. But I think I require the accuracy of an expert. Consider an integral of the form: $$\int_a^b f(x)dx,$$ where $f(x)$ is analytic and real valued for ...
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26 views

Choice of finite element spaces in plasticity

I am planning to run numerical simulations in metal elastoplasticity (von-Mises yield condition with and without isotropic hardening). However, I am completely new to this subject and I am unsure ...
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1answer
99 views

Euler–Maclaurin formula in $\mathbb{Z}^d$

I was wondering whether there is a Euler–Maclaurin formula of sorts for expressions such as $$ \sum_{x \in [a,b]^d\cap \mathbb{Z}^d} f(x) - \int_{[a,b]^d}f(x) $$ where $d\ge 2$ is an integer, $a,b \...
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12 views

One-sided Jacobi SVD and Divide&Conquer SVD stability and cost [closed]

I'm studying SVD, in particularly the Jacobi SVD and Divide&Conquer SVD algorithms. I can't find anything on the stability and error analysis on these methods. Also can someone show show me what's ...
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1answer
68 views

Rate of convergence of Padé approximants

Let $f$ be an entire function of order $1$. Two questions: 1) Can one assert that the diagonal Padé approximants converge to $f$ (pointwise or uniformly over compacts of $\mathbb C$)? 2) if yes, can ...
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1answer
58 views

Discrete curve-shortening flow – numerical implementation

I need to investigate the properties of open curves which evolve according to the standard curve-shortening flow (Wikipedia link), but with fixed extremes as boundaries (si it should converge to the ...
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102 views

Intrinsic numerical methods on Riemannian manifolds

I am interested in numerical methods for ordinary differential equations on a Riemannian manifold $M$. The general form of such an equation is $\dot x(t)=V(x(t)), x(0)=x_0 \in M$, where $V$ is a ...
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56 views

Failure in numerical experiment of singular integral equation?

Define \begin{equation} G(t,s) := - \frac{1}{2\pi} \left[\ln \left(4 \sin^2 \frac{t-s}{2}\right) -1 \right] \quad (t \neq s) \end{equation} and \begin{equation} K_0 \Psi := \int^{2\pi}_0 G(t,s) \Psi(...
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75 views

Solving numerically a linear ODE

I start by saying that I do not have a strong background in numerical analysis, so I may miss some basic things or make trivial mistakes. Motivated by some problems in digital signal processing, I ...
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26 views

ADMM for solving linear systems

I would like to use ADMM for solving $Mx=b$, where $M\in \mathbb{R}^{R\times R}$ is symmetric and positive definite. I know that a lot of methods will do for me in this case, but I'm specially ...
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4answers
222 views

algorithm for convex $C^2$ interpolation

Let $x_0<x_1<\ldots<x_n$ and $f_0,f_1,\ldots,f_n$ be real numbers and $$s_i=(f_i-f_{i-1})/(x_i-x_{i-1}),~~~c_i=(s_{i+1}-s_i)/(x_{i+1}-x_{i-1}).$$ If $f$ is a convex function defined on $[x_0,...
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34 views

First eigenfunction of the p-Laplacian in an interval $(a,b) \subset \mathbb R$

What is the explicit expression of the first eigenfunction $u$ of the $p$-Laplacian ($p>1$) in a bounded interval $(a,b) \subset \mathbb R$ (up to multiplicative constant)? \begin{equation} \begin{...
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39 views

How to solve a system of second order ODE from time t = T to t = 0

I have a system of second-order ODEs $$ \mathbf{M\ddot{x} + C\dot{x} + Kx = f} $$ I want to know some good numerical methods to solve this system of the equation given the initial conditions at time $...
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27 views

Consistency results of optimal control solutions using direct transcription

I have read a number of books on discrete time solutions to continuous time optimal control problems. What is not clear to me is what consistency results exist with respect to showing the discretized ...
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59 views

Compute weights (analytically or numerically) so that they minimize an integral

I have a measurable function $f:E\to[0,\infty)$, measurable weights $w_1,\ldots,w_k:E\to[0,1]$ with $\sum_{i=1}^kw_i=1$, positive probability densities $q_1,\ldots,q_k:E\to[0,\infty)$ and measurable ...
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66 views

ODE operator splitting with second order time discretization not possible?

I am trying to solve an ordinary differential equation (ODE) using an operator splitting approach: $\frac{\partial f}{\partial t} = A(f) + B(f)$ Let's assume that $A$ and $B$ are very simple: $\...
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600 views

I found a (probably new) family of real analytic closed Bezier-like curves; is it publishable?

Given $n$ distinct points $\mathbf{x} = (\mathbf{x}_1, \ldots, \mathbf{x}_n)$ in the plane $\mathbb{R}^2$, I associate a real analytic map: $f_{\mathbf{x}}: S^1 \to \mathbb{R}^2$ with the following ...
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74 views

Existence of the inverse Fourier transform, Carr Madan

I have a function $C_T(k)$ that is not $L_1$, because its limit in negative infinity is a constant. So I dampened it by $ e^{\alpha k} $. Let's call the transformed function (of the dampened function) ...
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204 views

Numerical analysis with p-adic numbers

How should one go about doing numerical analysis with $p$-adic numbers? By that I mean, how should one go about implementing numerical integration (using analogues of Newton-Cotes or perhaps Gaussian ...
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117 views

Rigorous error estimate for semi-discrete heat equation in bounded domain

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb R^N$ and $u_h$ be a solution of $$ \begin{cases} \partial_t u_h -\Delta_h u_h = f(x) & \text{ in } \Omega_h\\ u_h=0 &\text{ in } \...
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33 views

Bounding the working precision required in Spouge's Approximation

Spouge's approximation for the gamma function is $\Gamma(z+1) = (z+a)^{z+\frac{1}{2}}e^{-z-a} \left(c_0 + \sum_{k=1}^{a-1} \frac{c_k}{z+k} + \epsilon_a(z) \right)$ where the coefficients are given ...
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86 views

Hardness results for approximating Hölder continuous functions

Let $f \in \mathrm{Lip}^{L,\alpha}[a,b]$, and let $f_{h} \in C^{L}$ be a spline which interpolates $f$ at $a + ih$. Then standard theorems show that \begin{align*} \left\| f - f_{h} \right\|_{\infty} \...
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1answer
105 views

History- calculating convolution by tabular method

I often see a trick for calculating convolution of discrete data by a so-called Tabular method. There are a lot of Youtube videos and many Indian textbooks on Signal Processing [Books].1 Basically, ...
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24 views

Reference request: numerical methods for HJB free boundary problems

Suppose $r: \mathbb{R}^{d+1}\to \mathbb{R}, \ g: \mathbb{R}^d \to \mathbb R,\ b: \mathbb{R}^{d+1}\to \mathbb{R}^d$ and $ \sigma: \mathbb{R}^d \to \mathbb R^d, d \ge 2$, and consider an optimal ...
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1answer
157 views

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 ...
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188 views

analytic approximations of the min and max operators

Question: What is the state of the art on analytic approximations of $\min$ and $\max$? My hunch is that numerical analysts probably have a better solution than the one I propose here. For any $\...
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1answer
143 views

Numerical problems floating-point arithmetic

I am trying to calculate the following function in floating-point arithmetic. $$f(c,z)=\frac{(c-1)z}{(z-1)^2}\left( \sum_{k=2}^{c-1}\frac{1}{c-k}\left(\frac{z-1}{z}\right)^k-\left(\frac{z-1}{z}\right)...
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82 views

Smallest eigenvalue for large kernel matrix

I am interested in the the asymptotics of the minimum eigenvalue $\lambda_n^n$ of a class of kernel matrix $P = [ K(x_i - x_j) ]_{i,j}$, with $x_i$ equally spaced in the unit cube of $\mathbb{R}^d$. ...
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1answer
149 views

A numerical calculation for an integral

I am interested in the numerical calculation of $$ F(\eta)=\frac{2}{π}\int_0^{+\infty}\sin(t\eta-t^3)\frac{dt}{t}\quad\text{for $\eta\ge 0$}. $$ I believe that the function $F$ is bounded, but I do ...
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125 views

Accuracy of Richardson's error estimate in the presence of rounding errors

Richardson extrapolation is a well known technique in scientific computing. It is used to compute error estimates when our approximations can be viewed as a function of a parameter $h$. Common ...
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1answer
103 views

summation of oscillating functions

Consider series of the form $S=\sum_{n\ge1}f(n)P(n)$, where $f$ is some smooth function, and $P$ is a periodic or quasi-periodic function (e.g., $P$ can be a trigonometric function, so $S$ a Fourier ...
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111 views

Why is this identity about commutators of Lie derivatives true?

I am reading the paper "On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations" by Lubich. On page 2147 the author claims $$[T,V](\psi) = T'(\psi) V(\psi) - V'(\psi) T(...
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51 views

Convergence of Quasi-Newton method with fixed derivative

Consider the Newton iteration $x^{(k+1)} = x^{(k)} - DF( x^{(k)} )^{-1} \cdot F( x^{(k)} )$ to find a zero of a function $F : \mathbb R^k \rightarrow \mathbb R^k$. If we freeze the first derivative,...
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1answer
92 views

Non-trivial examples of regular Lagrangian flow in BV case

What is a concrete example of BV vector field $v$ with $\mathrm{div}\, v = 0$ that makes Ambrosio's theory of regular Lagrangian flow relevant? With concrete I mean that we can compute the flow ...
4
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1answer
338 views

Taylor expansion of exponential of a Lie derivative

In this paper on page 8 the author claims that the Taylor expansion for the expression $e^{tD_V}$ where $D_V$ is the Lie derivative with respect to a vector field $V$ (defined by $(D_VG)(x) = \frac{d}{...
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1answer
286 views

Questions about a return map

Consider the following map in the interval $u\in[-1,1]$ ($U\in[-1,1]$ also) $$U=u-\frac{64}{3}u^{3}+64u^{5}-64u^{7}+\frac{64}{3}u^{9}$$ It has 3 fixed points at $u=0,\pm 1$. If we compute the ...
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1answer
84 views

Numerical method for simultaneous computation of eigenvalues of a family of commuting matrices

I have a problem where I have $n$ commuting matrices $M_1,\dots,M_n$. It is a well-known fact that commuting matrices are simultaneously diagonalizable/triangularizable. I need to find the eigenvalues ...
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0answers
89 views

Is there an easy way to solve an “almost quadratic” equation?

I have an equation of the form: $ax^{k_1} + bx^{k_2} + cx^{k_3} + dx^{k_4} + e = 0$ where $a$, $b$, $c$ and $d$ are arbitrary real numbers; $k_1$ and $k_2$ are positive reals in the range of 1.92......
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1answer
70 views

Convergence of Chebyshev interpolation in L^1

Let $f\in C^0([-1,1])$ and $P_n(f)$ its interpolation polynomial at the Chebyshev nodes. I would be interested to know about any existing results (positive or negative) about the convergence of $P_n(...
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57 views

Can we numerically solve this saddle-point problem?

Let $(E,\mathcal E,\lambda)$ be a measure space; $f:E\to[0,\infty)^3$ be $\mathcal E$-measurable with $\|f\|\in\mathcal L^2(\lambda)$; $\tilde p:=\alpha_1f_1+\alpha_2f_2+\alpha_3f_3$ for some $\...

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