Questions tagged [na.numerical-analysis]
Numerical algorithms for problems in analysis and algebra, scientific computation
977
questions
0
votes
0answers
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 ...
0
votes
0answers
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^{...
0
votes
0answers
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 ...
3
votes
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 ...
6
votes
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 $A$ ...
1
vote
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 ...
2
votes
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 ...
0
votes
0answers
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 ...
1
vote
0answers
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 $A$...
3
votes
0answers
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/...
1
vote
0answers
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 ...
1
vote
0answers
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 ...
3
votes
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 \...
1
vote
0answers
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 ...
4
votes
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 ...
2
votes
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 ...
4
votes
0answers
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 ...
2
votes
0answers
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(...
1
vote
0answers
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 ...
0
votes
0answers
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 ...
4
votes
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,...
0
votes
0answers
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{...
0
votes
0answers
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 $...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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:
$\...
18
votes
0answers
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 ...
0
votes
0answers
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) ...
6
votes
0answers
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 ...
1
vote
0answers
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
$$
\begin{cases}
\partial_t u_h -\Delta_h u_h = f(x) & \text{ in } \Omega_h\\
u_h=0 &\text{ in } \...
0
votes
0answers
33 views
Bounding the working precision required in Spouge's Approximation
Spouge's approximation for the gamma function is
$\Gamma(z+1) = (z+a)^{z+\frac{1}{2}}e^{-z-a} \left(c_0 + \sum_{k=1}^{a-1} \frac{c_k}{z+k} + \epsilon_a(z) \right)$
where the coefficients are given ...
3
votes
0answers
86 views
Hardness results for approximating Hölder continuous functions
Let $f \in \mathrm{Lip}^{L,\alpha}[a,b]$, and let $f_{h} \in C^{L}$ be a spline which interpolates $f$ at $a + ih$. Then standard theorems show that
\begin{align*}
\left\| f - f_{h} \right\|_{\infty} \...
2
votes
1answer
105 views
History- calculating convolution by tabular method
I often see a trick for calculating convolution of discrete data by a so-called Tabular method. There are a lot of Youtube videos and many Indian textbooks on Signal Processing [Books].1
Basically, ...
0
votes
0answers
24 views
Reference request: numerical methods for HJB free boundary problems
Suppose $r: \mathbb{R}^{d+1}\to \mathbb{R}, \ g: \mathbb{R}^d \to \mathbb R,\ b: \mathbb{R}^{d+1}\to \mathbb{R}^d$ and $ \sigma: \mathbb{R}^d \to \mathbb R^d, d \ge 2$, and consider an optimal ...
4
votes
1answer
157 views
Non-polynomial splines, a non-linear problem
I'm looking for references on how to construct spline-like functions from a basis that does not include piecewise polynomials.
To be specific, given a class of functions such as "decaying ...
4
votes
0answers
188 views
analytic approximations of the min and max operators
Question:
What is the state of the art on analytic approximations of $\min$ and $\max$? My hunch is that numerical analysts probably have a better solution than the one I propose here.
For any $\...
0
votes
1answer
143 views
Numerical problems floating-point arithmetic
I am trying to calculate the following function in floating-point arithmetic.
$$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)...
0
votes
0answers
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 $\mathbb{R}^d$.
...
2
votes
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 ...
3
votes
0answers
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 ...
3
votes
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 ...
3
votes
0answers
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
$$[T,V](\psi) = T'(\psi) V(\psi) - V'(\psi) T(...
2
votes
0answers
51 views
Convergence of Quasi-Newton method with fixed derivative
Consider the Newton iteration
$x^{(k+1)} = x^{(k)} - DF( x^{(k)} )^{-1} \cdot F( x^{(k)} )$
to find a zero of a function $F : \mathbb R^k \rightarrow \mathbb R^k$. If we freeze the first derivative,...
2
votes
1answer
92 views
Non-trivial examples of regular Lagrangian flow in BV case
What is a concrete example of BV vector field $v$ with $\mathrm{div}\, v = 0$ that makes Ambrosio's theory of regular Lagrangian flow relevant?
With concrete I mean that we can compute the flow ...
4
votes
1answer
338 views
Taylor expansion of exponential of a Lie derivative
In this paper on page 8 the author claims that the Taylor expansion for the expression $e^{tD_V}$ where $D_V$ is the Lie derivative with respect to a vector field $V$ (defined by $(D_VG)(x) = \frac{d}{...
4
votes
1answer
286 views
Questions about a return map
Consider the following map in the interval $u\in[-1,1]$ ($U\in[-1,1]$ also)
$$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 ...
6
votes
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 ...
2
votes
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......
0
votes
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(...
0
votes
0answers
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 $\...
