# Questions tagged [na.numerical-analysis]

Numerical algorithms for problems in analysis and algebra, scientific computation

977
questions

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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 ...

**3**

votes

**1**answer

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

**2**answers

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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vote

**1**answer

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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votes

**2**answers

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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vote

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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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**1**answer

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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vote

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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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votes

**1**answer

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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votes

**1**answer

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 ...

**4**

votes

**4**answers

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:
$\...

**18**

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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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votes

**1**answer

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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**1**answer

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**

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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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votes

**1**answer

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 $\mathbb{R}^d$.
...

**2**

votes

**1**answer

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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votes

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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 ...

**3**

votes

**1**answer

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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votes

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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,...

**2**

votes

**1**answer

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

**1**answer

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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votes

**1**answer

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

**1**answer

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

**0**answers

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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votes

**1**answer

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 $\...