Questions tagged [numerical-integration]

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119 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 ...
2
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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(...
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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 $...
2
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0answers
66 views

Minimizing $\int\lambda({\rm d}y)\frac{\left|g(y)-\frac{p(y)}c\lambda g\right|^2}{r((i,x),y)}$ with respect to discrete parameter $i$

Let $I\subseteq\mathbb N$ be finite and nonempty, $(E,\mathcal E,\lambda)$ be a $\sigma$-finite measure space, $$\lambda f:=\int f\:{\rm d}\lambda$$ for $\lambda$-integrable $f:E\to\mathbb R$, $p:E\to(...
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0answers
97 views

How can we draw samples of this discrete probability distribution?

I'm running the Metropolis-Hastings algorithm with target distribution $\hat\mu$ (see definitions below) and proposal kernel $\hat Q$ on the product state space $\hat E:=I\times E$ and need to ...
2
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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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48 views

Any good references on the decay rate of Legendre coefficient?

Let $P_n:[-1,1]\rightarrow\mathbb{R}$ be the $n$-th Legendre Polynomial. and, let $$a_n:=\int_{-1}^1 f(t) P_n(t)\, dt$$ for some $f:[-1,1]\rightarrow\mathbb{R}$. Are there any good references on the ...
1
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0answers
157 views

Adaptive Simpsons Quadrature Algorithm for Double Integrals? [closed]

I'm currently using Numerical Analysis 10th edition by Richard L Burden as a reference for approximate Integration techniques. In there it describes the Adaptive Simpsons Quadrature rule that inputs ...
2
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0answers
43 views

Numerical evaluation of KL divergence for SDE

Consider the SDE $$ dX_t = v(X_t)dt + dW_t $$ where $W_t$ is a standard Brownian motion. Girsanov's theorem tells us that the Radon-Nikodym derivative of the measure $\mathbb{P}_v$ of $X_t$ with ...
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1answer
45 views

What is the minimum number of stages $s$ required for a Runge-Kutta type numerical method of given order $p$?

These slides (slide 42) give a table (same as Table 1.6 given in Butcher's General Linear Methdos of the minimum number of stages $s$ for a Runge-Kutta type numerical method of order $p$ (the slides ...
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0answers
55 views

Convergence of Gaussian quadrature rules for integration

I would like to discuss some issues about convergence of Gaussian quadrature rules for integration. I asked this question in Mathematics Stack Exchange here with a bounty period but received no answer....
2
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1answer
102 views

Proving that $\lim_{j \to i} Z_{ij} = [\ln(\frac{\Delta s_i}{2})-1]\Delta s_i$

If I have the following integral equation $$\phi(\vec{x})=\frac{1}{\pi}\int [\phi\frac{\partial (\ln r)}{\partial n} -\ln(r) \frac{\partial \phi}{\partial n}] ds$$ An approximate solution of $\phi$ ...
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0answers
118 views

Computing the difficult integral $\int_0^\infty J_0(x)^4\log(x)dx$

Computing numerically integrals of oscillating functions from $0$ to $\infty$ is a well-known and difficult problem. Here is an example for which I do not know a solution: I know how to compute (to ...
4
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1answer
127 views

Intractability of an integral by deterministic numerical methods

Suppose $X_1,\ldots,X_n$ is an i.i.d. sample from a probability distribution with continuous c.d.f. $F.$ Let $F_n$ be the empirical c.d.f. $$ F_n(x) = \frac 1 n \sum_{k=1}^n \mathbf 1_{X_n\le x} = \...
2
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1answer
165 views

Numerical methods for IDE [closed]

I would like to read a popular literature on the topic "Numerical methods for integro-differential equations". Could you recommend me any articles or book with a brief overview of some methods (maybe ...
1
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0answers
93 views

On functions obtained from Gaussian Quadrature integration

Fix an integer $n \ge 2$. Let $x_1,...,x_n$ s and $w_1,...,w_n$ s be the Gauss Quadrature nodes and weights respectively in the interval $[0,1]$ (https://en.wikipedia.org/wiki/Gaussian_quadrature) . ...
2
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1answer
137 views

On the continuity and injective-ness of Gauss quadrature scheme for numerical integration, with weight function identically $1$

Fix an integer $n\ge 2$. Let $[a,b]$ be an interval and $f: [a,b]\to \mathbb R$ be a continuous function and for $x_1,...,x_n$ being the Gaussian Quadrature nodes in $[a,b]$, and Gaussian Quadrature ...
6
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1answer
126 views

Any convergence rule for ${\mathbf X}_k={\mathbf A}{\mathbf X}_{k-1}{\mathbf B}$?

We know iteration ${\mathbf X}_k=\mathbf{A}{\mathbf X}_{k-1}$ converges if the spectral radius of $\mathbf A$ is smaller than 1 (see here). Is there any known rule for iteration ${\mathbf X}_k={\...
1
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0answers
47 views

Approximating norms using numerical integration? [closed]

I have a sequence of functions $u_m$ in $H^1(\Omega)$, where $\Omega$ is Lipschitz such that $u_m(x)=\int_{|y|\le \epsilon} \, f_m(x,y) \, dy$, but the integral cannot be expressed in terms of ...
1
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0answers
64 views

Reference request on numerical integration

Let $\rho:\mathbb R^d\to\mathbb R_+$ be a density function with finite first moment, i.e. $$\int_{\mathbb R^d}~ \rho(x)dx~=~1 \quad \mbox{ and }\quad \int_{\mathbb R^d}~ |x|\rho(x)dx<+\infty.$$ ...
4
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0answers
93 views

For noisy or fine-structured data, are there better quadratures than the midpoint rule?

Only the first two sections of this long question are essential. The others are just for illustration. Background Advanced quadratures such as higher-degree composite Newton–Cotes, Gauß–Legendre, ...
4
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1answer
196 views

Gaps between roots of consecutive Hermite polynomials

Let $H_k(x)$ be (probabilists' or physicists', does not matter for this question) Hermite polynomials. It is well-known that all the gaps between consecutive roots of $H_k(x)$ are at least a multiple ...
5
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0answers
60 views

Numerical approximation of the $\ell_p$ surface area

What numerical method can approximately compute the $(n-1)$-dimensional surface area of the $\ell_p$ ball $\{x\in\mathbb R^n: \sum_{i=1}^n |x_i|^p=1\}$, for $p\in[1,\infty)$? Ideally the method should ...
-1
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1answer
580 views

Applications of Fourier Transforms in Number Theory [closed]

I'm looking for applications of Fourier Transforms in number theory.
4
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1answer
273 views

How to integrate the $L^2$ function $1/|x|$ numerically

Let $f=\frac{1}{|x|},x\in\mathbb{R^3}$ and $\Omega=[-b,b]^3$. How to construct a quadrature scheme to solve $$ \int_\Omega f\phi\psi dx\quad ? $$ where $\phi\psi$ is smooth function. I know there ...
1
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1answer
486 views

Gauss quadrature for products of multilinear functions on a simplex

All, I am looking for Gauss quadrature formulas for a particular geometric setting. That is, I am integrating functions over the standard simplex (triangle in dimension 2 and tetrahedron in ...
1
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0answers
67 views

Best method for approximating rigid body rotation equations

I'm trying to numerically approximate the solution to the rigid body rotation problem, given by the equations $$I_1u_1^\prime=(I_2-I_3)u_2u_3$$ $$I_2u_2^\prime=(I_3-I_1)u_3u_1$$ $$I_3u_3^\prime=(I_1-...
1
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3answers
326 views

Finding energy minimizing path

I'm trying to find an approximation for the optimal path for a material point, minimizing the integral associated with the total energy. I managed to write the exact formula for the energy along a ...
3
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0answers
144 views

When (if ever) are the weights from Smolyak (sparse grid) cubature positive?

Are there any $1$-dimensional quadrature rules of arbitrary accuracy, on either $[0,1]$ or $\mathbb{R}$, with any non-trivial weight function, such that the associated $N$-dimensional cubature rule ...
12
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1answer
577 views

Expected number of lines meeting four given lines or “what is 1.72…”

Given four random lines in $\mathbb{R}P^3$, how many lines intersect all of those lines? In the recent paper Probabilistic Schubert Calculus, Peter Bürgisser and Antonio Lerario discuss this question ...
5
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2answers
1k views

Real world example of use of Monte Carlo method for high dimensional integrals

The Monte Carlo method for numerical integration is usually presented as a method invented to efficiently compute high dimensional integrals numerically. However, I haven't found any source which has ...
2
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0answers
182 views

Difficult integral - speed up numerical integration using a trick?

I have the following integral that I need to solve (with high precision) millions of times in my simulations. This is time consuming and is prohibiting me from proceeding from proceeding forward. I ...
2
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0answers
55 views

Similar matrix for numerical computations [closed]

I compute numerically a symmetric matrix $W$ from the flow of a ode. I have to check numerically if this matrix is definite-positive. Two cases: either I use the Cholesky algorithm :ok or I compute ...