# Questions tagged [numerical-integration]

The tag has no usage guidance.

33 questions
Filter by
Sorted by
Tagged with
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 ...
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(...
39 views

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 ...
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 ...
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 ...
157 views

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 ...
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 ...
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 ...
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....
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 $\phi$ ...
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 ...
127 views

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 ...
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 ...
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 ...
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 ...
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 ...