# Questions tagged [numerical-integration]

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### Numerical calculation of a double integral from the slowly-decaying oscillating function

Let us consider the following integral $$I = \int\limits_{0}^{+\infty}dx\int\limits_{-\infty}^{+\infty}dy \left[f(x,y) + g(x,y) \right].$$ We know several properties of these functions. There are ...
65 views

### Approximate solution of nonlinear ODE

Investigating some problem in optics I am faced with a nonlinear differential equation of the form $$- y(x)\frac{{{d^2}}}{{d{x^2}}}\left( {\frac{1}{{y(x)}}} \right) + {y^2}(x) = f(x)$$ with initial ...
1 vote
349 views

### Integrating a B-Spline basis function with respect to the standard normal PDF

I am looking for ways to evaluate exactly (i.e. analytically or semi-analytically) integrals of the type: $$\int_{-\infty}^{+\infty}B_{i}^k(u)e^{-\frac{(u-\mu)^2}{2\sigma^2}}du,$$ where $B_i^k$ is a ...
171 views

### Approximation of integral of gaussian function over a parallelepiped

Remark: I posted this question in math stackexchange here and computer science stackexchange https://cs.stackexchange.com/ few weeks ago but obtain no answer. Given a multi-dimensional gaussian ...
1 vote
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1 vote
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### 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 vote
66 views

### Reference request on numerical integration

Let $\rho:\mathbb R^d\to\mathbb R_+$ be a density function with finite first moment, i.e. $$\int_{\mathbb R^d}~ \rho(x)dx~=~1 \quad \mbox{ and }\quad \int_{\mathbb R^d}~ |x|\rho(x)dx<+\infty.$$ ...
215 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, ...
320 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 ...
72 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 ...
811 views

### Applications of Fourier Transforms in Number Theory [closed]

I'm looking for applications of Fourier Transforms in number theory.
341 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 ...
631 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 vote
75 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 vote
602 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 ...
232 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 ...
965 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 ...