# Questions tagged [numerical-integration]

The numerical-integration tag has no usage guidance.

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### 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) . ...

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123 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 ...

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42 views

### The limit of infinite ODE solver iteration with zero time step

Suppose I am trying to find a solution of an ordinary differential equation:
\begin{equation}
\begin{aligned}
y'(x) &= f(y(x))\\
y(0) &= y_0
\end{aligned}
\end{equation}
on ...

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17 views

### Numerical solution of two coupled nonlinear eigenvalue problems

I would like to numerically solve the following system of coupled nonlinear differential equations:
$$
-\frac{\hbar^2}{2m_a} \frac{\partial^2}{\partial x^2}\psi_a + V_{ext}\psi_a +
\left( g_a |...

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32 views

### Expression for the Markov Chain CLT variance for an arbitrary initial distribution

Let
$(\Omega,\mathcal A,\operatorname P)$ and $(E,\mathcal E,\pi)$ be probability spaces
$(X_n)_{n\in\mathbb N}$ be a $(E,\mathcal E)$-valued stochastic process on $(\Omega,\mathcal A,\operatorname P)...

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117 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={\...

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44 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 ...

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50 views

### Reference request on numerical integration

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

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43 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, ...

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139 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 ...

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55 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 ...

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360 views

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

I'm looking for applications of Fourier Transforms in number theory.

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30 views

### Truncation error of product and composition of functions?

I'm trying to solve the following problem numerically $$\int_0^a\alpha(x)\frac{dw(x)}{dx}\frac{d v(x)}{dx}dx.$$ For any $\alpha$,w and v functions of x.
The problem is that I don´t know what the ...

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259 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 ...

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255 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 ...

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55 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-...

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251 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 ...

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90 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 ...

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537 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 ...

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58 views

### references for numerical lookup table interpolation of P/ODE(s) RHS

I'm not sure that this *overflow is right place to ask.... Sorry if it is off top.
Does anyone know name for numerical dirty trick when we tabulate a right-hand side of given differential equation(s) ...

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780 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 ...

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159 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 ...

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### 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 ...

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80 views

### Basic Monte Carlo Integral Approximation

On the very first page of a well-known book on Monte Carlo techniques, there is the following statement. Let
\begin{equation}
I = \int_D g(\textbf{x})d\textbf{x},
\end{equation}
where $D \subset \...