# Questions tagged [numerical-integration]

The numerical-integration tag has no usage guidance.

33
questions

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vote

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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**

votes

**0**answers

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(...

**0**

votes

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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 $...

**2**

votes

**0**answers

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(...

**0**

votes

**0**answers

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**

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 ...

**0**

votes

**0**answers

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**

vote

**0**answers

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**

votes

**0**answers

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 ...

**0**

votes

**1**answer

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 ...

**0**

votes

**0**answers

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**

votes

**1**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 $\phi$ ...

**6**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

vote

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

vote

**0**answers

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**

vote

**0**answers

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

580 views

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

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

**4**

votes

**1**answer

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**

vote

**1**answer

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**

vote

**0**answers

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**

vote

**3**answers

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**2**answers

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**

votes

**0**answers

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**

votes

**0**answers

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 ...