# Questions tagged [numerical-integration]

The numerical-integration tag has no usage guidance.

40
questions

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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 ...

2
votes

0
answers

65
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### 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

1
answer

349
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### 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 ...

4
votes

0
answers

171
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### 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
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0
answers

48
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### Explicit growth rate estimation of Gauss-Laguerre quadrature

The $n$-th Gauss-Laguerre quadrature scheme aims to approximate integral of exponentially decreassing function over $[0;+ \infty[$ by a finite sum, according to:
$ \displaystyle { \int _0 ^{+ \infty} ...

30
votes

4
answers

5k
views

### How does Mathematica do symbolic integration?

I suppose there was at least once in our lifetime the point where we resorted to mathematica for help with an integral.-Unless you chose not to have the pleasure of using the continuum in your ...

2
votes

1
answer

212
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### Error in Gauss-Laguerre numerical quadrature scheme

The $n$-th Gauss-Laguerre quadrature scheme aims to approximate integral of exponentially decreassing function over $[0 ; \infty[$ by a finite sum, according to:
$$ \int _0
^{+ \infty}
...

0
votes

1
answer

124
views

### Finding numerical solution for nonlinear Poisson-like equation using finite difference method

I am trying to use finite difference method to solve for $u(x,t)$ in the equation:
\begin{align}
\frac{\partial^2u}{\partial x^2} = \frac{au}{1+bu},
\end{align}
which is actually part of a system of ...

0
votes

1
answer

244
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### Gaussian quadrature, with no exact result over polynomial, but on inverse functions

Generally, a Gaussian quadrature of degree $n$ over an interval $I$ is defined so that it integrates exactly polynomials up to degree $2n - 1$. The main tool are the orthogonal polynomials.
When $I$ ...

1
vote

1
answer

188
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### Quadrature methods for high-dimensional Gaussian integration

Suppose that $f$ is the density of a high(-$d$)-dimensional Gaussian measure with mean $\mu$ and non-singular covariance matrix $\Sigma$. Let $g:\mathbb{R}^d\rightarrow \mathbb{R}$ be a continuous ...

0
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0
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71
views

### Reverse Inequality

I was doing some numerical integration when I figured the function I was dealing with (i.e., the function I was integrating) evaluated to big numbers on a tiny portion of the interval (over which I ...

2
votes

0
answers

70
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(...

2
votes

0
answers

68
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### 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(...

2
votes

1
answer

166
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

103
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

279
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 ...

3
votes

0
answers

147
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

63
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 ...

2
votes

1
answer

105
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

155
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### 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

147
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

209
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### 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

96
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

146
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

131
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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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

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.
$$\int_{\mathbb R^d}~ \rho(x)dx~=~1 \quad \mbox{ and }\quad \int_{\mathbb R^d}~ |x|\rho(x)dx<+\infty.$$
...

4
votes

0
answers

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, ...

4
votes

1
answer

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 ...

5
votes

0
answers

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 ...

0
votes

1
answer

811
views

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

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

4
votes

1
answer

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 ...

2
votes

1
answer

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

0
answers

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

3
answers

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 ...

3
votes

0
answers

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 ...

14
votes

2
answers

965
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### 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
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### 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

217
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### 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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0
answers

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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 ...