# Questions tagged [numerical-integration]

The numerical-integration tag has no usage guidance.

59
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### How can I calculate the derivative of an integral with respect to a parameter if Leibniz's formula gives a divergent integral?

We are working on the problem related to a magnetic field in an axially symmetric magnetic plasma trap. Let's express the vector potential through the magnetic flux function
\begin{gather}
\label{1:01}...

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72
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### Resolving singularities in numerical integration

I am now trying to compute numerically the following integral.
$$
\begin{split}
L_1^s(\hat{\phi}_s)(r,\zeta,\theta_\zeta) &=\frac{1}{\sqrt{2}\pi}
\int\limits_{0}^{2\pi} d\varphi
\int\limits_0^{\pi}...

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1
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### Numerical method with rational nodes and weights to compute exact value of definite integral?

Description
Let $p(x)$ be a polynomial of degree $n$ and rational coefficients.
I'm interested in computing numerically the exact value of the integral $I$, which is also rational
$$I = \int_{a}^{b} p(...

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### How to calculate the weights for Discrete Laplacian Operator?

I am following this paper step by step and want to build an isotropic Laplacian kernel. As shown in the following figure, I can understand until using Taylor to expand the 2D discrete Laplacian ...

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1
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182
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### Approximation for a Bessel function integral

I'm trying to calculate hit probabilities on a dart board if the dart thrower has some Gaussian angle distribution function with width $\Delta$ and some systematic angle offsets $\phi_0, \theta_0$. ...

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### Dealing with noise that is white in time, colored in space numerically

I am broadly working on a dynamic process where we want to see how a field $\rho(r)$ changes in space in time with thermal noise. The system is biased around a thermodynamic saddle point dictated by $...

1
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1
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99
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### How to numerically solve differential equations involving sines, cosines and inverses of the unknown function? [closed]

Crossposted at SciComp SE
I'm very new to finite difference method and I am just introduced to methods of solving differential equation using finite difference method via sparse matrix method.
I find ...

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39
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### finding weak form of nonlinear differential equation for FEM simulation

The following is the well-known nonlinear differential equation for director's distribution at static equilibrium in liquid crystal displays(LCD). I want to obtain weak form of the given differential ...

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132
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### Lower bound $|\sum_{x \in X} \phi(x) - \int_{\mathbb{R^2}} \phi(x) \, dx | \geq C f(\phi)$

I asked this question on math.stackexchange before, but with a bad formulation. I think the problem is quite complicated, so I decided to ask it here. Tell me if I shouldn't.
Very recently, I ...

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### Fast numerical integration of $\int_{[0,\:1)^d}\left|f_x(y)-g(y)\right|^p\:{\rm d}y$ for varying $x\in[0,1)^d$

Let $k\in\mathbb N$ and $y_1,\ldots,y_k\in[0,1)^d$ with $$\frac1k\sum_{i=1}^kh(y_i)\approx\int_{[0,\:1)^d}h(y)\:{\rm d}y\tag1$$ for every nice enough function $h:[0,1)^d\to\mathbb R$.
Now let $p\ge1$, ...

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### Symplectic (or alike) integrator for system with Coulomb singularity and time-dependent potentials

I am trying to calculate classical trajectories for a single a ion and a single electron inside an RF trap. Therefore, I am dealing with a two-body system that possesses:
Coulomb potential with a ...

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0
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12
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### How to solve numerically nonlinear fractional differential equation containing at least two derivatives of non integer order?

Let's consider differential equation:
$$x^{(n)}(t) = f(t,x(t),x'(t),\ldots,x^{(n-1)}(t))$$
with initial conditions $x(0) = x_0, x'(0) = x_1, \ldots, x^{(n-1)}(0) = x_{n-1}$. It is easy to solve this ...

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1
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99
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### Numerical solution to some functional equation

Let $z>0$ be fixed. Consider the function $p_a: \mathbb R^2_+\to\mathbb R_+$ given as
$$
p_a(t,x):=\frac{1}{\sqrt{2\pi N_a(t)}}\left[\exp\left(-\frac{(x-z)^2}{2N_a(t)}\right)-\exp\left(-\frac{(x+z)^...

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46
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### Numerically integrating close to a singularity

Suppose you got a function $f(x)$ with a singularity $s$, point $a$ and a small number $\epsilon$.
For what $b$ does this equation hold?
$$\int_{s-a}^{s-\epsilon}f(x) dx + \int_{s+\epsilon}^{s+b}f(x) ...

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120
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### Numerically compute the Schwarz-Christoffel mapping to the square

I want to map the upper-half plane $$\mathbb H:=\{z\in\mathbb C:\Im(z)>0\}$$ to $[0,1)^2$ by a conformal map. If I got this right, then such a mapping is given by the Schwarz-Christoffel mapping to ...

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49
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### Error estimation for numeric quadrature on an $n$-simplex

I asked this on Math SE but got no interactions. Thinking for a bit this might be better suited for this site.
Suppose I have a sufficiently nice function from a simplex $S\subseteq\mathbb R^n$ with ...

3
votes

1
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300
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### Fourier series of $e^{(\cos(\pi x) - m)^2}$

I'm looking for the Fourier coefficient of a "periodic Gaussian", which writes
$$
f(x) = e^{-\frac{1}{2s}(\cos(\pi x) - m)^2}
$$
It is a real even 2-periodic function, so its Fourier ...

2
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89
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### Approximating a probability density with a point set

Let $f$ be a "nice" probability density on $\mathbb{R}^2$, let $p=1/k$ for some fixed positive integer $k$, and let $\epsilon>0$. Are there any known statements of the following form?
&...

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### Flux that can be represented by low and high resolution schemes

In the wiki page of Flux limiter, it writes:
If these edge fluxes can be represented by low and high resolution schemes, then a flux limiter can switch between these schemes depending upon the ...

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111
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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 ...

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75
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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
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1
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825
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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
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0
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227
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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 ...

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0
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76
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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} ...

32
votes

4
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6k
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### 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 ...

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1
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468
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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
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1
answer

151
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### 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

319
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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 $I$ ...

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1
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492
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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 ...

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78
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### 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 ...

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76
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### 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(...

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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
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171
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### 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 ...

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0
answers

165
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### 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
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0
answers

326
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### 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 ...

4
votes

0
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356
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### 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

89
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### 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
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109
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### 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 $\phi$ ...

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0
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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

152
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### 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
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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 ...

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0
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108
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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) . ...

2
votes

1
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### 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
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1
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155
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### 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
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48
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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
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0
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68
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### 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

254
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### 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

358
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### 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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votes

0
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86
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### 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

971
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### Applications of Fourier Transforms in Number Theory [closed]

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