Questions tagged [numerical-integration]
The numerical-integration tag has no usage guidance.
67 questions
-4
votes
2
answers
173
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Why exactly is Simpson's rule better than the Trapezoidal rule? [closed]
I am reading up on numerical integration and have trouble to really understand why or rather in what sense Simpson's rule is better than the Trapezoidal rule in general. There is a lot of stuff ...
2
votes
0
answers
68
views
Two-terms Euler-Maclaurin formula for concave functions over polytopes
Let $P\subset\mathbb{R}^{n}$ be a lattice polytope (vertices are
in $\mathbb{Z}^{n}$). Set $P_{k}=P\cap k^{-1}\mathbb{Z}^{n}$ for
$k\geq1$. Given a concave function $\phi:P\rightarrow\mathbb{R}$
(...
- 73
1
vote
1
answer
141
views
Understanding quadrature rule of a function multiplied by another $C^{\infty}$ function
Define a function $f \in C^m[-1,1],m \in \mathbb{N}$ and $g\in C^{\infty}[-1,1]$. Also define a quadrature rule $Q$ for approximating the integral $\int_{-1}^1 h(x)dx$ for some function integrable ...
- 69
0
votes
1
answer
97
views
Numerically bounding a Exponential-Trigonometric Integral [closed]
I am having some trouble with this (undergrad) problem. The Twitter account I found this from was deleted so I unfortunately have not found an answer.
I have tried decomposing into Riemann sum and ...
- 13
1
vote
0
answers
65
views
Discretization of oscillating integral
Suppose I am interested in computing
$$
I \equiv \int_0^B dx \, g(x) f(x)
$$
where $B$ is a known upper bound for the integral,
$g(x)$ is a known oscillating function and
$f(x)$ is a smooth function ...
- 33
0
votes
1
answer
135
views
Numerical integration with integrable singularity
Suppose I have a numerical estimation of discrete samples of a smooth function $C(t)$ at $t = a, \dots, T = Na$ and I want to (numerically) compute the integral of $f(t) = \frac{C(t)}{\sqrt{t}}$. In ...
- 33
0
votes
0
answers
136
views
Antiderivatives via Taylor series and the FT of Calculus
If $f$ is a real function on an interval $[a,b]$ such that
$f$ is computationally tractable on $[a,b]$: you can calculate $f(x)$ to $n$ bits of precision using an algorithm which is polynomial in $n$ ...
- 827
0
votes
1
answer
58
views
Integration algorithm and analytic property
This question is the continuation of the previous one.
In the article about the integration of analytical polynomial - time computable function $f(x)$ with the Taylor series $$f(x) = \sum_{n=0}^{\...
- 81
1
vote
0
answers
127
views
Integration in polynomial time
The work of Friedman and Ko and
Müller guarantee the polynomial time computability of the integrals of analytic functions inside the circle of convergence. But do algorithms have practical value? Is ...
- 81
2
votes
2
answers
272
views
Numerical integration method that doesn't involve derivative in the error bound
Consider the integral $\int_a^b f(t)dt$.
There are many numerical integration methods, like trapezoidal rule, Simpson's rule, Gaussian quadrature, but all they involve derivative in the error bound.
...
- 49
1
vote
0
answers
66
views
How can I calculate the derivative of an integral with respect to a parameter if Leibniz's formula gives a divergent integral?
We are working on the problem related to a magnetic field in an axially symmetric magnetic plasma trap. Let's express the vector potential through the magnetic flux function
\begin{gather}
\label{1:01}...
- 241
1
vote
0
answers
128
views
Resolving singularities in numerical integration
I am now trying to compute numerically the following integral.
$$
\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}...
- 11
2
votes
1
answer
104
views
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
$$I = \int_{a}^{b} p(...
- 253
1
vote
1
answer
250
views
Approximation for a Bessel function integral
I'm trying to calculate hit probabilities on a dart board if the dart thrower has some Gaussian angle distribution function with width $\Delta$ and some systematic angle offsets $\phi_0, \theta_0$. ...
- 19
3
votes
0
answers
122
views
Dealing with noise that is white in time, colored in space numerically
I am broadly working on a dynamic process where we want to see how a field $\rho(r)$ changes in space in time with thermal noise. The system is biased around a thermodynamic saddle point dictated by $...
- 31
1
vote
1
answer
106
views
How to numerically solve differential equations involving sines, cosines and inverses of the unknown function? [closed]
Crossposted at SciComp SE
I'm very new to finite difference method and I am just introduced to methods of solving differential equation using finite difference method via sparse matrix method.
I find ...
- 31
2
votes
0
answers
52
views
finding weak form of nonlinear differential equation for FEM simulation
The following is the well-known nonlinear differential equation for director's distribution at static equilibrium in liquid crystal displays(LCD). I want to obtain weak form of the given differential ...
- 31
1
vote
0
answers
145
views
Lower bound $|\sum_{x \in X} \phi(x) - \int_{\mathbb{R^2}} \phi(x) \, dx | \geq C f(\phi)$
I asked this question on math.stackexchange before, but with a bad formulation. I think the problem is quite complicated, so I decided to ask it here. Tell me if I shouldn't.
Very recently, I ...
- 183
3
votes
0
answers
95
views
Fast numerical integration of $\int_{[0,\:1)^d}\left|f_x(y)-g(y)\right|^p\:{\rm d}y$ for varying $x\in[0,1)^d$
Let $k\in\mathbb N$ and $y_1,\ldots,y_k\in[0,1)^d$ with $$\frac1k\sum_{i=1}^kh(y_i)\approx\int_{[0,\:1)^d}h(y)\:{\rm d}y\tag1$$ for every nice enough function $h:[0,1)^d\to\mathbb R$.
Now let $p\ge1$, ...
- 167
2
votes
0
answers
53
views
Symplectic (or alike) integrator for system with Coulomb singularity and time-dependent potentials
I am trying to calculate classical trajectories for a single a ion and a single electron inside an RF trap. Therefore, I am dealing with a two-body system that possesses:
Coulomb potential with a ...
- 21
1
vote
1
answer
108
views
Numerical solution to some functional equation
Let $z>0$ be fixed. Consider the function $p_a: \mathbb R^2_+\to\mathbb R_+$ given as
$$
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)^...
- 1,399
1
vote
0
answers
49
views
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?
$$\int_{s-a}^{s-\epsilon}f(x) dx + \int_{s+\epsilon}^{s+b}f(x) ...
- 21
1
vote
0
answers
234
views
Numerically compute the Schwarz-Christoffel mapping to the square
I want to map the upper-half plane $$\mathbb H:=\{z\in\mathbb C:\Im(z)>0\}$$ to $[0,1)^2$ by a conformal map. If I got this right, then such a mapping is given by the Schwarz-Christoffel mapping to ...
- 167
3
votes
0
answers
69
views
Error estimation for numeric quadrature on an $n$-simplex
I asked this on Math SE but got no interactions. Thinking for a bit this might be better suited for this site.
Suppose I have a sufficiently nice function from a simplex $S\subseteq\mathbb R^n$ with ...
- 141
3
votes
1
answer
425
views
Fourier series of $e^{(\cos(\pi x) - m)^2}$
I'm looking for the Fourier coefficient of a "periodic Gaussian", which writes
$$
f(x) = e^{-\frac{1}{2s}(\cos(\pi x) - m)^2}
$$
It is a real even 2-periodic function, so its Fourier ...
- 31
2
votes
0
answers
93
views
Approximating a probability density with a point set
Let $f$ be a "nice" probability density on $\mathbb{R}^2$, let $p=1/k$ for some fixed positive integer $k$, and let $\epsilon>0$. Are there any known statements of the following form?
&...
- 4,049
1
vote
0
answers
75
views
Flux that can be represented by low and high resolution schemes
In the wiki page of Flux limiter, it writes:
If these edge fluxes can be represented by low and high resolution schemes, then a flux limiter can switch between these schemes depending upon the ...
- 111
1
vote
0
answers
132
views
Numerical calculation of a double integral from the slowly-decaying oscillating function
Let us consider the following integral
$$
I = \int\limits_{0}^{+\infty}dx\int\limits_{-\infty}^{+\infty}dy \left[f(x,y) + g(x,y) \right].
$$
We know several properties of these functions.
There are ...
- 335
2
votes
0
answers
94
views
Approximate solution of nonlinear ODE
Investigating some problem in optics I am faced with a nonlinear differential equation of the form
$$ - y(x)\frac{{{d^2}}}{{d{x^2}}}\left( {\frac{1}{{y(x)}}} \right) + {y^2}(x) = f(x)$$
with initial ...
- 21
1
vote
1
answer
1k
views
Integrating a B-Spline basis function with respect to the standard normal PDF
I am looking for ways to evaluate exactly (i.e. analytically or semi-analytically) integrals of the type:
$$
\int_{-\infty}^{+\infty}B_{i}^k(u)e^{-\frac{(u-\mu)^2}{2\sigma^2}}du,
$$
where $B_i^k$ is a ...
- 13
4
votes
0
answers
311
views
Approximation of integral of gaussian function over a parallelepiped
Remark: I posted this question in math stackexchange here and computer science stackexchange https://cs.stackexchange.com/ few weeks ago but obtain no answer.
Given a multi-dimensional gaussian ...
- 250
1
vote
0
answers
91
views
Explicit growth rate estimation of Gauss-Laguerre quadrature
The $n$-th Gauss-Laguerre quadrature scheme aims to approximate integral of exponentially decreassing function over $[0;+ \infty[$ by a finite sum, according to:
$ \displaystyle { \int _0 ^{+ \infty} ...
- 97
32
votes
4
answers
6k
views
How does Mathematica do symbolic integration?
I suppose there was at least once in our lifetime the point where we resorted to mathematica for help with an integral.-Unless you chose not to have the pleasure of using the continuum in your ...
- 536
2
votes
1
answer
918
views
Error in Gauss-Laguerre numerical quadrature scheme
The $n$-th Gauss-Laguerre quadrature scheme aims to approximate integral of exponentially decreassing function over $[0 ; \infty[$ by a finite sum, according to:
$$ \int _0
^{+ \infty}
...
- 97
0
votes
1
answer
221
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 ...
- 65
0
votes
1
answer
417
views
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 $I$ ...
- 97
1
vote
1
answer
853
views
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 ...
- 5,405
0
votes
0
answers
82
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 ...
- 369
2
votes
0
answers
80
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(...
- 269
2
votes
0
answers
76
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(...
- 167
2
votes
1
answer
174
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 ...
- 16.2k
0
votes
0
answers
206
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 ...
- 469
1
vote
0
answers
374
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 ...
- 11
4
votes
0
answers
532
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 ...
- 1,038
0
votes
1
answer
138
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
110
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 $\phi$ ...
- 111
6
votes
0
answers
202
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 ...
- 13.1k
4
votes
1
answer
159
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
282
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
110
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) . ...
- 1,209