Questions tagged [numerical-integration]
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70 questions
0 votes
1 answer
83 views
Confusion in creating bounds over step size in numerical integration for error calculation?
This question is linked to: Computational complexity of integration in two dimensions Here in a suggested answer by John Gunnar Carlsson it is mentioned that $|E| \leq K_1 h^2 |\Omega| M_2$ for some ...
- 103
1 vote
0 answers
248 views
Runge-Kutta integration on spherical manifold for Neural ODEs (with Tensorflow implementation) [closed]
I am confused about the correct implementation of Runge-Kutta integration on a d-dimensional hypersphere manifold. (Additionally I am locked into using tensorflow and am trying to implement a ...
- 11
-4 votes
2 answers
710 views
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
99 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 $\phi:P\rightarrow\mathbb{R}$ (...
- 63
1 vote
1 answer
185 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 ...
- 39
0 votes
1 answer
134 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
4 votes
0 answers
517 views
Computing the second Vassiliev invariant only using the configuration spaces of triples and quadruples of points on a knot
In a previous question I had asked about the second Vassiliev invariant, $\nu_2(\gamma)$ defined for a smooth embedding $\gamma(t):\mathbb{S}^1\rightarrow\mathbb{R}^3$. The numerous equivalent ways of ...
- 189
1 vote
0 answers
104 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
243 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
169 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 $n$ ...
- 847
0 votes
1 answer
94 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 $$f(x) = \sum_{n=0}^{\...
- 81
1 vote
0 answers
170 views
Integration in polynomial time
The work of Friedman and Ko and Müller guarantee the polynomial time computability of the integrals of analytic functions inside the circle of convergence. But do algorithms have practical value? Is ...
- 81
2 votes
2 answers
400 views
Numerical integration method that doesn't involve derivative in the error bound
Consider the integral $\int_a^b f(t)dt$. There are many numerical integration methods, like trapezoidal rule, Simpson's rule, Gaussian quadrature, but all they involve derivative in the error bound. ...
- 49
1 vote
0 answers
104 views
How can I calculate the derivative of an integral with respect to a parameter if Leibniz's formula gives a divergent integral?
We are working on the problem related to a magnetic field in an axially symmetric magnetic plasma trap. Let's express the vector potential through the magnetic flux function \begin{gather} \label{1:01}...
- 241
1 vote
0 answers
236 views
Resolving singularities in numerical integration
I am now trying to compute numerically the following integral. $$ \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
164 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 $$I = \int_{a}^{b} p(...
- 273
1 vote
1 answer
347 views
Approximation for a Bessel function integral
I'm trying to calculate hit probabilities on a dart board if the dart thrower has some Gaussian angle distribution function with width $\Delta$ and some systematic angle offsets $\phi_0, \theta_0$. ...
- 19
3 votes
0 answers
175 views
Dealing with noise that is white in time, colored in space numerically
I am broadly working on a dynamic process where we want to see how a field $\rho(r)$ changes in space in time with thermal noise. The system is biased around a thermodynamic saddle point dictated by $...
- 31
1 vote
1 answer
129 views
How to numerically solve differential equations involving sines, cosines and inverses of the unknown function? [closed]
Crossposted at SciComp SE I'm very new to finite difference method and I am just introduced to methods of solving differential equation using finite difference method via sparse matrix method. I find ...
- 31
2 votes
0 answers
76 views
finding weak form of nonlinear differential equation for FEM simulation
The following is the well-known nonlinear differential equation for director's distribution at static equilibrium in liquid crystal displays(LCD). I want to obtain weak form of the given differential ...
- 31
1 vote
0 answers
164 views
Lower bound $|\sum_{x \in X} \phi(x) - \int_{\mathbb{R^2}} \phi(x) \, dx | \geq C f(\phi)$
I asked this question on math.stackexchange before, but with a bad formulation. I think the problem is quite complicated, so I decided to ask it here. Tell me if I shouldn't. Very recently, I ...
- 53
3 votes
0 answers
111 views
Fast numerical integration of $\int_{[0,\:1)^d}\left|f_x(y)-g(y)\right|^p\:{\rm d}y$ for varying $x\in[0,1)^d$
Let $k\in\mathbb N$ and $y_1,\ldots,y_k\in[0,1)^d$ with $$\frac1k\sum_{i=1}^kh(y_i)\approx\int_{[0,\:1)^d}h(y)\:{\rm d}y\tag1$$ for every nice enough function $h:[0,1)^d\to\mathbb R$. Now let $p\ge1$, ...
- 249
2 votes
0 answers
82 views
Symplectic (or alike) integrator for system with Coulomb singularity and time-dependent potentials
I am trying to calculate classical trajectories for a single a ion and a single electron inside an RF trap. Therefore, I am dealing with a two-body system that possesses: Coulomb potential with a ...
- 21
1 vote
1 answer
169 views
Numerical solution to some functional equation
Let $z>0$ be fixed. Consider the function $p_a: \mathbb R^2_+\to\mathbb R_+$ given as $$ 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)^...
- 1,693
1 vote
0 answers
63 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? $$\int_{s-a}^{s-\epsilon}f(x) dx + \int_{s+\epsilon}^{s+b}f(x) ...
- 21
1 vote
0 answers
346 views
Numerically compute the Schwarz-Christoffel mapping to the square
I want to map the upper-half plane $$\mathbb H:=\{z\in\mathbb C:\Im(z)>0\}$$ to $[0,1)^2$ by a conformal map. If I got this right, then such a mapping is given by the Schwarz-Christoffel mapping to ...
- 249
3 votes
0 answers
99 views
Error estimation for numeric quadrature on an $n$-simplex
I asked this on Math SE but got no interactions. Thinking for a bit this might be better suited for this site. Suppose I have a sufficiently nice function from a simplex $S\subseteq\mathbb R^n$ with ...
- 141
3 votes
1 answer
502 views
Fourier series of $e^{(\cos(\pi x) - m)^2}$
I'm looking for the Fourier coefficient of a "periodic Gaussian", which writes $$ f(x) = e^{-\frac{1}{2s}(\cos(\pi x) - m)^2} $$ It is a real even 2-periodic function, so its Fourier ...
- 31
2 votes
0 answers
111 views
Approximating a probability density with a point set
Let $f$ be a "nice" probability density on $\mathbb{R}^2$, let $p=1/k$ for some fixed positive integer $k$, and let $\epsilon>0$. Are there any known statements of the following form? &...
- 4,131
1 vote
0 answers
103 views
Flux that can be represented by low and high resolution schemes
In the wiki page of Flux limiter, it writes: If these edge fluxes can be represented by low and high resolution schemes, then a flux limiter can switch between these schemes depending upon the ...
- 111
1 vote
0 answers
151 views
Numerical calculation of a double integral from the slowly-decaying oscillating function
Let us consider the following integral $$ I = \int\limits_{0}^{+\infty}dx\int\limits_{-\infty}^{+\infty}dy \left[f(x,y) + g(x,y) \right]. $$ We know several properties of these functions. There are ...
- 335
2 votes
0 answers
120 views
Approximate solution of nonlinear ODE
Investigating some problem in optics I am faced with a nonlinear differential equation of the form $$ - y(x)\frac{{{d^2}}}{{d{x^2}}}\left( {\frac{1}{{y(x)}}} \right) + {y^2}(x) = f(x)$$ with initial ...
- 21
1 vote
1 answer
2k views
Integrating a B-Spline basis function with respect to the standard normal PDF
I am looking for ways to evaluate exactly (i.e. analytically or semi-analytically) integrals of the type: $$ \int_{-\infty}^{+\infty}B_{i}^k(u)e^{-\frac{(u-\mu)^2}{2\sigma^2}}du, $$ where $B_i^k$ is a ...
- 13
4 votes
0 answers
373 views
Approximation of integral of gaussian function over a parallelepiped
Remark: I posted this question in math stackexchange here and computer science stackexchange https://cs.stackexchange.com/ few weeks ago but obtain no answer. Given a multi-dimensional gaussian ...
- 250
1 vote
0 answers
127 views
Explicit growth rate estimation of Gauss-Laguerre quadrature
The $n$-th Gauss-Laguerre quadrature scheme aims to approximate integral of exponentially decreassing function over $[0;+ \infty[$ by a finite sum, according to: $ \displaystyle { \int _0 ^{+ \infty} ...
- 373
33 votes
4 answers
7k views
How does Mathematica do symbolic integration?
I suppose there was at least once in our lifetime the point where we resorted to mathematica for help with an integral.-Unless you chose not to have the pleasure of using the continuum in your ...
- 548
3 votes
1 answer
1k views
Error in Gauss-Laguerre numerical quadrature scheme
The $n$-th Gauss-Laguerre quadrature scheme aims to approximate integral of exponentially decreassing function over $[0 ; \infty[$ by a finite sum, according to: $$ \int _0 ^{+ \infty} ...
- 373
0 votes
1 answer
268 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
490 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 $I$ ...
- 373
1 vote
1 answer
1k 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 ...
- 4,942
0 votes
0 answers
124 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
99 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
91 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(...
- 249
2 votes
1 answer
221 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 ...
- 16.6k
0 votes
0 answers
268 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 ...
- 479
1 vote
0 answers
394 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
651 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
194 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
135 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
248 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 ...
- 13.9k