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Questions tagged [approximate-integration]

Use this tag for questions related to approximate integration, which constitutes a broad family of algorithms for calculating the numerical value of a definite integral.

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Numerical solution for finding mean of a random variable with error function terms [closed]

I'm trying to compute the expected value $E[z]$ associated with this a random variable whose CDF is given by $F_Z(z)=\int_{\infty}^z \left [(1-\epsilon)\Phi \left(\frac {z-\mu}{\sigma}\right)+\epsilon ...
user3101457's user avatar
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Most efficient quadrature formula

On the integral I am looking to obtain an approximation with machine precision, I've thought about applying a Gauss-Jacobi formula or Gauss-Legendre for: $$ \int_{0}^{2}\sqrt{2-x}dx $$ I am not sure ...
Jorge Ávila Balmaceda's user avatar
1 vote
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50 views

Numerical integration with modified Bessel function of second kind

I am working with the so-called screened Poisson PDE, whose solutions in two-dimensions are given in terms of the modified Bessel function of the second kind, $K_0$, for Dirichlet boundary conditions ...
Woe's user avatar
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0 answers
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Integrating products of many oscillating functions

I'm working with the circular random matrix ensembles, in particular the circular unitary ensemble (CUE). For unitaries drawn from the CUE with dimension $N$, the distribution of its eigenvalues is ...
miggle's user avatar
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1 answer
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Prove that $ \int_0^1 x(\sec x)^2 \,dx < 1.$

Prove that $ \int_0^1 x(\sec x)^2 \,dx < 1.$ I tried a few combinations but nothing seems to work, as the integral is quite close to 1 itself. And question strictly inhibits use of calculators to ...
OpateItZOpatoOpate's user avatar
1 vote
0 answers
22 views

Stationary phase approximation, compact support needed

I am trying to understand the stationary phase approximation as described on Wikipedia https://en.wikipedia.org/wiki/Stationary_phase_approximation. As necessary condition, they mention a compact ...
Sebastian 's user avatar
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Applications of highly oscillatory integrals

I was reading a series of articles on numerical integration of highly oscillatory functions, e.g., S. Olver, Numerical approximation of highly oscillatory integrals S. Xiang, H. Wang, Fast ...
Vl F's user avatar
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4 votes
2 answers
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Asymptotic expansion of $\int_0^1 e^{x(tp-\frac{1}{2}t^2)}dt$ as $x\to\infty$ for different $p$

Let $p\in \mathbb{R}$, I would like to investigate the asymptotic behavior of the following integral: $$\int_0^1 e^{x(tp-\frac{1}{2}t^2)}dt$$ as $x\to\infty$. In particular, I would like to know how ...
John's user avatar
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0 answers
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Numerical method to integrate an exponentiated polynomial

Let $P_{n}$ be the set of polynomials of degree $n$. Then if $f \in P_{2n + 1}$ we can compute $$\int_{-1}^1 f(x) dx$$ precisely using $n$-point Gauss-Legendre quadrature. I am interested in computing ...
digbyterrell's user avatar
2 votes
1 answer
139 views

Huge error bounds for midpoint rule in calculating integral

[the value of K is 195]. 1I have this problem an integration approximation problem of: $$\int_0^{4\sqrt{\pi}}\sin(x^2)dx$$ with n = 4. The result is 5.01. But when I check the error bound using the ...
Lê Hoàng Ân's user avatar
1 vote
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Midpoint rule integration for a matrix-vector product

Consider a function $F=F(q)$ which is a symmetric, positive-definite matrix form of dimensions $n\times n$, where $q \in \mathbb{R}^n$ and $q=q(t)$. When applying the midpoint integration rule on $F$, ...
Meclassic's user avatar
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Is the Monte Carlo integration method(s) actually a viable way to accurately integrate functions?

Recently, I was playing around with some Monte Carlo simulations using Python to evaluate integrals of functions such as f(x) = x(1-x)sin²[200x(1-x)]. I am aware that Monte Carlo integration methods ...
KibalchishTheCoder's user avatar
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How to integrate arbitrary discrete-time linear and angular body fixed velocity to world space?

I have body fixed angular velocity values and linear acceleration values streaming in to my application. at some interval $\delta t$. I need to get a world position from these, assuming the start ...
craigB's user avatar
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Iterative formula for the integration of autonomous differential equations

Consider the following autonomous differential equation: $$y'=f(y)$$ where $y=y(t)$ with $y_0 = y(0)$. Let us suppose that $y^{[0]}(t)$ is a good approximation for $y(t)$. Using this function in the ...
TobiR's user avatar
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2 votes
2 answers
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Do numerical integration methods ever utilize evaluations of the derivative of the function?

Suppose one is performing numerical integration of some function $f(x)$, but in addition to being able to evaluate its value at points $f(x_1), \dots, f(x_n)$, one also can additionally obtain its ...
Betterthan Kwora's user avatar
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1 answer
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Proof that the Trapezoidal Method for solving IVP is $O(h^3)$

My question comes from Chapter 6 of the book Introduction to Numerical Analysis, 2nd edition, by Kendall Atkinson. In section 6.5, p.368-369, the author proves that the Trapezoidal method for solving ...
Leonidas's user avatar
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1 vote
0 answers
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numerical spherical integration

I am trying to find a numerical sperical integration methods with high precision. I have been using the numerical integration based on lebedev quadrature of 131th order. Is there any numerical code ...
Yujie Zhang's user avatar
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1 answer
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How can I (analytically or numerically) integrate the product of exp(-x) and fractionally degreed polynomials?

Are there any closed form solution to this integral? If not are there any good numerical methods to evaluate it? $$ \int_{0}^{\infty} (x+3)^{2.5} \times (1+x)^{3.5} \times exp(-x) dx$$ I have tried ...
ILSH's user avatar
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Approximation of integration of piecewise continuous function at discrete points

I have an integral of the form \begin{equation} \int_{0}^{1} f(x)dx, \end{equation} where $f$ is a continuous function that is not differentiable at countable number of points, defined on $[0,1]$. ...
Phoenix8128's user avatar
4 votes
1 answer
81 views

Calculate the integrals $\int\limits_{0}^{\infty }e^{-\lambda (x-1)^2}dx+\int\limits_{0}^{\infty }e^{-\lambda (x-2)^2}dx$

In the $\lambda \to \infty $ limit, approximate the integral $$I(\lambda )=\int\limits_{0}^{\infty }e^{-\lambda (x-1)^2(x-2)^2}dx$$ I understand that the function $-\lambda (x-1)^2(x-2)^2$ reaches a ...
Partim's user avatar
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1 vote
1 answer
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In the limit $\lambda \to \infty $ approximate the integral $I(\lambda )=\int_{-\infty }^{\infty }e^{-x^2}(\cosh x)^{\lambda }dx$

With the help of the previous answer (here) I decided to do the same with this integral In the limit $\lambda \to \infty $ approximate the integral $$I(\lambda )=\int_{-\infty }^{\infty }e^{-x^2}(\...
Partim's user avatar
  • 891
2 votes
2 answers
141 views

Method of steepest descent approximate the integral $I(\lambda )=\int\limits_{1}^{\infty }\left ( \frac{\ln x}{x} \right )^{\lambda }dx$

At $\lambda \to \infty $ Method of steepest descent approximate the integral $$I(\lambda )=\int\limits_{1}^{\infty }\left ( \frac{\ln x}{x} \right )^{\lambda }dx$$ My attempt: The method itself and ...
Partim's user avatar
  • 891
3 votes
1 answer
111 views

Asymptotic of the integral $\int\limits_{0}^{\infty }\cos \left ( ax+\frac{2b}{\sqrt{x}} \right )dx$

At $ab^2\gg 1$, investigate the leading asymptotic of the integral $$\int\limits_{0}^{\infty }\cos \left ( ax+\frac{2b}{\sqrt{x}} \right )dx$$ I have a general formula that I have used $$\int e^{if(x)}...
Partim's user avatar
  • 891
2 votes
1 answer
125 views

Approximate the integral $I(x)=\int\limits_{0}^{\infty }\cos \left ( x\left ( t^2-t^4 \right ) \right )dt$

At $x\to \infty $ approximate the integral $$I(x)=\int\limits_{0}^{\infty }\cos \left ( x\left ( t^2-t^4 \right ) \right )dt$$ My attempt: $$f(t) = t^2-t^4\Rightarrow f'(t) = 2t - 4t^3 = 2t(1 - 2t^2) =...
Partim's user avatar
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2 votes
1 answer
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Minimizing $\frac{1-\int_{1-\int_{0}^1 F(r)dr }^1 F(t) dt }{1-\int_{0}^1 F(t)^2 dt}$ for increasing function subject to $F(0)=0, F(1)=1$

Let $F:[0,1]\to [0,1]$ be an increasing function with $F(0)=0, F(1)=1$. Define $A(x)=1-\int_{x}^1 F(t) dt$. I am trying to approximately minimize the following ratio across all $F$ (i.e find a lower ...
AspiringMat's user avatar
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0 answers
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Quantifying numerical error of the approximated square wave

From a numerical point of view, it makes sense the following statement \begin{equation} \lim_{A\to \infty}\int_0^{2\pi}\frac{\partial \arcsin^2\sin{x}}{\partial x} - \tanh^2\left( A\cos(x)\right)\,...
struct's user avatar
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3 votes
0 answers
112 views

Is the approximation of the integral of $f$ by $\sum_{i=1}^kw_if(x_i)$ exact assuming $f$ has no high frequency components?

Let $d\in\mathbb N$, $f:\mathbb R^d\to\mathbb R$ be Lebesgue integrable and vanishing outside $[0,1)^d$. Moreover, let $k\in\mathbb N$ and $x_1,\ldots,x_k\in[0,1)^d$. Let $w_1,\ldots,w_k\ge0$ with $\...
0xbadf00d's user avatar
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0 votes
1 answer
82 views

Solution to $\int_0^\infty \frac{e^{a u}}{2\mathrm{cosh}\left(\frac{u^2}{2}-b\right)+1} du$, where $a>0$ and $b>0$

Can someone please help me in finding the solution to the following integral: $$\int_0^\infty \frac{e^{a u}}{2\mathrm{cosh}\left(\frac{u^2}{2}-b\right)+1} du,$$ where $a>0$ and $b>0$. I have ...
Nik's user avatar
  • 33
3 votes
0 answers
83 views

Existence of inverse fourier transform

Is it possible to evaluate an inverse fourier transform of these functions? $f(\omega)=\exp(-(k^2-\omega^2)^{1/2})$, $g(\omega)=\frac{\exp(-(k^2-\omega^2)^{1/2})}{(k^2-\omega^2)^{1/2}}$, where k is a ...
gebegb's user avatar
  • 31
0 votes
1 answer
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Strange results for the error of numerical integration: The Newton-Cotes rules

I was using the Maple software for the computation of the numerical integration error. I'm obtaining two strange properties which I'm unable to explain. First for items $(72)$ and $(73)$ below while ...
user122424's user avatar
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1 vote
1 answer
360 views

Bilinear Transform vs Standard Numerical Methods

I am not very familiar with control theory but have a decent bit of experience with classical numerical integration. I am looking at a the equation $$\dot{x}(t) = Ax(t) + Bu(t) \hspace{1cm} x(0) =0 $$ ...
Jason's user avatar
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1 vote
0 answers
281 views

Numerical integration: The composite Newton-Cotes formulas, uniqueness and inductive definition for a given order of exactneness

I have a question on Rabinowitz and Davis: Methods of numerical Integration. They start to give a sequence for what they call The (composite) Integration Newton-Cotes formulas. This together with my ...
user122424's user avatar
  • 3,978
0 votes
2 answers
140 views

Discrepancy in value of $\int_0^\frac\pi2 \delta(\tan(t)-xt)dt$

The goal would have been for the smallest positive solution of $y\cot(y)=x$ using a Dirac $\delta(x)$ Fourier series and the Bateman function: $$y\cot(y)=x\mathop\implies^?\frac1{\sec^2(y)-x}=\int_0^\...
Тyma Gaidash's user avatar
3 votes
1 answer
114 views

How can I asymptotically expand $\frac{1}{\theta}\int_{-\infty}^{t}\exp\left(\frac{s-t}{\theta}\right)\frac{d}{ds}D(s)\mathrm{d}s?$ to linear order?

How do you asymptotically expand the following to linear order? $$\frac{1}{\theta}\int_{-\infty}^{t}\exp\left(\frac{s-t}{\theta}\right)\frac{d}{ds}D(s)\mathrm{d}s$$ I need to show that in the limit $\...
Frustrated_Mathematician's user avatar
-1 votes
2 answers
233 views

Please help in calculating Integral on the unit ball.

For an arbitrary vector R with length $$R= |\overrightarrow{R}| > 1$$, we define the integral I(R), which is taken over a ball of unit radius: $$I(R)=\int_{|r|\leq 1} \frac{dxdydz}{|R-r|^{2}}$$ ...
vansh saxena's user avatar
0 votes
1 answer
177 views

Numerical Integration over Triangles in 2D

I'm comparing different quadrature formulas over triangular regions in 2D on MATLAB. My professor gave me the article from Xiao-Gimbutas (2010): A numerical algorithm for the construction of efficient ...
Don Abbondio's user avatar
1 vote
1 answer
98 views

Solving ODE with derivative boundary condition with finite difference method by central approximation

I am trying to solve the following ODE: $$ \frac{d^2y}{dx^2}=y(x) $$ Where I have two boundary conditions: $ y(0)=10 $; and $ \frac{dy(x\rightarrow\inf)}{dx}=0 $ I am trying to solve the problem ...
HWIK's user avatar
  • 53
2 votes
2 answers
59 views

Name of integration technique where product term is near constant over the interval?

Consider $$\int_{\theta-\epsilon}^{\theta+\epsilon} g(x)f(x) dx$$ where f(x) is near constant on the interval $(\theta-\epsilon, \theta+\epsilon)$, and g(x) is not. It follows: $$\int_{\theta-\epsilon}...
Dylan Madisetti's user avatar
1 vote
0 answers
148 views

How do I perform a saddle point approximation to approximately find the integral $\int_{-\pi/2}^{\pi/2} \cos^n(x) dx$?

I am supposed to apply the saddle point approximation (i.e. method of steepest descent) to calculate the following integral: $$\int_{-\pi/2}^{\pi/2} \cos^n(x) dx$$ But I have no idea how to write the ...
Volodymyr's user avatar
1 vote
0 answers
45 views

What is the most efficient way to calculate $I =\int_0^1 (1-x)^{1/2}(x^3+1) $and what is the degree of the interpolating polynomial used? $

I have some doubts about the following exercise What is the most efficient way to calculate $I =\int_0^1 (1-x)^{1/2}(x^3+1)dx $ ? Solution: Let $g(x)=x^3+1$ and $w(x)=(1-x)^{1/2}$. With a change of ...
some_math_guy's user avatar
2 votes
3 answers
113 views

Find the value of $x(t)$ when $x'(t)=0$ given $x''(t) = a\cdot x'(t) / x(t)$ and some initial conditions [closed]

I am considering the differential equation $$ x''(t) = a\frac{x'(t)}{x(t)} $$ where $a>0$ is an arbitrary constant. I cannot solve this equation directly, but I am wondering if there's still any ...
MaxHeart's user avatar
1 vote
0 answers
49 views

Numerical integration, which quadrature formula use?

Back when I was still student (last year) I studied/performed numerical integrations, everytime the quadrature formula was provided within the exercise. The exercise time came to an end and now I am ...
SoleGoodman's user avatar
0 votes
3 answers
268 views

Evaluate $\iint(x^{2} + y^{2})dx\,dy$ over the area in the first quadrant bounded by the circle $x^2 + y^2 = a^2$.

Well I am stuck from the beginning. I have two methods: Method 1: $y=\sqrt{a^2−x^2}$ here $y$ varies from $\sqrt{a^2−x^2}$ to $0$ $x$ varies from $a$ to $0$ $\iint(x^2+y^2) dx\, dy= \int\left(x^2y+\...
Sashilina Choudhury's user avatar
3 votes
1 answer
83 views

Follow up question: asymptotics of a two dimensional integral

This is a follow up question of Asymptotics of a two dimensional integral about asymptotics of integrals. The problem is to find the leading order term of this integral. $$\int_0^1d\epsilon\int_{-\...
Yu Tian's user avatar
  • 137
2 votes
1 answer
106 views

Asymptotics of a two dimensional integral

I am working on the following integral $$\int_0^1d\epsilon\int_{-\epsilon}^\epsilon dt\left(\sqrt{1-(\rho+t)^2}-\sqrt{1-\rho^2}\right)e^{-N t^2},$$ where $\rho=1-\epsilon$, $N\rightarrow \infty$. The ...
Yu Tian's user avatar
  • 137
4 votes
3 answers
280 views

Asymptotic behavior of integral with Laplace's method

I am working on the following integral $\int_0^1 dx\int_0^1 dT \sqrt{1-(1-\sqrt{x}+\sqrt{xT})^2} e^{-n xT},$ as $n\rightarrow \infty$. The goal is to find the asymptotic behavior of the integral to ...
Yu Tian's user avatar
  • 137
3 votes
0 answers
67 views

Advantage of "integrating by differentiating" ie.$\def\e{\varepsilon}\int_a^b f(x)dx=\lim_{\e\to0}f(d/d\e)\frac{e^{\e b}-e^{\e a}}\e$

I just stumbled upon a (german) article that features the following formula to compute integrals by differentiating: $$\def\e{\varepsilon}\int_a^b\!\! f(x)dx\ =\ \lim_{\e\to0}f\left(\frac d{d\e}\right)...
emacs drives me nuts's user avatar
0 votes
1 answer
56 views

Approximating an integral in the from $e^{1/(a+bx/c)}$

I want to integrate the equation $e^{1/(a+bx/c)}$ with respect to x, where a, b, c are all constants, but I can't seem to find an approximation for it. Is there a way to approximate this integral or ...
Limona2000's user avatar
0 votes
0 answers
72 views

An complicated integral involving Erfi function

Sir, While, studying the diffraction of the Gaussian beam through apertures, I have faced the following integral (as an expression of diffracted field), $$I= \int_0^R \exp(-\alpha r^2+i\beta r)\bigg[\...
R. Bhattacharya's user avatar
0 votes
1 answer
122 views

How do you find weights for unevenly spaced quadrature rules?

I'm taking a Numerical-mathematics class right now and am fighting with a certain problem. We got this following exercise for Quadrature rules: Find the weights $w_1,w_2$ for the Quadrature rule : $$ ...
Jakob Sachs's user avatar

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