# 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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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 ... 1 vote 0 answers 52 views ### 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 ... 1 vote 0 answers 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 ... • 111 0 votes 0 answers 34 views ### 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 ... • 285 1 vote 1 answer 125 views ### Prove that$ \int_0^1 x(\sec x)^2 \,dx < 1.$Prove that$ \int_0^1 x(\sec x)^2 \,dx < 1.$I tried a few combinations but nothing seems to work, as the integral is quite close to 1 itself. And question strictly inhibits use of calculators to ... 1 vote 0 answers 22 views ### Stationary phase approximation, compact support needed I am trying to understand the stationary phase approximation as described on Wikipedia https://en.wikipedia.org/wiki/Stationary_phase_approximation. As necessary condition, they mention a compact ... 0 votes 0 answers 42 views ### Applications of highly oscillatory integrals I was reading a series of articles on numerical integration of highly oscillatory functions, e.g., S. Olver, Numerical approximation of highly oscillatory integrals S. Xiang, H. Wang, Fast ... • 11 4 votes 2 answers 113 views ### Asymptotic expansion of$\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 ... • 13.3k 0 votes 0 answers 85 views ### 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 ... • 300 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 ... 1 vote 0 answers 39 views ### 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$F$, ... • 435 0 votes 0 answers 80 views ### 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 ... 0 votes 0 answers 27 views ### 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 ... • 133 1 vote 0 answers 38 views ### 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 ... • 528 2 votes 2 answers 77 views ### 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 ... 0 votes 1 answer 96 views ### 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 ... • 1,054 1 vote 0 answers 59 views ### 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 ... 0 votes 1 answer 61 views ### 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 ... • 1 0 votes 0 answers 45 views ### Approximation of integration of piecewise continuous function at discrete points I have an integral of the form $$\int_{0}^{1} f(x)dx,$$ where$f$is a continuous function that is not differentiable at countable number of points, defined on$[0,1]$. ... 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 ... • 891 1 vote 1 answer 117 views ### 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 $$I(\lambda )=\int_{-\infty }^{\infty }e^{-x^2}(\... • 891 2 votes 2 answers 141 views ### Method of steepest descent approximate the integral I(\lambda )=\int\limits_{1}^{\infty }\left ( \frac{\ln x}{x} \right )^{\lambda }dx At \lambda \to \infty Method of steepest descent approximate the integral$$I(\lambda )=\int\limits_{1}^{\infty }\left ( \frac{\ln x}{x} \right )^{\lambda }dx$$My attempt: The method itself and ... • 891 3 votes 1 answer 111 views ### Asymptotic of the integral \int\limits_{0}^{\infty }\cos \left ( ax+\frac{2b}{\sqrt{x}} \right )dx At ab^2\gg 1, investigate the leading asymptotic of the integral$$\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)}... • 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: $$f(t) = t^2-t^4\Rightarrow f'(t) = 2t - 4t^3 = 2t(1 - 2t^2) =... • 891 2 votes 1 answer 97 views ### Minimizing \frac{1-\int_{1-\int_{0}^1 F(r)dr }^1 F(t) dt }{1-\int_{0}^1 F(t)^2 dt} for increasing function subject to F(0)=0, F(1)=1 Let F:[0,1]\to [0,1] be an increasing function with F(0)=0, F(1)=1. Define A(x)=1-\int_{x}^1 F(t) dt. I am trying to approximately minimize the following ratio across all F (i.e find a lower ... • 2,447 0 votes 0 answers 44 views ### Quantifying numerical error of the approximated square wave From a numerical point of view, it makes sense the following statement \lim_{A\to \infty}\int_0^{2\pi}\frac{\partial \arcsin^2\sin{x}}{\partial x} - \tanh^2\left( A\cos(x)\right)\,... • 41 3 votes 0 answers 112 views ### Is the approximation of the integral of f by \sum_{i=1}^kw_if(x_i) exact assuming f has no high frequency components? Let d\in\mathbb N, f:\mathbb R^d\to\mathbb R be Lebesgue integrable and vanishing outside [0,1)^d. Moreover, let k\in\mathbb N and x_1,\ldots,x_k\in[0,1)^d. Let w_1,\ldots,w_k\ge0 with \... • 13.9k 0 votes 1 answer 82 views ### Solution to \int_0^\infty \frac{e^{a u}}{2\mathrm{cosh}\left(\frac{u^2}{2}-b\right)+1} du, where a>0 and b>0 Can someone please help me in finding the solution to the following integral:$$\int_0^\infty \frac{e^{a u}}{2\mathrm{cosh}\left(\frac{u^2}{2}-b\right)+1} du,$$where a>0 and b>0. I have ... • 33 3 votes 0 answers 83 views ### Existence of inverse fourier transform Is it possible to evaluate an inverse fourier transform of these functions? f(\omega)=\exp(-(k^2-\omega^2)^{1/2}), g(\omega)=\frac{\exp(-(k^2-\omega^2)^{1/2})}{(k^2-\omega^2)^{1/2}}, where k is a ... • 31 0 votes 1 answer 89 views ### Strange results for the error of numerical integration: The Newton-Cotes rules I was using the Maple software for the computation of the numerical integration error. I'm obtaining two strange properties which I'm unable to explain. First for items (72) and (73) below while ... • 3,978 1 vote 1 answer 360 views ### Bilinear Transform vs Standard Numerical Methods I am not very familiar with control theory but have a decent bit of experience with classical numerical integration. I am looking at a the equation$$\dot{x}(t) = Ax(t) + Bu(t) \hspace{1cm} x(0) =0 $$... • 131 1 vote 0 answers 281 views ### Numerical integration: The composite Newton-Cotes formulas, uniqueness and inductive definition for a given order of exactneness I have a question on Rabinowitz and Davis: Methods of numerical Integration. They start to give a sequence for what they call The (composite) Integration Newton-Cotes formulas. This together with my ... • 3,978 0 votes 2 answers 140 views ### Discrepancy in value of \int_0^\frac\pi2 \delta(\tan(t)-xt)dt The goal would have been for the smallest positive solution of y\cot(y)=x using a Dirac \delta(x) Fourier series and the Bateman function:$$y\cot(y)=x\mathop\implies^?\frac1{\sec^2(y)-x}=\int_0^\... • 12.5k 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$\...
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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: $$I(R)=\int_{|r|\leq 1} \frac{dxdydz}{|R-r|^{2}}$$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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I just stumbled upon a (german) article that features the following formula to compute integrals by differentiating: $$\def\e{\varepsilon}\int_a^b\!\! f(x)dx\ =\ \lim_{\e\to0}f\left(\frac d{d\e}\right)... • 10.5k 0 votes 1 answer 56 views ### Approximating an integral in the from e^{1/(a+bx/c)} I want to integrate the equation e^{1/(a+bx/c)} with respect to x, where a, b, c are all constants, but I can't seem to find an approximation for it. Is there a way to approximate this integral or ... 0 votes 0 answers 72 views ### An complicated integral involving Erfi function Sir, While, studying the diffraction of the Gaussian beam through apertures, I have faced the following integral (as an expression of diffracted field),$$I= \int_0^R \exp(-\alpha r^2+i\beta r)\bigg[\...
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 :  ...