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.
249
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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 $F$, ...
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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 ...
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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 ...
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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 ...
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2
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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 ...
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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 ...
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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 ...
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Determine whether rectangle/trapezoid area will be "overall" underestimate or overestimate in an interval
When estimating the area under a curve using left-bound rectangles, I know there will be an underestimate in a given interval if the function is increasing, $f'\left(x\right)>0$, and an ...
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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 ...
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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 $[0,1]$. ...
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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 ...
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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
$$I(\lambda )=\int_{-\infty }^{\infty }e^{-x^2}(\...
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Method of steepest descent approximate the integral $I(\lambda )=\int\limits_{1}^{\infty }\left ( \frac{\ln x}{x} \right )^{\lambda }dx$
At $\lambda \to \infty $ Method of steepest descent approximate the integral
$$I(\lambda )=\int\limits_{1}^{\infty }\left ( \frac{\ln x}{x} \right )^{\lambda }dx$$
My attempt:
The method itself and ...
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Asymptotic of the integral $\int\limits_{0}^{\infty }\cos \left ( ax+\frac{2b}{\sqrt{x}} \right )dx$
At $ab^2\gg 1$, investigate the leading asymptotic of the integral
$$\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)}...
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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:
$$f(t) = t^2-t^4\Rightarrow f'(t) = 2t - 4t^3 = 2t(1 - 2t^2) =...
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Approximation of integral on a grid of given values in R
Suppose that $f()$ is a mixture of two exponentials which is given by:
\begin{equation}
f(x; \Theta) = w_1 \lambda_1 e^{-\lambda_1 x} + w_2\lambda_2 e^{-\lambda_2 x},\; x> 0
\end{equation}
I aim to ...
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Minimizing $\frac{1-\int_{1-\int_{0}^1 F(r)dr }^1 F(t) dt }{1-\int_{0}^1 F(t)^2 dt}$ for increasing function subject to $F(0)=0, F(1)=1$
Let $F:[0,1]\to [0,1]$ be an increasing function with $F(0)=0, F(1)=1$. Define $A(x)=1-\int_{x}^1 F(t) dt$. I am trying to approximately minimize the following ratio across all $F$ (i.e find a lower ...
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Quantifying numerical error of the approximated square wave
From a numerical point of view, it makes sense the following statement
\begin{equation}
\lim_{A\to \infty}\int_0^{2\pi}\frac{\partial \arcsin^2\sin{x}}{\partial x} - \tanh^2\left( A\cos(x)\right)\,...
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Is the approximation of the integral of $f$ by $\sum_{i=1}^kw_if(x_i)$ exact assuming $f$ has no high frequency components?
Let $d\in\mathbb N$, $f:\mathbb R^d\to\mathbb R$ be Lebesgue integrable and vanishing outside $[0,1)^d$. Moreover, let $k\in\mathbb N$ and $x_1,\ldots,x_k\in[0,1)^d$. Let $w_1,\ldots,w_k\ge0$ with $\...
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Solution to $\int_0^\infty \frac{e^{a u}}{2\mathrm{cosh}\left(\frac{u^2}{2}-b\right)+1} du$, where $a>0$ and $b>0$
Can someone please help me in finding the solution to the following integral:
$$\int_0^\infty \frac{e^{a u}}{2\mathrm{cosh}\left(\frac{u^2}{2}-b\right)+1} du,$$ where $a>0$ and $b>0$.
I have ...
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3
votes
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Existence of inverse fourier transform
Is it possible to evaluate an inverse fourier transform of these functions?
$f(\omega)=\exp(-(k^2-\omega^2)^{1/2})$,
$g(\omega)=\frac{\exp(-(k^2-\omega^2)^{1/2})}{(k^2-\omega^2)^{1/2}}$,
where k is a ...
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Strange results for the error of numerical integration: The Newton-Cotes rules
I was using the Maple software for the computation of the numerical integration error.
I'm obtaining two strange properties which I'm unable to explain. First for items $(72)$
and $(73)$ below while ...
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Bilinear Transform vs Standard Numerical Methods
I am not very familiar with control theory but have a decent bit of experience with classical numerical integration.
I am looking at a the equation
$$\dot{x}(t) = Ax(t) + Bu(t) \hspace{1cm} x(0) =0 $$
...
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Numerical integration: The composite Newton-Cotes formulas, uniqueness and inductive definition for a given order of exactneness
I have a question on Rabinowitz and Davis: Methods of numerical Integration. They start to give a sequence for what they call The (composite) Integration Newton-Cotes formulas. This together with my ...
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Estimating integral $\int f(x)pdf(x)^{\alpha} dx$ involving some power of a probability distribution
Suppose a random variable $X$ has an unknown probability distribution $p(x)$ which we can draw samples $X_1,\ldots,X_n$ from. For a known function $f$, under some sufficiently nice conditions, we may ...
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Discrepancy in value of $\int_0^\frac\pi2 \delta(\tan(t)-xt)dt$
The goal would have been for the smallest positive solution of $y\cot(y)=x$ using a Dirac $\delta(x)$ Fourier series and the Bateman function:
$$y\cot(y)=x\mathop\implies^?\frac1{\sec^2(y)-x}=\int_0^\...
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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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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:
$$I(R)=\int_{|r|\leq 1} \frac{dxdydz}{|R-r|^{2}}$$
...
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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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2
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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:
$$\int_{\theta-\epsilon}...
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How do I perform a saddle point approximation to approximately find the integral $\int_{-\pi/2}^{\pi/2} \cos^n(x) dx$?
I am supposed to apply the saddle point approximation (i.e. method of steepest descent) to calculate the following integral: $$\int_{-\pi/2}^{\pi/2} \cos^n(x) dx$$ But I have no idea how to write the ...
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What is the most efficient way to calculate $I =\int_0^1 (1-x)^{1/2}(x^3+1) $and what is the degree of the interpolating polynomial used? $
I have some doubts about the following exercise
What is the most efficient way to calculate $I =\int_0^1 (1-x)^{1/2}(x^3+1)dx $ ?
Solution:
Let $g(x)=x^3+1$ and $w(x)=(1-x)^{1/2}$. With a change of ...
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Find the value of $x(t)$ when $x'(t)=0$ given $x''(t) = a\cdot x'(t) / x(t)$ and some initial conditions [closed]
I am considering the differential equation
$$ x''(t) = a\frac{x'(t)}{x(t)} $$
where $a>0$ is an arbitrary constant. I cannot solve this equation directly, but I am wondering if there's still any ...
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Numerical integration, which quadrature formula use?
Back when I was still student (last year) I studied/performed numerical integrations, everytime the quadrature formula was provided within the exercise. The exercise time came to an end and now I am ...
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Evaluate $\iint(x^{2} + y^{2})dx\,dy$ over the area in the first quadrant bounded by the circle $x^2 + y^2 = a^2$.
Well I am stuck from the beginning. I have two methods:
Method 1:
$y=\sqrt{a^2−x^2}$
here $y$ varies from $\sqrt{a^2−x^2}$ to $0$
$x$ varies from $a$ to $0$
$\iint(x^2+y^2) dx\, dy= \int\left(x^2y+\...
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Follow up question: asymptotics of a two dimensional integral
This is a follow up question of Asymptotics of a two dimensional integral about asymptotics of integrals.
The problem is to find the leading order term of this integral.
$$\int_0^1d\epsilon\int_{-\...
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1
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105
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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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3
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249
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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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0
answers
67
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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:
$$\def\e{\varepsilon}\int_a^b\!\! f(x)dx\ =\ \lim_{\e\to0}f\left(\frac d{d\e}\right)...
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Approximating an integral in the from $e^{1/(a+bx/c)}$
I want to integrate the equation $e^{1/(a+bx/c)}$ with respect to x, where a, b, c are all constants, but I can't seem to find an approximation for it. Is there a way to approximate this integral or ...
0
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0
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61
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An complicated integral involving Erfi function
Sir,
While, studying the diffraction of the Gaussian beam through apertures, I have faced the following integral (as an expression of diffracted field),
$$I= \int_0^R \exp(-\alpha r^2+i\beta r)\bigg[\...
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1
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106
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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 :
$$ ...
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0
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33
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Approximation of a function using n intervals
Consider the function 𝑓 𝑥 = 𝑠𝑖𝑛 𝑛𝜋𝑥 and divide the domain 0 ≤ 𝑥 ≤ 1 into m intervals. For the “exact” approximation of the function, we will use 𝑚 = 100 intervals.
Plot the “exact” function ...
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1
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79
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Numerical quadrature with preassigned points
I have been looking for a numerical quadrature that might be possible to pre-assign specific nodes.
For instance, I need to numerically calculate the integral of $f(x)$ in the interval $[a,b]$ but the ...
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3
votes
1
answer
34
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Spectrally-Accurate Quadrature of Singular Integrand
I have a set of PDEs governing some function $f(r)$ which I desire to solve via a psuedospectral method (we can consider $f$ to be smooth). It is defined on the interval $r\in[0,\infty)$ with symmetry ...
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An integral approximation for $Q = \sum_x f(x) $ when $\tan(C x) = x$
I have a sum of the form
$$Q = \sum_{x}f(x).$$
Here, the $x$ entering the sum are the countably infinite set of solutions to
$$ \tan(C x) = x.$$
The $x$ are not integers, but they do satisfy
$$ \frac{\...
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1
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52
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Numerical Integration on Some Points in xy-plane
I have an assignment to finding the result of the integration of some points given in the closed interval $[0,1.2]$ with the method of Trapezoidal Rule, Simpson 1/3 and Simpson 3/8 rule. The problem ...
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To determine the integration of $ \int_{0}^{+\infty} \exp\!\Big(-\Big(\frac{ax^2+bx+c}{gx+h}\Big)\Big) dx$.
What is the integration of the following function:
$$ \int_\nolimits{0}^{+\infty} \exp\!\bigg(-\bigg(\frac{ax^2+bx+c}{gx+h}\bigg) \bigg)dx.$$
What I have done is as follows:
Here, $\kappa=c-\Big(\...
2
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0
answers
126
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LaGrange Multiplier constrain with discretised functions (Matlab)
I am a little stuck with my optimization assignment. It's, no surprise, the catenary but the approach we are looking for is a little different than whatever I could track down on different forums ...
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