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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19 views

Approximating function by its step approximation

$f$ be any bounded measurable function on $[0,1]$. Define the step function $f_n(x)=f(\frac{i}{n})$, if $x \in [\frac{i-1}{n},\frac{i}{n}]$. Is it always true that $$\int |f(x)- f_n(x)|dx \rightarrow ...
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29 views

Integrating over roots

(I'm an independent learner, so no professor to ask.) There is a short example given in a youtube video not in given in any textbooks that I don't really follow. Yet it seems very useful, and possibly ...
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0answers
7 views

The background to gaussian quadrature for double integrals

A college assignment in numerical analysis requires that I research into the historical and mathematical background of the gaussian quadrature for double integral approximation, among other things. ...
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1answer
84 views

Numeric calculation of infinite Fourier integral in 2D

Consider a 2D function $f(x,y)$ on $\mathbb{R}^2$, which is finite and decays on some finite interval. I don't have a nice analytical/closed-form expression for $f(x,y)$, but can evaluate it at any $(...
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57 views

Approximating an integral involving Gaussians

I'm trying to evaluate the integral $$ \int_{-\infty}^{\infty}\frac{\left[\sum_{j}c_{j}\left(x-x_{j}\right)\exp\left(-\left(x-x_{j}\right)^{2}\right)\right]^{2}}{\sum_{i}\exp\left(-\left(x-x_{i}\right)...
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1answer
50 views

Laplace's method for integration

I am trying to compute $$I(\lambda) = \int_{0}^1 \frac{x}{\sqrt{1+x^4}}e^{\lambda x}\mathrm{d}x $$ for large, real, positive $\lambda$. I'm attempting this with Laplace's method as suggested, however ...
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27 views

Saddle Point Approximation for Multiple Contour Integrals

Using the multinomial theorem $$ \left(\sum_{j=1}^{N} x_j \right)^N = \sum_{m_1 + \cdots + m_n = N} \frac{N!}{m_1! \cdots m_N!} x^{m_1} \cdots x^{m_N}, $$ and the contour integral expression of the ...
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1answer
40 views

Asymptotic expansion of integral of airy function

In this question I am given that the asymptotic expansion of the Airy function for large $z$ is given by $$Ai(z) = \frac{1}{2}\pi^{-\frac{1}{2}}z^{-1/4}\exp\left(-\frac{2}{3}z^{\frac{3}{2}}\right)\...
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0answers
32 views

approximation of $\int \ln(1+\frac{2k}{a+b-k}) $

I'm trying to calculate an integration over an arbitrary polygon $P$, and a point $p$ is on $P$. $k$ is a constant, and $A(x,y,z)$ and $B(x,y,z)$ are arbitrary functions returning a scalar. As $\int_P ...
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60 views

Estimate the error in the trapezoidal rule

I have the following exercise: We numerically approximate the integral $\int_{x_{L}}^{x_{R}}f(x)dx$ with the trapezoidal rule, using $x_{L}$ and $x_{R}$ as the only nodes. Next, we halve the interval $...
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2answers
96 views

Why is the Gaussian-Legendre Quadrature so effective?

I understand how it works, how its derived, etc. The proof of it has been shown to me. That is to say, I know how Legendre polynomials are derived, I know they are orthogonal, I know we sample a ...
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0answers
20 views

Piecewise Gauss-Legendre quadrature order of convergence

Given a definite integral $\int_a^bf$, If we increase the number of nodes and weights of the Gaussian quadrature, we would get closer to the exact integral. But another way to get more exact is to ...
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21 views

Romberg Integration: What are the rules to using it, as my implementation fails?

Im attempting to numerically evaluate $\int\limits_{-1}^1 \cos^2(x) dx$. I know the exact answer ahead of time: $1 + \cos(1)\sin(1)\approx 1.4546487134\ldots$. Im doing this in Microsoft Excel, to ...
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1answer
25 views

Simpson's Rule Question with Example

I'm just learning Simpson's Rule for integral approximation and I have a question. $ \int _a^b\:f\left(x\right)dx\:approx=\frac{\frac{b-a}{n}}{3}\left[\left(1f\left(x0\right)\right)+4f\left(x1\right)+...
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0answers
29 views

Solution or approximation of $\int_0^r exp(-g*\sqrt{x^2+2*b*x+c})dx$

I'm trying to find a solution or approximation of $$\int_0^r \exp(-g\sqrt{x^2+2bx+c})dx$$ where g > 0, c > 0 and r is from 0 to infinity. I found that $$\frac{1.5}{b*\sqrt{c+1.5}+c+1.5}$$ Works ...
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1answer
31 views

Approximate the same integral on different range of same length

Let $f(x)=(\tan x)^{\frac{3}{2}}-3\tan x+\sqrt{\tan x}.$ Consider the three integrals $$ I_1=\int_0^1f(x)\ dx, I_2=\int_{0.3}^{1.3}f(x)\ dx, I_3=\int_{0.5}^{1.5}f(x)\ dx $$ Then how to show that $I_1&...
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4answers
66 views

Approximating the integral $\int_1^\frac{3}{2}\frac{\ln(x+1)-\ln(2)}{x-1}dx$ to accuracy of $0.001$

I need to approximate the integral $\int\limits_1^\frac{3}{2}\frac{\ln(x+1)-\ln(2)}{x-1}dx$ with an accuracy of $0.001.$ How don't know how I can do it, since the integrand doesn't have an elementary ...
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0answers
16 views

Find weights and coefficients such that quadarature formula is exact

I'm struggling with the following problem: What is the maximum degree of exactness that we can obtain with the following quadrature >formula $$\int_0^1 f(x)\frac{1}{\sqrt{x}}dx \approx w_0 f(x_0) +...
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13 views

Arbitrarily large degree of exactness for a precise quadrature formula

I don't know how to solve the second part of the following exercise: Consider the quadrature formula $Q(f)=w_1 f(x_1)$ for the computation of the weighted integral $$W(f)=\int_0^1 x^{\alpha} f(x)dx$$ ...
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1answer
48 views

Taylor expansion of a Lorentzian integral to second order

I am considering a Lorentzian integral of the following type: \begin{equation} \int_{-\infty}^\infty\mathrm dx\,\frac{\gamma^2}{(x-x_0)^2+\gamma^2}f(x), \end{equation} where $f(x)$ is a "nice&...
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26 views

Is there an analytic formula or good fitting function for the Hankel transform of a Sersic profile?

I'm trying to find an analytic form for the Hankel transform of a Sersic profile: $$ \begin{align} f_n(r) &= \frac{e^{-r^{1/n}}}{n \Gamma(2n)}\\ F_n(k) &= \int_0^\infty f(r) r J_0(k r) dr \end{...
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2answers
46 views

Approximate the value of the intergral

Suppose that $\arctan(x)=\sum_{n=0}^{\infty} \frac{(-1)^nx^{2n+1}}{2n+1}$ for all x $\in$ $[-1,1]$. Use the least number of terms to approximate the value of the integral $$\int_0^{1/2} \frac{x-\...
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3answers
75 views

$\int_{1}^{2}\frac{e^x+e^{4/x}}{x}dx$

$I=\int_{1}^{2}\frac{e^x+e^{4/x}}{x}dx$ For $ x \in (1,2), \frac{e^x+e^{4/x}}{x}>e^x$ $\Rightarrow I=\int_{1}^{2}\frac{e^x+e^{4/x}}{x}dx$ >$\int_{1}^{2}e^xdx$ $\Rightarrow I>e^2-e$ And $I<...
3
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1answer
49 views

Integral of a product of Bessel functions of the first kind

I want to do this integral $H(\rho)=\int_{0}^{\infty} J_1(2 \pi Lr)J_0(2\pi \rho r)dr$, where $J_1$ and $J_0$ are Bessel functions of the first kind and $L\in \mathbb{R}$ is a constant, so I tried to ...
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0answers
25 views

Compute the Posterior from a Hierarchical Bayesian Graph

Suppose that I have the following Bayesian network: I want to calculate $p(\omega | \mathbf{x})$, where $\mathbf{x} = [x_{1}, \ldots, x_{N}]$ is my observed data. My distributions are: $$ \mathbf{x}...
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0answers
34 views

Asymptotic form of Complicated Integral

I want to approximate a integral given like $$\langle x\rangle=\int\limits_0^1 \mathrm dxx \int\limits_0^1\mathrm dy f(y,x)g(y)$$ And $f(y,x)$ is $$f(x,y)=\frac m{h(x)}\exp\Bigg(L\int\limits_0^x \...
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1answer
210 views

Integrate a weighted Bessel function over the unit disk

I would like to evaluate a complex-valued integral of the form $$ I_e = \int_0^1 x e^{iax} J_0(b \sqrt{1-x^2}) dx $$ where $a$ and $b$ are real numbers (not necessarily positive) and $J_0(z)$ is the ...
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20 views

Approximating $\int_{-\infty}^\infty e^{-f(q)/\hbar} dq$

I'm a Physics student (self-studying Zee's Quantum Field Theory in a Nutshell) hoping to get the following mathematical query solved. It is mentioned that to approximate $I=\int_{-\infty}^\infty e^{-f(...
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28 views

Monte Carlo Approximation of Integral Transform

I'm looking for bounds over the following Monte Carlo approximation. Define a subset $\Omega$ of R^n (compact, not sure if it is important) and probability density function $P(X)$ over it. Further, ...
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0answers
40 views

Find values of $p$ such that trapezoidal rule gives second order approximation

I have the following question, which should be easy, but I'm puzzled now: Consider $f(x)=|x|^\frac{p}{2}$, where $p$ is a non-negative integer and the integral $\int_{-1}^{1}f(x)dx$. For which ...
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0answers
16 views

changing variables for numerical integration

I want to numerically integrate a set of data in Python, but I think my question might a math than a Python question. where I have discrete data points L, f(L) and I need to perform this integral: ...
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0answers
105 views

Newton-Cotes quadrature formula being “correct” for polynomials

So I'm going through a chapter on Newton-Cotes quadrature formulas and everything's clear up until a certain point. First let me just get the notation stuff out of the way so we can be on the same ...
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1answer
108 views

Integral $ \int_{-\infty}^{+\infty} e^{-t^2}\ln(1+e^t) \, dt $ [closed]

$$ \int_{-\infty}^{+\infty} e^{-t^2}\ln(1+e^t) \, dt $$ Hello everyone,i would like to know the result of the above integral and how to calculate or estimate it. background and progress so far:(1)...
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1answer
19 views

Error Estimation for Gauss Quadrature

I have an equation $$ \int_{0}^{1}sin(x)dx $$ and I want to find the error estimation for Gaussian quadrature using the above equation. I saw this formula for finding the error of the gaussian ...
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0answers
16 views

Error Estimate for f(x,y) using trapezoidal rule

Please I need help. How do I find the error estimate of the trapezoidal rule of the function $$ \int_{0}^{1}\frac{x^{2}}{1+y^{3}}dy $$ using $$ -\frac{h^{2}}{12}[f'(b)-f'(a)] $$ where $$x,y\in(0,1)$$ ...
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1answer
79 views

Asking for help in evaluation of an integral in a research paper

I need Help in evaluating an integral. I have a function $$\rho_0(x) = \min_{ y\in\mathbb R} \rho( x,y) = \min_{0\leq y < 1} \rho(x, y), $$ which is $1$- periodic. $\rho_0(x) $ can be ...
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0answers
31 views

Operation count for Trapezoidal Integration vs Romberg Integration

I have been working with computing and numerical integration, and I have to juxtapose the operations required for Trapezoidal and Romberg for $n$ points to a certain error tolerance. For trapezoidal ...
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2answers
31 views

Maximum value of the integral

The maximum value of the integral $$\int_{a-1}^{a+1} \frac{1}{1+x^8}dx $$ is attained A) at exactly two values of $a$ B) only at one value of $a$, which is positive C) only at one value of $a$ ...
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1answer
43 views

Evaluate $\int_{0}^{2}(x^2 + x^{3/2} + x + x^{1/2})dx$ using Simpson's Rule

I'm running into some troubles while trying to evaluate $$\int_{0}^{2}(x^2 + x^{3/2} + x + x^{1/2})dx$$ using Simpson's Rule Simpson's Rule states $$Q(f) = \frac{b-a}{6}(f(a) + 4f(\frac{a+b}{2}) +f(...
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0answers
32 views

(Feedback to) Two-point Gaussian Quadrature for $\int_{0}^{2}(x^2 + 3x -1)dx$

May I ask you for feedback ? Thanks ! We're asked to evaluate the integral $$\int_{0}^{2}(x^2 + 3x -1)dx$$ using the Gaussian-Quadrature Formula with $n=2$ points. We're not allowed to use that the ...
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0answers
28 views

Need to show $||S_nf||_2 \leq ||f||_2$ approximation theory

I am considering the inner product space $L_{2 \pi}^2$ of square integrable 2$\pi$-periodic functions on $\Re$, with the inner product defined by: $<f,g>$ = $\int_{0}^{2\pi} f(x)g(x) dx$. For $...
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0answers
28 views

Bayesian Neural Network

I am reading Bayesian Neural Network from PRML, section 5.7. I understand that posterior distribution is approximated by using the Laplace approximation. Then, for predictive distribution, the author ...
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0answers
8 views

Approximate formulas for the linear transformations

The upper half-plane is mapped onto the unit disk so that the point $z=hi\,\,\, (h>0)$ passed into the center of the circle. Find the length $\Gamma$ of the image of the segment $[0, a]$ of the ...
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1answer
46 views

Evaluate $\int_{-1}^{0}(x^4-x^2+2)dx$ using Simpson's Rule

We're asked to evaluate the integral $$\int_{-1}^{0}(x^4-x^2+2)dx$$ using Simpson's Rule.
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28 views

How can I estimate error when computing improper integral with a finite interval?

I'm trying to evaluate integrals of the form $$I = \int ^{\infty }_{a} f( x) dx$$ Where $a$ is any real number. In order to estimate this integral, I pick some large positive number $M$ and instead ...
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1answer
27 views

Given the quadrature rule $Q[f]=f(x_1)+f(x_2) \approx \int_{-1}^{1}f(x)dx = I[f]$ , find the nodes $x_1, x_2$

Given is the following quadrature rule : $$Q[f]=f(x_1)+f(x_2) \approx \int_{-1}^{1}f(x)dx = I[f]$$ and the degree of accuracy $q \geq2$. We're asked to find the nodes $x_1, x_2$. May I ask you to ...
4
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1answer
75 views

What's the name for this numerical integration algorithm?

Suppose I estimate $\int_0^1 f(x)dx$ as $\frac12(f(a)+f(b))$, with $a,\,b$ chosen to achieve the lowest-order possible error. We assume $f$ equals its Maclaurin series and $\int$ commutes with $\sum$, ...
4
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0answers
99 views

Pointwise Convergence of Convolution with Approximate Identity

Here are some background settings. Let $K\in L^1$ s.t. $\int_{\mathbb R^n}K(x)dx=1$ and let $K_{\varepsilon}(x):=\varepsilon^{-n}K(x/\varepsilon)$ for all $\varepsilon\gt0.$ Then we will ...
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0answers
15 views

Approximating the Solution of a Second Kind Integral Equation

I am defining the approximation $k_N$ to $k$ by the following construction: Let $h:=1/N $ and, $n=$ {$0, \frac{1}{2}, 1 , \dots , N$} let $t_n := nh = \frac{n}{N}$ And for $0 \leq x \leq 1$ and $n=1,...
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2answers
56 views

Integral of modified Bessel function of second kind first order multiply by incomplete gamma function?

Is there any possible solution or approximation for that given integral? $$\int_0^\infty {\big(v^{\frac{m}{2}-\frac{1}{4}}\big)}K_1\Bigg[\frac{2\sqrt[4]{v}}{l}\Bigg]\Gamma\left[m,-\frac{a+b v}{c}\...