Questions tagged [quadrature]

Quadrature refers to techniques in numerical integration, such as Riemann sum approximations, Simpson's rule and Gaussian quadrature.

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Approximating the first moment of h(x) where $x$~Lognomal($\mu, \sigma$)

What is the best way to approximate $E(h(X))$, where $X$ ~ Lognomal($\mu, \sigma$). So far, I can think of Monte Carlo Methods and Gaussian Hermite quadrature as below: \begin{align} E(h(X)) &= ...
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51 views

The Distribution of Abscissae and Sums of Weights in Gaussian Quadrature

Let $n$ be a positive integer, and for $i = 1, 2, \ldots, n$, let $x_i$ be the $i^\text{th}$ abscissa for $n$-point Gaussian quadrature, and $w_i$ the associated weight, so that $-1 < x_1 < x_2 &...
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1answer
47 views

Change of Interval for Chebyshev–Gauss quadrature

I am curretly working to numerically evaluate an integral of the form: $$\int_{-1}^{1} f(x) \sqrt{1-x^2} dx$$ For this issue Gauss-Chebysehv integration of second kind seems ideal as it uses the ...
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1answer
87 views

Gaussian quadrature: Orthogonal polynomial for chi distribution

I have a problem involving numerical integration of the form: $$I = \int_0^\infty \!dx \, w(x) f(x)$$ where the weighting function is a chi distribution of degree 2, i.e., $$w(x) = x \, e^{\frac{-...
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137 views

Cubic Bézier curve arc length parametrization reversal: find t given a length

I am following this paper Approximate Arc Length Parametrization, M. Walter & A. Fournier, 1996 and have succesfully implemented the direct solution, as in finding the length $s(t)$ given $t$. ...
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122 views

References on quadrature of $\int_0^{\infty} f(x) \exp(-x^a) dx$

I am aware of Hermite-Gaussian quadrature techniques for integrals of the form $$ \int_0^{\infty} f(x) \exp(-x^2) dx $$ However, am I looking for references on quadrature where the exponent is more ...
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30 views

CDF Approximation in Two Dimensions

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be our respective probability space and $X$ a two-dimensional random variable with values on $[0,1]^2$ and probability density function $f(x_1,x_2)$ for $x_1, ...
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38 views

Orthogonal polynomials on triangles: Connection with quadrature rules?

In a 1D interval $[a, b]$, quadrature rules and orthogonal polynomials are tightly interconnected. For example, given the $n$ roots $t_j$ of the $n$th orthogonal polynomial $p_n$, one has the ...
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56 views

Can using Gauss-Hermite quadrature for estimation, introduce a bias?

I am using Gauss-Hermite quadrature to estimate E[h(x)] where x is a log normal random variable. I happen to see some bias in my final results. I was wondering if there is a formula specifying an ...
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1answer
98 views

Expressing a classic integral having no elementary primitive.

Some functions do not have primitive functions which are easy to express with elementary functions. A famous example is $$\int_a^be^{-x^2}dx$$ But what about if we can write $$=\int_a^b1\cdot e^{-...
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83 views

Approximate Solution for $\int_{E_T}^\infty \frac{2 E b^\nu}{\Gamma (\nu)} \int_0^\infty x^{\nu -2}\exp(-b x)\exp(-(A^2 + E^2)/x) I_0(2EA/x) dx dE$

(This question is more about solving the integral than what the integral represents.) The homodyned K distribution has the following probability density function (PDF): $$ f(E | A, \nu, b) = \frac{2 ...
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86 views

3D Gauss-Hermite Quadrature

Is it possible to examine a 3D integral by using Gauss-Hermite quadrature type technique? I mean there might be an equation like this (with analogy to 1D Gauss-Hermite quadrature): $\int_{-1}^{-1} \...
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84 views

Proof of Newton–Cotes quadrature rules, I guess?

I've been searching for the proof of this formula, I think the main problem is that the book doesn't really give it a name, therefore it's hard to search it up even online. I read the topics suggested ...
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239 views

Integration using Gauss-Laguerre quadrature

I need to solve integral using Gauss-Lagerre'e quadrature: $\int_0^\infty e^{-10x}(5x^3-\pi)$ on interval [0, $\infty$). I am not really familiar with this kind of methods, but I need to solve it ...
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323 views

Gaussian Quadrature Error Estimate

According to Chapter 5 of Numerical Methods and Software by Kahaner, et al. (1989), it can be shown that the error associated with Gaussian quadrature is $\displaystyle\int_a^b f(x)\,dx - \sum_{i=1}^...
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264 views

how to implement adaptive gaussian (kronrod?) quadrature (technicalities)

I am trying to figure out what is the best way to implement an adaptive quadrature scheme which preferentially makes use of guassian quadrature. Guassian quadrature doesn't really give any easily ...
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116 views

How to calculate the mean and the variance for the function with Multidimensional Polynomial Chaos Approach?

I would like to calculate the mean and the variance for the function f(x1,x2)=pi*(x1-1)*sin(pi*x1)*(1-x2^2); (x1,x2)->Uniform distribution [-1,1] using the ...
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21 views

Sampling Method to Calculate Result of High Dimension Integral

I need to calculate the average outputs ($Y_j$) from a non-linear simulation model ($M_j$) which takes a large number of possible input strings ($B_i$). $$Y_j = E(M_j(B_i))$$ Each $B_i=(b_{i,1},b_{...
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1answer
87 views

Integrate over a triangle in the 2D normal distribution

I'd like to evaluate following expression efficiently (numerically). $$g_a(x) := \int_0^x e^{-t^2} \int_0^{at} e^{-s^2} dsdt$$ If I want a given fixed accuracy, and evaluate both integrals using e.g....
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37 views

Gauss Quadrature Error in 2D

Could someone tell me what is the error when applying the Gauss Quadrature rule in 2 dimensions? I know that for one dimension the error is $$\frac{(n!)^{4}}{(2n+1)[(2n)!]^{3}} \cdot f(\xi)^{2n}(b-a)...
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32 views

Gauss-Chebyshev Integration on arbitrary intervals

So, I would like to do a numerical integration of something of the form: $\int_{-1}^1 dx f(x) \sqrt{1-x^2}$ I tried this using Gauss-Legendre only to find out that the last second factor keeps the ...
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43 views

Change numerical integration range when using Gaussian quadrature

I have a little problem related to numerical integration. If someone knows the solution I would be very grateful for sharing. I'm calculating numerical integrals using Gaussian points $x_{1} , x_{2} ,...
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38 views

Minimum number of sub-intervals for the composite trapezoid rule.

I have been working on the following problem. Given a uniform partition of the interval [-2,2] the following integral is approximated using the composite trapezoid rule $$ I = \int_{-2}^{2} ln(1 + x^2)...
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24 views

Determination of Weights for Hierarchical Sparse Girds Quadrature

For the solving a $d$-dimensional SDE with a sparse grid algorithm related to an option pricing problem I ended up having to solve the following integral equation. $g_{\bar{l},\bar{i}} = \int_D p(e^{...
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1answer
40 views

Are there quadrature methods allowing to track current progress, converging faster than Monte-Carlo?

Consider an integral to evaluate: $$I=\int_a^b f(x)\,\mathrm dx.\tag1$$ In the Monte-Carlo quadrature, if $I_n$ is $n$th estimate of $I$, then for $(n+1)$th estimate we have: $$I_{n+1}=\frac{nI_n+f(...
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27 views

When can one trust that an adaptive quadrature scheme is not missing nontrivial areas of the domain?

hcubature http://ab-initio.mit.edu/wiki/index.php/Cubature_(Multi-dimensional_integration) seems to be a useful package for computing multidimensional integrals numerically. However, I have come upon ...
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48 views

Why are isolated zeros allowed in the weight function?

In an exercise I have determined the Gaussian Quadrature formula and I have applied that also for a specific function. Then there is the following question: Explain why isolated zeros are allowed ...
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32 views

Gaussian Quadrature and Degree of Precision

how to prove a interpolatory quadrature has degree of precision k = 2n + 1 if and only if it is a Gaussian Quadrature
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23 views

Roughness of $\int\sqrt{1-t^2}f(t)\,dt$ in $[-1,1]$

Why is this integrand non-smooth in $[-1,1]$? $$\int\sqrt{1-t^2}f(t)\,dt$$ where $f$ possesses an analytic extinction to a complex neighbourhood $[-1,1]$ More general, how can I find it out? Thanks ...
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43 views

Determining constants to make a quadrature formula with degree of precision equal to three.

I just need an explanation on how these linear equations are found when plugging and testing $f(x) = x^k. $ For example, in the first linear equation, what happens to the constants c and d. Why are ...
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1answer
40 views

Bisection Newton - Quadratures

The problem states the following: Find with at least 10 digitis of precisión the roots of the following equation: $\int_x^{x^2} \!e^{-t^2}\,\mathrm{d}t = x^5 -3x^2 + 1 $ in the closed Interval [-1,1]. ...
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74 views

Comparison in the accuracy of Romberg Integration and Second Order Newton-Cotes Quadrature

Context I ask this question because I'm currently working on a program that solves the Cahn-Hilliard equation in 2D. For this project I need a subroutine to calculate the free energy functional by ...
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1answer
22 views

Error of a quadrature applied to an approximation of a function with its own error

This has come up with a math modeling project I am doing. To try and boil it down, suppose I am using a numerical quadrature such that $ \int_0^1 f(x) dx = \sum_{i=1}^n f(x_i) + O(h^n)$ where $O(h^n)...
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49 views

Can a recursive, continuous integral be approximated with Gauss-Legendre or similar?

Maybe I need to reformulate(?). Suppose there is this simple function: $$f(x)=\int_a^b{x \text{d}x}$$ If it were to be discretized, there would be losses due to sampling. In order for the errors to ...
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1answer
133 views

Degree of accuracy for the Gauss-Lobatta quadrature formula

We would like to approximate the integral of $f$ over an interval $[a,b]$ with positive measure $\mu$. To this end, we choose $n$ points $x_1, x_2, \ldots, x_n$ inside the open interval $(a,b)$ ...
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2answers
260 views

Error of composite trapezium rule

Given the integral $$\int_{1}^{\infty} \frac{e^{-x}}{x^2}dx$$ I want to find a value of $b >1$, and a number of intervals $n$ such that the error in $$\int_{1}^{b} \frac{e^{-x}}{x^2}dx$$ is less ...