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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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Quadrature Rule Error and Peano Kernel

Consider the following quadrature: $$\int_0^1 x^cf(x)dx\approx Af(0)+B\int_0^1 f(x)dx, c>1, \neq 0$$ Determine A and B such that this rule has a degree of exactness 1. Let $E(f)$ be the error ...
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Why does $ \int_{\Omega} f\, dA = 2\pi \,f(0) $ and not zero?

I have been experimenting with quadrature domains. The most obvious one is a circle. Let $f(z)$ be holomorphic on a large enough region and $\Omega= \{|z| < 1 \}$ then: $$ \int_{\Omega} f\, dA = ...
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If the Gaussian Quadrature approximations are eventually all equal is the function a polynomial?

For a fixed function $f$ defined on $[-1, 1]$, let's define $$G_m(f)=\sum_{k=1}^m w_k^{(m)}f\left(x_k^{(m)}\right)$$ where $x_1^{(m)}, x_2^{(m)}, \ldots, x_m^{(m)}$ are the $m$ roots of the $m$th ...
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Approximation to $\int_{-1}^{1}f(x)dx$ using only $f(0)$, $f'(-1)$, and $f''(1)$

The exercise is Using only $f(0)$, $f'(-1)$, and $f''(1)$, compute an approximation to $\int_{-1}^{1}f(x)dx$ that is exact for all quadratic polynomials I have only seen guassian quadrature ...
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Show that the trapezoid rule is more precise than Simpson rule for $x^{5} - \alpha \cdot x^{4}$

I need to show that for $\int_{0}^{1}(x^{5}-\alpha \cdot x^{4})dx$, the trapezoid rule is more precise than Simpson when $\frac{15}{14} < \alpha < \frac{85}{74}$ What I have done : Trapezoid ...
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Gauss - Chebyshev Quadrature confusion. Please help.

So I'm reading my lecturer's notes on Gauss-Chebyshev Quadrature (lecturer uses the word Formulation instead of Quadrature) and there is a point where he lost me completely. Here are his notes and ...
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Numerical Convergence of Trapezoidal Rule

So I am implementing the trapezoidal rule as my choice of quadrature. Most functions seem to be second order convergence except for two: $$ f(x) = e^{cos(x)} - .1cos(x), \quad x \in [-7, -2] $$ The ...
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Gaussian quadrature with weight function x^2

I would like to get the points and weights of Gaussian quadrature formulas for $$ \int_{-1}^{+1} x^2 f(x)\;\text{d}x. $$ Is this tabulated anywhere yet?
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Simple Gaussian Quadrature Confusion

I'm trying to compute $\int_0^4 x^2 dx$ using 2-point Gaussian quadrature. This should give an exact result since for 2 points, we can exactly handle polynomials of degree 2*2-1=3. By the usual ...
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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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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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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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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 ...
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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....