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

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### 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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### Finding $x(t)$ for the mechanical system $x'' = -x$

Using energy conservation theorem and method of integration by quadrature, find $x(t)$ for the mechanical system $x'' = -x$, considering a spring with mass $1$ and elasticity constant $1$. My attempt:...
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### 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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### Gauss-Legendre vs Gauss-Chebyshev quadratures (and Clenshaw-Curtis)

There's a point I wish to elucidate about integrating polynomials. It is known that functions are better approximated by polynomials with non-uniformly spaced nodes, which is why Gauss quadratures ...
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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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### Why not simply use sine weights with Clenshaw-Curtis nodes?

Clenshaw-Curtis quadrature is based on writing $$\int_{-1}^{1} f(x)dx=\int_{0}^{\pi}f(\cos y)\sin y dy$$ and then replacing $f(\cos y)$ by a truncated Fourier series, so that the integral can be ...
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### High accuracy root finder of Legendre polynomials' derivatives?

I need GLL (Gauss-Legendre-Lobatto) nodes for the Legendre-Galerkin-NI spectral method. It requires me to find the roots of the derivatives of Legendre polynomials. My Matlab program calculates the ...
Consider the quadrature formula $\int _{-1}^1\vert x \vert f(x)dx \approx \frac{1}{2}[f(x_0)+f(x_1)]$, where $x_0$ and $x_1$ are quadrature points. Then the highest degree of the polynomial, for which ...