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This is an old source (it gives code in Fortran), but it might still serve your purpose: Calculation of Canterbury approximants, Computer Physics Communications 10 (1975) 234. A somewhat more recent s …

Math journals that have recently published papers on "quadrature" include Applied Numerical Mathematics IMA Journal of Numerical Analysis Journal of Approximation Theory Journal of Computational and …

To solve $\int d^3 y\, K(x,y,\lambda) f(y) = f(x)$ I would discretize the coordinates $x\mapsto x_n$, $y\mapsto y_m$, $K(x,y,\lambda)\mapsto K(x_n,y_m,\lambda)\equiv K_{nm}(\lambda)$ and solve the det …

The maximum $x_n$ of $$f_n(x):=e^{-1/x}\Bigl(1+\frac{1}{n^2 x^n} \Bigr)$$ is the smallest solution in $(0,1)$ of the equation $$x=n x^n+\frac{1}{n}.$$ For $n\gg 1$ this gives $x_n\rightarrow 1/n$. The …

The two weak formulations of the Stokes equation, $$\int_\Omega\bigl(\nabla\mathbf{v}\cdot\nabla\psi_i+\psi_i\nabla p \bigr) dV=0,$$ and $$\int_\Omega\bigl(\nabla\mathbf{v}\cdot\nabla\psi_i-p \nabla\p …

With Mathematica I can first sum the series over $\ell$ to get a closed-form expression in terms of polygamma functions, $$Z(2,2)= \sum_{k,\ell \geq 0} \frac{2k+3}{\binom{k+2}{2}^2} \frac{2\ell+3}{(\ …

An easy and reliable way to share code is via Zenodo --- works much like arXiv, you get a DOI, can update your files, and it's free. We use it regularly to document computer simulations in physics, I …

Since $\ln x = - \ln(1/x)$, to evaluate the logarithm near zero is equivalent to evaluating it for large argument. You can then use the result $$\ln y=\frac{\pi}{2a}\left(1+{\cal O}(y^{-2})\right),$$ …

Let me insert a factor $N^{-1/2}$, so that the Fourier transform is unitary: $$U_{mn}=N^{-1/2}e^{-2\pi i(m-1)(n-1)/N},\quad m,n=1,\ldots,N.$$ We truncate the $N\times N$ matrix $U$ to the $k\times k$ …

See, for example, these lecture notes, or, for a more formal exposition, consult Levinson's Algorithm, Wold's Decomposition, and Spectral Estimation by A. Papoulis (1985).

The probability $P(f)$ that the spacing $s$ of two eigenvalues of a random real symmetric matrix is smaller than the average spacing $\bar{s}$ by a fraction $f$ is given by $$P(f)=1-e^{-\pi f^2/4},$$ …

An overview by one of the developers of Mathematica, focusing on definite integrals, is at Symbolic definite integration: methods and open issues. Mathematica knows all the entries in Gradshteyn-Ryzhi …

If you work in the frequency domain there is no need to zero-pad the data to achieve a minimum length. In the time domain the resulting signal will then be periodically extended beyond the boundaries. …

You may want to use a stochastic algorithm. Entry points to the literature (which is large) could be A stochastic algorithm for high-dimensional integrals over unbounded regions with Gaussian weight …

In the field of hydrodynamics the first calculation by a human computer was carried out around 1920 for a project to transform an open sea into a closed lake, with the aim to protect Holland from floo …