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Convolve the data (simplest thing is to use a box) and renormalize. Use the conv command in MATLAB. EDIT: You might also consider splines. Keep the dicontinuity of higher derivatives in mind if you d …

Interval arithmetic methods will permit rigorous bounds. You might try INTLAB. There are various books on rigorous numerics, e.g., Warwick Tucker's Validated Numerics, and the journal Reliable Computi …

Your system of equations is of the form $Az = b$, where $z_{i(\alpha)} = x^\alpha$ and $i(\alpha)$ is the grlex index of $\alpha \in$ $ X_{n,k} \equiv$ {$\beta \in \mathbb{Z}^n: \sum_j \beta_j = k$} …

Considering a "flop" as a real arithmetic operation and ignoring precision, and apropos of @skbmoore's reference, the paper [Johnson, S. G. and Frigo, M. "A Modified Split-Radix FFT With Fewer Arithme …

More generally, black-box linear algebra is an entire subfield of linear algebra, in which the matrix-vector multiplication is treated as an oracle. Generally if this oracle has subquadratic complexit …

You might consider iterative and black-box methods, and particularly Arnoldi.

There is an interval algorithm for finding global extrema. See, e.g. section 5.5 of Jaulin et al. or 5.2 of Tucker for overviews, with some C++ source code.

How about the fast multipole method? (Or does that count as multigrid?)

If your $f$ is a probability distribution, then you can use a kernel density estimate to estimate the derivative. For a bit more detail and relevant references, see section 2.2 of A Tutorial on Kernel …

The recent state of the art is described in section 7.2 of Simoncini, V. "Computational methods for linear matrix equations." SIAM Rev. 58, 377 (2016), https://doi.org/10.1137/130912839. Your equation …

Let $R = \{x_1,\dots,x_n\}$. An integer relation between $x_j$ and $-1$ is a pair $(a_j,a_{j*}) \in \mathbb{Z}^2$ satisfying $a_j x_j = a_{j*}$. Suppose there are such integer relations for all $j \in …

$\pi$ can be computed with the hexadecimal BBP series, though apparently there are faster known ways to compute all of the bits to some level. Knuth attributes to Brent JACM 23, 242 (1976) the result …

The FFT is an important part of the fast multipole method, which is probably what you would want to use.