This namespace contains implementations of various approaches to [numerical
differentiation](https://en.wikipedia.org/wiki/Numerical_differentiation).

Each of these methods uses [Richardson
extrapolation](https://en.wikipedia.org/wiki/Richardson_extrapolation) to
accelerate convergence, and roundoff error estimation to bail out of the
computation when roundoff error threatens to overwhelm the calculation.

For an implementation of [forward-mode automatic
differentiation](https://en.wikipedia.org/wiki/Automatic_differentiation),
see [[sicmutils.differential]] (for the backing implementation)
and [[sicmutils.calculus.derivative]] (for the [[sicmutils.operator/Operator]]
instances that make it pleasant to use this method.)

This namespace contains function to compute various [elliptic
integrals](https://en.wikipedia.org/wiki/Elliptic_integral) in [Carlson
symmetric form](https://en.wikipedia.org/wiki/Carlson_symmetric_form), as well
as the [Jacobi elliptic
functions](https://en.wikipedia.org/wiki/Jacobi_elliptic_functions).

This namespace contains a discussion of polynomial interpolation, and different
methods for fitting a polynomial of degree `N-1` to `N` points and evaluating
that polynomial at some different `x`.

This namespace contains a discussion of rational function interpolation, and
different methods for fitting rational functions to `N` points and evaluating
them at some value `x`.

Richardson interpolation is a special case of polynomial interpolation; knowing
the ratios of successive `x` coordinates in the point sequence allows a more
efficient calculation.

Entrypoint for univariate and multivariate minimization routines.

ODE solvers for working with initial value problems.

Implements utilities shared by all integrators, for example:

- code to wrap a sequence of progressively better estimates in a common `integrator` interface
- data structures implementing various integration intervals.

### U Substitution and Variable Changes

This namespace provides implementations of functions that accept an
`integrator` and perform a variable change to address some singularity, like
an infinite endpoint, in the definite integral.

The strategies currently implemented were each described by Press, et al. in
section 4.4 of ['Numerical
Recipes'](http://phys.uri.edu/nigh/NumRec/bookfpdf/f4-4.pdf).

Trapezoid method.

This namespace contains an implementation of Brent's method for finding the
minimum of a real-valued function.