This is a library of special functions of statistics, mathematical physics, or other applied mathematics and numerical methods that support their evalution.

Style

Since often only 4 byte IEEE single precision is needed and since this uses both 1/2 the space and, in a SIMD vectorized context, 1/2 the time, these routines are also usually generic over a floating point type F which may be float32 or float64. If the language ever supports it this might be extended to float16 or float128. std/rationals.Rational[T] is not crazy, but would likely require re-implementing many base transcendental functions (sin,exp,..).

If you read pretty much any material on numerical methods then you will soon know that almost all methods have run-time performance which can be very sensitive to error tolerances & input parameters. Even the simplest example of how many terms to sum in a power series exhibits this. Lack of caution in "nesting" evaluations, e.g. as in numerical root finding or numerical integration, can really multiply the time cost of wanton precision.

Therefore, most library entry points have a request-reply protocol for error. I.e., they accept an optional requested error or convergence tolerance and they reply with an estimate on the achieved error. If the achieved error is too big then the method has failed, and the value is not to be trusted naively.

In Nim, this is all modeled by two extra default parameters to generics for the calls: err=1e-7, est: var float64=privateDummy. To receive the reply estimate you must declare and pass a float64. If you do not care about these details then you can simply not pass the extra parameters and hope for the best.

The requested error is relative since the caller does not know the answer ahead of time. The reply estimate is absolute since additive adjustments seem more common than multiplicative and these do not alter the error. { This may be up for debate/change in early stages. Perhaps relative error returns are best. E.g., since a caller need only multiply to convert, not divide. }

Scope

It'd be good to add hypergeometric functions (which often subsume or are used by so many others), spherical harmonics, bessel functions, elliptic, exponential, and trigonometric integrals, as well as the density/cdf/quantile functions for the more common probability distributions such as uniform, exponential, normal, Chi-square, F, logistic, gamma & beta, Cauchy, Weibull (maybe GEV), Poisson, Kolmogorov-Smirnov, maybe log Normal. PRs are very welcome.