\begin{hcarentry}[updated]{Numeric prelude}
\label{numericprelude}
\report{Henning Thielemann}%11/08
\status{experimental, active development}
\participants{Dylan Thurston, Mikael Johansson}
\makeheader

The hierarchy of numerical type classes is revised and oriented at algebraic
structures.  Axiomatics for fundamental operations are given as
QuickCheck properties, superfluous super-classes like
\texttt{Show} are removed, semantic and representation-specific operations are
separated, the hierarchy of type classes is more fine grained, and identifiers
are adapted to mathematical terms.

There are both certain new type classes
representing algebraic structures
and new types of mathematical objects.
Currently supported algebraic structures are
% \begin{compactitem}
group (additive),
ring,
principal ideal domain,
field,
algebraic closures,
transcendental closures,
module and vector space,
normed space,
lattice,
differential algebra,
monoid.
% \end{compactitem}

There is also a collection of mathematical object types,
which is useful both for applications and
testing the class hierarchy.
The types are
% \begin{compactitem}
lazy Peano number,
arbitrarily quantified non-negative lazy number (generalization of Peano number),
complex number, quaternion,
residue class,
fraction,
partial fraction,
number equipped with physical units in two variants
(dynamically and statically checked)
fixed point number with respect to arbitrary bases and numbers of fraction digits,
infinite precision number in an arbitrary positional system as lazy lists of digits
supporting also numbers with terminating representations,
polynomial, power series, \textsc{Laurent} series
root set of a polynomial,
matrix (basics only),
algebra, e.g., multi-variate polynomial (basics only),
Gaussians for exact Fourier transform,
permutation group.
% \end{compactitem}

Due to Haskell's flexible type system,
you can combine all these types,
e.g., fractions of polynomials, residue classes of polynomials,
complex numbers with physical units,
power series with real numbers as coefficients.

Using the revised system requires hiding some of the standard functions
provided by Prelude, which is fortunately supported by GHC.
The library has basic Cabal support
and a growing test-suite of QuickCheck tests
for the implemented mathematical objects.

Each data type now resides in a separate module.
Cyclic dependencies could be eliminated by fixing some types in class methods.
E.g., power exponents became simply Integer instead of Integral,
which has also the advantage of reduced type defaulting.

% \FuturePlans
% Collect more Haskell code related to mathematics, e.g., for linear algebra.
% Study of alternative numeric type class proposals and
% common computer algebra systems.

% A still unsolved problem arises for
% residue classes, matrix computations, infinite precision numbers,
% fixed point numbers, and others.
% It should be possible to assert statically
% that the arguments of a function are residue classes with
% respect to the same divisor, or that they are vectors of the same size.
% Possible ways out are encoding values in types or local type
% class instances. The latter one is still neither proposed nor
% implemented in any Haskell compiler.
% The modules are implemented in a way to keep all options open.
% That is, for each number type there is one module
% implementing the necessary operations
% which expect the context as a parameter.
% Then there are several modules
% which provide different interfaces through type class instances
% to these operations.

News:
  Added two wrapper types that allow types
  to be used with both Haskell98 and NumericPrelude type classes
  that are initially implemented for only one of them.


\FurtherReading
\url{http://www.haskell.org/haskellwiki/Numeric_Prelude}
\end{hcarentry}
