# Applications and libraries/Mathematics

## 1 Applications

### 1.1 Physics

Meep
Meep (or MEEP) is a free finite-difference time-domain (FDTD) simulation software package developed at MIT to model electromagnetic systems.
Numeric Quest
Jan Skibinski's Numeric Quest library provides modules that are useful for Quantum Mechanics, among other things.

## 2 Libraries

### 2.1 Linear algebra

hs-linear-algebra
Patrick Perry's linear algebra library, built on BLAS.
Wrapper to CLAPACK
Digital Signal Processing
Modules for matrix manipulation, Fourier transform, interpolation, spectral estimation, and frequency estimation.
Index-aware linear algebra
Frederik Eaton's library for statically checked matrix manipulation in Haskell
Numeric Quest
Jan Skibinski's Numeric Quest library provides several modules that are useful for linear algebra in general, among other things.
vector-space
The vector-space package defines classes and generic operations for vector spaces and affine spaces. It also defines a type of infinite towers of generalized derivatives (linear transformations).
HMatrix
By Alberto Ruiz. From the project website:
This library provides a purely functional interface to linear algebra and other numerical computations, internally implemented using GSL, BLAS and LAPACK.
All linear algebra functions mentioned in GNU-Octave's Quick Reference (except syl) are already available both for real and complex matrices: eig, svd, chol, qr, hess, schur, inv, pinv, expm, norm, and det. There are also functions for numeric integration and differentiation, nonlinear minimization, polynomial root finding, etc.
Vec
By Scott E. Dillard. Static dimension checking:
Vectors are represented by lists with type-encoded lengths. The constructor is :., which acts like a cons both at the value and type levels, with () taking the place of nil. So x:.y:.z:.() is a 3d vector. The library provides a set of common list-like functions (map, fold, etc) for working with vectors. Built up from these functions are a small but useful set of linear algebra operations: matrix multiplication, determinants, solving linear systems, inverting matrices.

### 3.1 Physical units

Dimensionalized numbers
Working with physical units like second, meter and so on in a type-safe manner.
NumericPrelude: Physical units
Numeric values with dynamically checked units.
CalDims
This is not simply a library providing a new type of
Num
class, but stand-alone calculation tool that supports user defined functions and units (basic and derived), so it can provide dimension-safe calculation (not embedded but via shell). Calculations can be modified/saved via shell. It uses rational numbers to avoid rounding errors where possible.
Dimensional
Library providing data types for performing arithmetic with physical quantities and units. Information about the physical dimensions of the quantities/units is embedded in their types and the validity of operations is verified by the type checker at compile time. The boxing and unboxing of numerical values as quantities is done by multiplication and division of units.

### 3.2 Number representations

#### 3.2.1 Decimal numbers

Decimal arithmetic library
An implementation of real decimal arithmetic, for cases where the binary floating point is not acceptable (for example, money).

#### 3.2.2 Real and rational numbers

There are several levels of handling real numbers and according libraries.

##### 3.2.2.1 Arbitrary precision
• Numbers have fixed precision
• Rounding errors accumulate
• Sharing is easy, i.e. in
sqrt pi + sin pi
,
pi
is computed only once
• Fast, because the routines can make use of the fast implementation of
Integer
operations
Numeric Quest
Jan Skibinski's Numeric Quest library provides, among other things, a type for arbitrary precision rational numbers with transcendental functions.
FixedPoint.hs
part of NumericPrelude project
AERN-Basics AERN-Real AERN-Real-Interval AERN-Real-Double
contains type classes that form a foundation for rounded arithmetic and interval arithmetic with explicit control of rounding and the possibility to increase the rounding precision arbitrarily for types that support it. At the moment there are instances for Double floating point numbers where one can control the direction of rounding but cannot increase the rounding precision. In the near future instances for MPFR arbitrary precision numbers will be provided. Intervals can use as endpoints any type that supports directed rounding in the numerical order (such as Double or MPFR) and operations on intervals are rounded either outwards or inwards. Outwards rounding allows to safely approximate exact real arithmetic while a combination of both outwards and inwards rounding allows one to safely approximate exact interval arithmetic. Inverted intervals with Kaucher arithmetic are also supported.
AERN-RnToRm
contains arithmetic of piecewise polynomial function intervals that approximate multi-dimensional (almost everywhere) continuous real functions to arbitrary precision
hmpfr
hmpfr is a purely functional haskell interface to the MPFR library
##### 3.2.2.2 Dynamic precision
• You tell the precision, an expression shall be computed to and the computer finds out, how precise to compute the input values.
• Rounding errors do not accumulate
• Sharing of temporary results is difficult, that is in
sqrt pi + sin pi
,
pi
will certainly be computed twice, each time with the required precision.
• Almost as fast as arbitrary precision computation
ERA is an implementation (in Haskell 1.2) by David Lester.
It is quite fast, possibly the fastest Haskell implementation. At 220 lines it is also the shortest. Probably the shortest implementation of exact real arithmetic in any language.
The provided number type
CReal
is instance of the Haskell 98 numeric type classes and thus can be used whereever you used Float or Double before and encountered some numerical difficulties.
Here is a mirror: http://darcs.augustsson.net/Darcs/CReal/
IC-Reals is an implementation by Abbas Edalat, Marko Krznarć and Peter J. Potts.
This implementation uses linear fractional transformations.
Few Digits by Russell O'Connor.
This is a prototype of the implementation he intendeds to write in Coq. Once the Coq implementation is complete, the Haskell code could be extracted producing an implementation that would be proved correct.
COMP is an implementation by Yann Kieffer.
The work is in beta and relies on new primitive operations on Integers which will be implemented in GHC. The library isn't available yet.
Hera is an implementation by Aleš Bizjak.
It uses the MPFR library to implement dyadic rationals, on top of which are implemented intervals and real numbers. A real number is represented as a function `int -> interval` which represents a sequence of intervals converging to the real.
##### 3.2.2.3 Dynamic precision by lazy evaluation

The real numbers are represented by an infinite datastructure, which allows you to increase precision successively by evaluating the data structure successively. All of the implementations below use some kind of digit stream as number representation. Sharing of results is simple. The implementations are either fast on simple expressions, because they use large blocks/bases, or they are fast on complex expressions, because they consume as little as possible input digits in order to emit the required output digits.

BigFloat is an implementation by Martin Guy.
It works with streams of decimal digits (strictly in the range from 0 to 9) and a separate sign. The produced digits are always correct. Output is postponed until the code is certain what the next digit is. This sometimes means that no more data is output.
In "The Most Unreliable Technique in the World to compute pi" Jerzy Karczmarczuk develops some functions for computing pi lazily.
NumericPrelude: positional numbers
Represents a real number as pair
(exponent,[digit])
, where the digits are
Int
s in the open range
(-basis,basis)
. There is no need for an extra sign item in the number data structure. The
basis
can range from
10
to
1000
. (Binary representations can be derived from the hexadecimal representation.) Showing the numbers in traditional format (non-negative digits) fails for fractions ending with a run of zeros. However the internal representation with negative digits can always be shown and is probably more useful for further processing. An interface for the numeric type hierarchy of the NumericPrelude project is provided.
It features
• basis conversion
• basic arithmetic: addition, subtraction, multiplication, division
• algebraic arithmetic: square root, other roots (no general polynomial roots)
• transcendental arithmetic: pi, exponential, logarithm, trigonometric and inverse trigonometric functions

### 3.3 Type class hierarchies

There are several approaches to improve the numeric type class hierarchy.

Dylan Thurston and Henning Thielemann's Numeric Prelude
Experimental revised framework for numeric type classes. Needs hiding of Prelude, overriding hidden functions like fromInteger and multi-parameter type classes. Probably restricted to GHC.
Jerzy Karczmarczuk's approach
Serge D. Mechveliani's Basic Algebra proposal
Andrew Frank's approach
The proposal: ftp://ftp.geoinfo.tuwien.ac.at/frank/numbersPrelude_v1.pdf
Ongoing efforts for the language revision

### 3.4 Discrete mathematics

Number Theory Library
Andrew Bromage's Haskell number theory library, providing operations on primes, fibonacci sequences and combinatorics.
HGAL
An haskell implementation of Brendan McKay's algorithm for graph canonic labeling and automorphism group. (aka Nauty)
Computational Oriented Matroids
is a book by Jürgen G. Bokowski, where he develops Haskell code for Matroid computations.

### 3.5 Computer Algebra

DoCon - Algebraic Domain Constructor
A library for Algebra, turns GHCi into a kind of Computer Algebra System
Papers by Jerzy Karczmarczuk
Some interesting uses of Haskell in mathematics, including functional differentiation, power series, continued fractions.

hstats

### 3.7 Plotting

Gnuplot
Simple wrapper to gnuplot
gnuplot wrapper as part of GSL Haskell package
Chart
A library for generating 2D Charts and Plots, based upon the cairo graphics library.
plot
A library for generating figures, based upon the cairo graphics libary with

probability
the module Numeric.Probability.Visualize contains a wrapper to R

### 3.8 Miscellaneous libraries

The HaskellMath library is a sandbox for experimenting with mathematics algorithms. So far I've implemented a few quantitative finance models (Black Scholes, Binomial Trees, etc) and basic linear algebra functions. Next I might work on either computer algebra or linear programming. All comments welcome!
David Amos' collection of math libraries in Haskell - including number theory, commutative algebra, combinatorics, permutation groups and more.
Various math stuff by Henning Thielemann
This is some unsorted mathematical stuff including: gnuplot wrapper (now maintained as separate package), portable grey map (PGM) image reader and writer, simplest numerical integration, differentiation, zero finding, interpolation, solution of differential equations, combinatorics, some solutions of math riddles, computation of fractal dimensions of iterated function systems (IFS)
Numeric Quest
Jan Skibinski wrote a collection of Haskell modules that are useful for Mathematics in general, and Quantum Mechanics in particular.
Some of the modules are hosted on haskell.org. They include modules for:
• Rational numbers with transcendental functions
• Roots of polynomials
• Eigensystems
• Tensors
• Dirac quantum mechanics
Other modules in Numeric Quest are currently only available via the Internet Archive. They include, among many other things:
Geometric Algorithms
A small Haskell library, containing algorithms for two-dimensional convex hulls, triangulations of polygons, Voronoi-diagrams and Delaunay-triangulations, the QEDS data structure, kd-trees and range-trees.
Hmm is a small Haskell library to parse and verify Metamath databases.
Probabilistic Functional Programming
The PFP library is a collection of modules for Haskell that facilitates probabilistic functional programming, that is, programming with stochastic values. The probabilistic functional programming approach is based on a data type for representing distributions. A distribution represent the outcome of a probabilistic event as a collection of all possible values, tagged with their likelihood. A nice aspect of this system is that simulations can be specified independently from their method of execution. That is, we can either fully simulate or randomize any simulation without altering the code which defines it.
Sinc function
Gamma and Beta function
Boolean
A general boolean algebra class and some instances for Haskell.
HODE
HODE is a binding to the Open Dynamics Engine. ODE is an open source, high performance library for simulating rigid body dynamics.
Ranged Sets
A ranged set is a list of non-overlapping ranges. The ranges have upper and lower boundaries, and a boundary divides the base type into values above and below. No value can ever sit on a boundary. So you can have the set $(2.0, 3.0] \cup (5.3, 6)$.
hhydra
Hhydra is a tool to compute Goodstein successions and hydra puzzles described by Bernard Hodgson in his article 'Herculean or Sisyphean tasks?' published in No 51 March 2004 of the Newsletter of the European Mathematical Society.

This page contains a list of libraries and tools in a certain category. For a comprehensive list of such pages, see Applications and libraries.