Applications and libraries/Mathematics
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Contents 
1 Applications
1.1 Physics
 Meep
 Meep (or MEEP) is a free finitedifference timedomain (FDTD) simulation software package developed at MIT to model electromagnetic systems. Meep used Haskell to generate C++ code, after Meep 1.0 Haskell generation droped in favor of handwritten C++ code.
 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
 bedandbreakfast
 A library that implements Matrix operations in pure Haskell using mutable arrays and the ST Monad. bedandbreakfast does not need any additional software to be installed and can perform basic matrix operations like multiplication, finding the inverse, and calculating determinants efficiently.
 hslinearalgebra
 Patrick Perry's linear algebra library, built on BLAS. hscblas seems to be a more uptodate fork.
 Digital Signal Processing
 Modules for matrix manipulation, Fourier transform, interpolation, spectral estimation, and frequency estimation.
 Indexaware 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.
 vectorspace
 The vectorspace 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:
 A purely functional interface to linear algebra and other numerical algorithms, internally implemented using LAPACK, BLAS, and GSL.
 This package includes standard matrix decompositions (eigensystems, singular values, Cholesky, QR, etc.), linear systems, numeric integration, root finding, etc.
 Vec
 By Scott E. Dillard. Static dimension checking:
 Vectors are represented by lists with typeencoded 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 listlike 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 See also
See also: Design discussions
3.1 Physical units
 Dimensionalized numbers
 Working with physical units like second, meter and so on in a typesafe manner.
 NumericPrelude: Physical units
 Numeric values with dynamically checked units.
 CalDims
 This is not simply a library providing a new type of class, but standalone calculation tool that supports user defined functions and units (basic and derived), so it can provide dimensionsafe calculation (not embedded but via shell). Calculations can be modified/saved via shell. It uses rational numbers to avoid rounding errors where possible.Num
 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 piis computed only oncepi
 Fast, because the routines can make use of the fast implementation of operationsInteger
 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
 AERNBasics AERNReal AERNRealInterval AERNRealDouble
 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.
 AERNRnToRm
 contains arithmetic of piecewise polynomial function intervals that approximate multidimensional (almost everywhere) continuous real functions to arbitrary precision
 numbers
 provides an uptodate, easytouse BigFloat implementation that builds with a modern GHC, among other things.
3.2.2.2 Dynamic precision
 You tell the precision and an expression shall be computed to, and the computer finds out, how precisely to compute the input values
 Rounding errors do not accumulate
 Sharing of temporary results is difficult, that is, in ,sqrt pi + sin piwill be computed twice, each time with the required precision.pi
 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 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.CReal
 Here is a mirror: http://darcs.augustsson.net/Darcs/CReal/
 ICReals 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 which represents a sequence of intervals converging to the real.Int > Interval
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 , where the digits are(exponent,[digit])s in the open rangeInt. There is no need for an extra sign item in the number data structure. The(basis,basis)can range frombasisto10. (Binary representations can be derived from the hexadecimal representation.) Showing the numbers in traditional format (nonnegative 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.1000
 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 multiparameter 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
 Haskell Prime
 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.
See also Libraries and tools/Cryptography
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.
 HCAS by Rob Tougher.
3.6 Statistics
 hstats
 Statistical Computing with Haskell
 hmatrixgslstats
 A binding to the statistics portion of GSL. Works with hmatrix
 hstatistics
 A library for doing statistics. Works with hmatrix
3.7 Plotting
 easyplot
 Simple and easy wrapper to gnuplot.
 Gnuplot
 Simple wrapper to gnuplot
 hmatrix
 contains a deprecated gnuplot wrapper
 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
a simple, monadic interface.
 probability
 the module Numeric.Probability.Visualize contains a wrapper to R
3.8 Numerical optimization
This classification is somewhat arbitrary. Something more systematic like GAMS might be helpful.
3.8.1 bindings
 Adaptive Simulated Annealing
 A Haskell interface to Lester Ingber's adaptive simulating annealing code.
 CMAES
 A wrapper for Covariance Matrix Adaptation Evolution Strategy
 nlopthaskell
 A lowlevel binding to the nlopt library
 ipopths
 A haskell binding to ipopt including automatic differentiation
 glpkhs
 A highlevel interface to GLPK's linear programming and mixed integer programming features.
3.8.2 pure haskell
 nonlinearoptimization
 A purehaskell CG_DESCENT method is implemented
3.9 Miscellaneous libraries
 HaskellMath
 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!
 HaskellForMaths
 David Amos' library for combinatorics, group theory, commutative algebra and noncommutative algebra, which is described in an accompanying blog.
 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:
 See the Numeric Quest page for more information.
 Geometric Algorithms
 A small Haskell library, containing algorithms for twodimensional convex hulls, triangulations of polygons, Voronoidiagrams and Delaunaytriangulations, the QEDS data structure, kdtrees and rangetrees.
 Hmm: Haskell Metamath
 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.
 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 nonoverlapping 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 .
 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.