Difference between revisions of "Applications and libraries/Mathematics"

From HaskellWiki
Jump to navigation Jump to search
(Numeric Prelude needs some GHC extensions)
(84 intermediate revisions by 32 users not shown)
Line 1: Line 1:
  +
== Applications ==
−
{{unknown copyright}}
  
  +
−
{{LibrariesPage}}
  
  +
=== Physics ===
  +
  +
;[http://ab-initio.mit.edu/meep/ Meep]
  +
:Meep (or MEEP) is a free finite-difference time-domain (FDTD) simulation software package developed at MIT to model electromagnetic systems. Meep used Haskell to generate C++ code, after Meep [http://ab-initio.mit.edu/wiki/index.php/Meep_release_notes 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.
   
−
== Libraries for numerical algorithms and mathematics ==
 +
== Libraries ==
   
 
=== Linear algebra ===
  
=== Linear algebra ===
   
−
;[http://dis.um.es/~alberto/GSLHaskell GSLHaskell]
 +
;[http://hackage.haskell.org/package/bed-and-breakfast bed-and-breakfast]
  +
:A library that implements Matrix operations in pure Haskell using mutable arrays and the ST Monad. bed-and-breakfast does not need any additional software to be installed and can perform basic matrix operations like multiplication, finding the inverse, and calculating determinants efficiently.
−
:High level functional interface to standard linear algebra computations and other numerical algorithms based on the GNU Scientific Library.
  
   
  +
;[https://github.com/patperry/hs-linear-algebra hs-linear-algebra]
−
;[http://www.isi.edu/~hdaume/HBlas/ Wrapper to CLAPACK]
  
  +
:Patrick Perry's linear algebra library, built on BLAS. [https://github.com/cartazio/hs-cblas hs-cblas] seems to be a more up-to-date fork.
  +
  +
;[http://www.cs.utah.edu/~hal/HBlas/index.html Wrapper to CLAPACK]
   
 
;[http://haskelldsp.sourceforge.net/ Digital Signal Processing]
  
;[http://haskelldsp.sourceforge.net/ Digital Signal Processing]
−
:Modules for matrix manpulation, digital signal processing, spectral estimation, and frequency estimation.
 +
:Modules for matrix manipulation, Fourier transform, interpolation, spectral estimation, and frequency estimation.
   
 
;[http://article.gmane.org/gmane.comp.lang.haskell.general/13561 Index-aware linear algebra]
  
;[http://article.gmane.org/gmane.comp.lang.haskell.general/13561 Index-aware linear algebra]
 
:Frederik Eaton's library for statically checked matrix manipulation in Haskell
  
:Frederik Eaton's library for statically checked matrix manipulation in Haskell
   
  +
;[[Numeric Quest]]
−
;Indexless linear algebra algorithms by Jan Skibinski, see below
  
  +
: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).
  +
  +
;[http://hackage.haskell.org/cgi-bin/hackage-scripts/package/hmatrix HMatrix]
  +
:By Alberto Ruiz. From the project [http://perception.inf.um.es/hmatrix/ 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.
  +
  +
;[http://hackage.haskell.org/cgi-bin/hackage-scripts/package/Vec 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.''
  +
  +
== See also ==
  +
  +
* [[Linear algebra]]
  +
* [[Mathematical prelude discussion]]
  +
  +
  +
See also: [[Linear algebra|Design discussions]]
  +
  +
=== [[Physical units]] ===
  +
  +
;[[Dimensionalized numbers]]
  +
: Working with physical units like second, meter and so on in a type-safe manner.
  +
  +
;[http://darcs.haskell.org/numericprelude/src/Number/SI.hs NumericPrelude: Physical units]
  +
: Numeric values with dynamically checked units.
  +
  +
;[[CalDims]]
  +
:This is not simply a library providing a new type of <hask>Num</hask> 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.
  +
  +
;[http://code.google.com/p/dimensional/ 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.
   
 
=== Number representations ===
  
=== Number representations ===
   
  +
==== Decimal numbers ====
−
;[http://www.n-heptane.com/nhlab/repos/Decimal Decimal arithmetic library]
  
  +
  +
;[http://src.seereason.com/decimal/ Decimal arithmetic library]
 
:An implementation of real decimal arithmetic, for cases where the binary floating point is not acceptable (for example, money).
  
:An implementation of real decimal arithmetic, for cases where the binary floating point is not acceptable (for example, money).
   
  +
==== Real and rational numbers ====
−
;[[Exact real arithmetic]]
  
−
:is an interesting area: it is a deep connection between numeric methods and deep theoretic fondations of algorithms (and mathematics). Its topic: ''computable real number''s raise a lot of interesting questions rooted in mathematical analysis, arithmetic, but also [http://en.wikipedia.org/wiki/Computability_theory computability theory] (see numbers-as-programs approaches). Computable reals can be achieved by many approaches -- it is not one single theory.
  
   
  +
There are several levels of [[Exact real arithmetic|handling real numbers]] and according libraries.
   
−
=== other ===
 +
===== Arbitrary precision =====
   
  +
* Numbers have fixed precision
−
;[http://www.dinkla.net/fp/cglib.html Geometric Algorithms]
  
  +
* Rounding errors accumulate
−
: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.
  
  +
* Sharing is easy, i.e. in <hask>sqrt pi + sin pi</hask>, <hask>pi</hask> is computed only once
  +
* Fast, because the routines can make use of the fast implementation of <hask>Integer</hask> operations
   
  +
;[[Numeric Quest]]
−
;[http://www.info.unicaen.fr/~karczma/arpap/ Papers by Jerzy Karczmarczuk]
  
  +
:Jan Skibinski's [[Numeric Quest]] library provides, among other things, a type for arbitrary precision rational numbers with transcendental functions.
−
:Some interesting uses of Haskell in mathematics, including functional differentiation, power series, continued fractions.
  
  +
  +
;[http://cvs.haskell.org/darcs/numericprelude/src/Number/FixedPoint.hs FixedPoint.hs]
  +
:part of NumericPrelude project
  +
  +
;[http://hackage.haskell.org/cgi-bin/hackage-scripts/package/AERN-Basics AERN-Basics] [http://hackage.haskell.org/cgi-bin/hackage-scripts/package/AERN-Real AERN-Real] [http://hackage.haskell.org/cgi-bin/hackage-scripts/package/AERN-Real-Interval AERN-Real-Interval] [http://hackage.haskell.org/cgi-bin/hackage-scripts/package/AERN-Real-Double 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.
  +
  +
;[http://hackage.haskell.org/cgi-bin/hackage-scripts/package/AERN-RnToRm AERN-RnToRm]
  +
:contains arithmetic of ''piecewise polynomial function intervals'' that approximate multi-dimensional (almost everywhere) continuous real functions to arbitrary precision
  +
  +
;[http://hackage.haskell.org/cgi-bin/hackage-scripts/package/hmpfr hmpfr]
  +
:hmpfr is a purely functional haskell interface to the [http://www.mpfr.org/ MPFR] library
  +
  +
;[http://hackage.haskell.org/package/numbers numbers]
  +
:provides an up-to-date, easy-to-use BigFloat implementation that builds with a modern GHC, among other things.
  +
  +
===== 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 <hask>sqrt pi + sin pi</hask>, <hask>pi</hask> ''will'' be computed twice, each time with the required precision.
  +
* Almost as fast as arbitrary precision computation
  +
  +
;[http://www.cs.man.ac.uk/arch/dlester/exact.html 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 <hask>CReal</hask> 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/
  +
  +
;[http://www.doc.ic.ac.uk/~ae/exact-computation/#bm:implementations IC-Reals] is an implementation by Abbas Edalat, Marko Krznar&#263; and Peter J. Potts.
  +
:This implementation uses linear fractional transformations.
  +
  +
;[http://r6.ca/ Few Digits] by Russell O'Connor.
  +
:This is a prototype of the implementation he intendeds to write in [http://coq.inria.fr/ Coq]. Once the Coq implementation is complete, the Haskell code could be extracted producing an implementation that would be proved correct.
  +
<!--
  +
Example:
  +
*Data.Real.CReal> answer 1000 (exp 1 + sqrt 2)
  +
-->
  +
  +
;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.
  +
  +
;[http://www2.arnes.si/~abizja4/hera/ Hera] is an implementation by Aleš Bizjak.
  +
:It uses the [http://www.mpfr.org/ MPFR] library to implement dyadic rationals, on top of which are implemented intervals and real numbers. A real number is represented as a function <hask>Int -> Interval</hask> which represents a sequence of intervals converging to the real.
  +
  +
===== 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.
  +
  +
;[http://medialab.freaknet.org/bignum/ 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 [http://medialab.freaknet.org/bignum/dudeney.html no more data is output].
  +
  +
;In [http://users.info.unicaen.fr/~karczma/arpap/lazypi.ps.gz "The Most Unreliable Technique in the World to compute pi"] Jerzy Karczmarczuk develops some functions for computing pi lazily.
  +
  +
;[http://darcs.haskell.org/numericprelude/src/Number/Positional.hs NumericPrelude: positional numbers]
  +
:Represents a real number as pair <hask>(exponent,[digit])</hask>, where the digits are <hask>Int</hask>s in the open range <hask>(-basis,basis)</hask>. There is no need for an extra sign item in the number data structure. The <hask>basis</hask> can range from <hask>10</hask> to <hask>1000</hask>. (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
  +
  +
=== Type class hierarchies ===
  +
  +
There are several approaches to improve the [[Mathematical prelude discussion|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 [http://www.haskell.org/pipermail/haskell-cafe/2001-February/001510.html approach]
  +
  +
;Serge D. Mechveliani's [ftp://ftp.botik.ru/pub/local/Mechveliani/basAlgPropos/ Basic Algebra proposal]
  +
  +
;Andrew Frank's [http://www.haskell.org/pipermail/haskell-cafe/2006-April/015326.html approach]
  +
:The proposal: ftp://ftp.geoinfo.tuwien.ac.at/frank/numbersPrelude_v1.pdf
  +
  +
;Haskell Prime: [http://hackage.haskell.org/trac/haskell-prime/ticket/112 Ongoing efforts for the language revision]
  +
  +
=== Discrete mathematics ===
  +
  +
;[http://andrew.bromage.org/darcs/numbertheory/ Number Theory Library]
  +
:Andrew Bromage's Haskell number theory library, providing operations on primes, fibonacci sequences and combinatorics.
  +
  +
;[http://users.skynet.be/jyp/HGAL/ HGAL]
  +
:An haskell implementation of Brendan McKay's algorithm for graph canonic labeling and automorphism group. (aka Nauty)
  +
  +
;[http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=0521849306 Computational Oriented Matroids]
  +
:is a book by [http://wwwopt.mathematik.tu-darmstadt.de/~bokowski/ Jürgen G. Bokowski], where he develops Haskell code for Matroid computations.
  +
  +
See also [[Libraries and tools/Cryptography]]
  +
  +
=== Computer Algebra ===
   
 
;[http://haskell.org/docon/ DoCon] - Algebraic Domain Constructor
  
;[http://haskell.org/docon/ DoCon] - Algebraic Domain Constructor
−
:A Computer Algebra System
 +
:A library for Algebra, turns GHCi into a kind of Computer Algebra System
  +
  +
;[http://www.info.unicaen.fr/~karczma/arpap/ Papers by Jerzy Karczmarczuk]
  +
:Some interesting uses of Haskell in mathematics, including [[functional differentiation]], power series, continued fractions.
  +
  +
;[http://www.robtougher.com/HCAS/ HCAS] by Rob Tougher.
  +
  +
=== Statistics ===
  +
;[http://hackage.haskell.org/package/hstats hstats]
  +
: Statistical Computing with Haskell
  +
  +
;[http://hackage.haskell.org/cgi-bin/hackage-scripts/package/hmatrix-gsl-stats hmatrix-gsl-stats]
  +
: A binding to the statistics portion of GSL. Works with hmatrix
  +
  +
;[http://hackage.haskell.org/cgi-bin/hackage-scripts/package/hstatistics hstatistics]
  +
: A library for doing statistics. Works with hmatrix
  +
  +
=== Plotting ===
  +
  +
;[http://hackage.haskell.org/package/easyplot easyplot]
  +
: Simple and easy wrapper to gnuplot.
  +
  +
;[[Gnuplot]]
  +
: Simple wrapper to gnuplot
  +
  +
;[http://hackage.haskell.org/packages/archive/hmatrix/latest/doc/html/Graphics-Plot.html hmatrix]
  +
: contains a deprecated gnuplot wrapper
  +
  +
;[http://hackage.haskell.org/cgi-bin/hackage-scripts/package/Chart Chart]
  +
: A library for generating 2D Charts and Plots, based upon the cairo graphics library.
  +
  +
;[http://hackage.haskell.org/cgi-bin/hackage-scripts/package/plot plot]
  +
: A library for generating figures, based upon the cairo graphics libary with
  +
a simple, monadic interface.
  +
  +
;[http://hackage.haskell.org/cgi-bin/hackage-scripts/package/probability probability]
  +
: the module Numeric.Probability.Visualize contains a wrapper to [http://www.r-project.org/ R]
  +
  +
=== Numerical optimization ===
  +
This classification is somewhat arbitrary. Something more systematic like GAMS might be helpful.
  +
  +
==== bindings ====
  +
;[http://repetae.net/john/recent/out/HsASA.html Adaptive Simulated Annealing]
  +
:A Haskell interface to Lester Ingber's adaptive simulating annealing code.
  +
  +
;[http://hackage.haskell.org/package/cmaes CMA-ES]
  +
:A wrapper for Covariance Matrix Adaptation Evolution Strategy
  +
  +
;[https://github.com/ghorn/nlopt-haskell nlopt-haskell]
  +
:A low-level binding to the nlopt library
  +
  +
;[http://hackage.haskell.org/package/ipopt-hs ipopt-hs]
  +
:A haskell binding to ipopt including automatic differentiation
  +
  +
;[http://hackage.haskell.org/package/glpk-hs glpk-hs]
  +
:A high-level interface to GLPK's linear programming and mixed integer programming features.
  +
  +
==== pure haskell ====
  +
;[http://hackage.haskell.org/package/nonlinear-optimization nonlinear-optimization]
  +
:A pure-haskell CG_DESCENT method is implemented
  +
  +
=== Miscellaneous libraries ===
   
 
;[http://www.robtougher.com/HaskellMath/ HaskellMath]
  
;[http://www.robtougher.com/HaskellMath/ 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!
  
: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!
  +
  +
;[http://hackage.haskell.org/package/HaskellForMaths HaskellForMaths]
  +
:David Amos' library for combinatorics, group theory, commutative algebra and non-commutative algebra, which is described in an [http://haskellformaths.blogspot.com/ accompanying blog].
   
 
;[http://darcs.haskell.org/htam/ Various math stuff by Henning Thielemann]
  
;[http://darcs.haskell.org/htam/ Various math stuff by Henning Thielemann]
−
:This is some unsorted mathematical stuff including: GNUPlot wrapper, portable grey map (PGM) image reader and writer, simplest numerical integration, differentiation, interpolation, solution of differential equations, combinatorics, some solutions of math riddles, computation of fractal dimensions of iterated function systems (IFS)
 +
: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]]
−
;[http://cvs.haskell.org/darcs/numericprelude/ Numeric Prelude]
  
  +
:Jan Skibinski wrote a collection of Haskell modules that are useful for Mathematics in general, and Quantum Mechanics in particular.
−
: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.
  
   
  +
:Some of the modules are hosted on [http://darcs.haskell.org/numeric-quest/ haskell.org]. They include modules for:
−
;[http://repetae.net/john/recent/out/HsASA.html Adaptive Simulated Annealing]
  
  +
:* Rational numbers with transcendental functions
−
:A Haskell interface to Lester Ingber's adaptive simulating annealing code.
  
  +
:* Roots of polynomials
  +
:* Eigensystems
  +
:* Tensors
  +
:* Dirac quantum mechanics
   
  +
:Other modules in Numeric Quest are currently only available via the [http://web.archive.org/web/20010605003250/http://www.numeric-quest.com/haskell/ Internet Archive]. They include, among many other things:
−
;[http://andrew.bromage.org/darcs/numbertheory/ Number Theory Library]
  
  +
:* [http://web.archive.org/web/*/http://www.numeric-quest.com/haskell/ State vector evolution]
−
:Andrew Bromage's Haskell number theory library, providing operations on primes, fibonacci sequences and combinatorics.
  
  +
:* [http://web.archive.org/web/*/http://www.numeric-quest.com/haskell/ Short study of fuzzy oscillator]
   
  +
:See the [[Numeric Quest]] page for more information.
−
;[http://www.solcon.nl/mklooster/repos/hmm/ Hmm: Haskell Metamath]
  
  +
  +
;[http://www.dinkla.net/fp/cglib.html 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.
  +
  +
;[http://home.solcon.nl/mklooster/repos/hmm/ Hmm: Haskell Metamath]
 
:Hmm is a small Haskell library to parse and verify Metamath databases.
  
:Hmm is a small Haskell library to parse and verify Metamath databases.
   
−
;[http://eecs.oregonstate.edu/~erwig/pfp/ Probabilistic Functional Programming]
 +
;[[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.
  
: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]]
  
;[[Sinc function]]
  +
  +
;[[Gamma and Beta function]]
   
 
;[http://repetae.net/john/recent/out/Boolean.html Boolean]
  
;[http://repetae.net/john/recent/out/Boolean.html Boolean]
Line 69: Line 287:
 
:HODE is a binding to the Open Dynamics Engine. ODE is an open source, high performance library for simulating rigid body dynamics.
  
:HODE is a binding to the Open Dynamics Engine. ODE is an open source, high performance library for simulating rigid body dynamics.
   
  +
;[http://sourceforge.net/projects/ranged-sets Ranged Sets]
−
;[http://cvs.haskell.org/darcs/numeric-quest/ Mirror of the following numeric modules by Jan Skibinski]
  
  +
: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 <math>(2.0, 3.0] \cup (5.3, 6)</math>.
−
:
  
  +
−
*[http://web.archive.org/web/*/http://www.numeric-quest.com/haskell/fractions.html Numerics with fractions]<em>(via Internet Archive since 10/06/2003).</em>
  
  +
;[http://code.google.com/p/hhydra/ hhydra]
−
*[http://web.archive.org/web/*/http://www.numeric-quest.com/haskell/Roots.html Roots of polynomials]<em>(via Internet Archive since 10/06/2003).</em> It implements the well known Laguerre's method for finding complex roots of polynomials.
  
  +
: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.
−
*[http://web.archive.org/web/*/http://www.numeric-quest.com/haskell/Orthogonals.html Indexless linear algebra algorithms]<em>(via Internet Archive since 10/06/2003).</em> Orthogonalization, solution of linear equations, eigenvalues and eigenvectors.
  
  +
−
* [http://web.archive.org/web/*/http://www.numeric-quest.com/haskell/ State vector evolution]<em>(via Internet Archive since 10/06/2003)</em>
  
  +
[[Category:Mathematics|*]]
−
* [http://web.archive.org/web/*/http://www.numeric-quest.com/haskell/ Short study of fuzzy oscillator]<em>(via Internet Archive since 10/06/2003)</em>
  
  +
{{LibrariesPage}}
−
* [http://web.archive.org/web/*/http://www.numeric-quest.com/haskell/Tensor.html N-dimensional tensors]<em>(via Internet Archive since 10/06/2003)</em>
  

Revision as of 20:26, 24 March 2014

Applications

Physics

Meep
Meep (or MEEP) is a free finite-difference time-domain (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.

Libraries

Linear algebra

bed-and-breakfast
A library that implements Matrix operations in pure Haskell using mutable arrays and the ST Monad. bed-and-breakfast does not need any additional software to be installed and can perform basic matrix operations like multiplication, finding the inverse, and calculating determinants efficiently.
hs-linear-algebra
Patrick Perry's linear algebra library, built on BLAS. hs-cblas seems to be a more up-to-date fork.
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:
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 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.

See also


See also: Design discussions

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.

Number representations

Decimal numbers

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

Real and rational numbers

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

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
numbers
provides an up-to-date, easy-to-use BigFloat implementation that builds with a modern GHC, among other things.
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 pi, pi will 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.
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 Ints 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

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
Haskell Prime
Ongoing efforts for the language revision

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

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.

Statistics

hstats
Statistical Computing with Haskell
hmatrix-gsl-stats
A binding to the statistics portion of GSL. Works with hmatrix
hstatistics
A library for doing statistics. Works with hmatrix

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

Numerical optimization

This classification is somewhat arbitrary. Something more systematic like GAMS might be helpful.

bindings

Adaptive Simulated Annealing
A Haskell interface to Lester Ingber's adaptive simulating annealing code.
CMA-ES
A wrapper for Covariance Matrix Adaptation Evolution Strategy
nlopt-haskell
A low-level binding to the nlopt library
ipopt-hs
A haskell binding to ipopt including automatic differentiation
glpk-hs
A high-level interface to GLPK's linear programming and mixed integer programming features.

pure haskell

nonlinear-optimization
A pure-haskell CG_DESCENT method is implemented

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 non-commutative 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 two-dimensional convex hulls, triangulations of polygons, Voronoi-diagrams and Delaunay-triangulations, the QEDS data structure, kd-trees and range-trees.
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.
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 .
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.

Retrieved from "https://wiki.haskell.org/index.php?title=Applications_and_libraries/Mathematics&oldid=57703"