diff --git a/CHANGELOG.md b/CHANGELOG.md --- a/CHANGELOG.md +++ b/CHANGELOG.md @@ -1,27 +1,34 @@ -# Changelog for `hspray` - -All notable changes to this project will be documented in this file. - -The format is based on [Keep a Changelog](https://keepachangelog.com/en/1.0.0/), -and this project adheres to the -[Haskell Package Versioning Policy](https://pvp.haskell.org/). - - -## 0.1.0.0 - 2022-12-11 - -First release. - - -## 0.1.1.0 - 2022-12-12 - -* New functions `toList`, `sprayTerms` and `bombieriSpray`. - -* New operation `.^`, to multiply a spray by an integer. - -* Added some unit tests. - - -## 0.1.2.0 - 2023-02-24 - -New function `derivSpray`, to differentiate a spray. - +# Changelog for `hspray` + +All notable changes to this project will be documented in this file. + +The format is based on [Keep a Changelog](https://keepachangelog.com/en/1.0.0/), +and this project adheres to the +[Haskell Package Versioning Policy](https://pvp.haskell.org/). + + +## 0.1.0.0 - 2022-12-11 + +First release. + + +## 0.1.1.0 - 2022-12-12 + +* New functions `toList`, `sprayTerms` and `bombieriSpray`. + +* New operation `.^`, to multiply a spray by an integer. + +* Added some unit tests. + + +## 0.1.2.0 - 2023-02-24 + +New function `derivSpray`, to differentiate a spray. + + +## 0.1.3.0 - 2023-08-29 + +* `Powers(..)` is now exported. + +* Completed the README to show how to deal with symbolic coefficients. + diff --git a/LICENSE b/LICENSE --- a/LICENSE +++ b/LICENSE @@ -1,674 +1,674 @@ - GNU GENERAL PUBLIC LICENSE - Version 3, 29 June 2007 - - Copyright (C) 2007 Free Software Foundation, Inc. - Everyone is permitted to copy and distribute verbatim copies - of this license document, but changing it is not allowed. - - Preamble - - The GNU General Public License is a free, copyleft license for -software and other kinds of works. - - The licenses for most software and other practical works are designed -to take away your freedom to share and change the works. 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If not, see . + +Also add information on how to contact you by electronic and paper mail. + + If the program does terminal interaction, make it output a short +notice like this when it starts in an interactive mode: + + Copyright (C) + This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. + This is free software, and you are welcome to redistribute it + under certain conditions; type `show c' for details. + +The hypothetical commands `show w' and `show c' should show the appropriate +parts of the General Public License. Of course, your program's commands +might be different; for a GUI interface, you would use an "about box". + + You should also get your employer (if you work as a programmer) or school, +if any, to sign a "copyright disclaimer" for the program, if necessary. +For more information on this, and how to apply and follow the GNU GPL, see +. + + The GNU General Public License does not permit incorporating your program +into proprietary programs. If your program is a subroutine library, you +may consider it more useful to permit linking proprietary applications with +the library. If this is what you want to do, use the GNU Lesser General +Public License instead of this License. But first, please read +. diff --git a/README.md b/README.md --- a/README.md +++ b/README.md @@ -1,74 +1,130 @@ -# hspray - - -[![Stack-lts](https://github.com/stla/hspray/actions/workflows/Stack-lts.yml/badge.svg)](https://github.com/stla/hspray/actions/workflows/Stack-lts.yml) -[![Stack-nightly](https://github.com/stla/hspray/actions/workflows/Stack-nightly.yml/badge.svg)](https://github.com/stla/hspray/actions/workflows/Stack-nightly.yml) - - -Simple multivariate polynomials in Haskell. - -___ - - -```haskell -import Math.Algebra.Hspray -x = lone 1 :: Spray Double -y = lone 2 :: Spray Double -z = lone 3 :: Spray Double -poly = (2 *^ (x^**^3 ^*^ y ^*^ z) ^+^ x^**^2) ^*^ (4 *^ (x ^*^ y ^*^ z)) -prettySpray show "X" poly --- "(4.0) * X^(3, 1, 1) + (8.0) * X^(4, 2, 2)" -``` - -More generally, one can use the type `Spray a` as long as the type `a` has -the instances `Eq` and `Algebra.Ring` (defined in the **numeric-prelude** -library). For example `a = Rational`: - -```haskell -import Math.Algebra.Hspray -import Data.Ratio -x = lone 1 :: Spray Rational -y = lone 2 :: Spray Rational -z = lone 3 :: Spray Rational -poly = ((2%3) *^ (x^**^3 ^*^ y ^*^ z) ^+^ x^**^2) ^*^ ((7%4) *^ (x ^*^ y ^*^ z)) -prettySpray show "X" poly --- "(7 % 4) * X^(3, 1, 1) + (7 % 6) * X^(4, 2, 2)" -``` - -Or `a = Spray Double`: - -```haskell -import Math.Algebra.Hspray -p = lone 1 :: Spray Double -x = lone 1 :: Spray (Spray Double) -y = lone 2 :: Spray (Spray Double) -poly = ((p *^ x) ^+^ (p *^ y))^**^2 -prettySpray (prettySpray show "a") "X" poly --- "((1.0) * a^(2)) * X^(0, 2) + ((2.0) * a^(2)) * X^(1, 1) + ((1.0) * a^(2)) * X^(2)" -``` - -Evaluation: - -```haskell -import Math.Algebra.Hspray -x = lone 1 :: Spray Double -y = lone 2 :: Spray Double -z = lone 3 :: Spray Double -poly = 2 *^ (x ^*^ y ^*^ z) --- evaluate poly at x=2, y=1, z=2 -evalSpray poly [2, 1, 2] --- 8.0 -``` - -Differentiation: - -```haskell -import Math.Algebra.Hspray -x = lone 1 :: Spray Double -y = lone 2 :: Spray Double -z = lone 3 :: Spray Double -poly = 2 *^ (x ^*^ y ^*^ z) ^+^ (3 *^ x^**^2) --- derivate with respect to x -prettySpray show "X" $ derivSpray 1 poly --- "(2.0) * X^(0, 1, 1) + (6.0) * X^(1)" -``` +# hspray + + +[![Stack-lts](https://github.com/stla/hspray/actions/workflows/Stack-lts.yml/badge.svg)](https://github.com/stla/hspray/actions/workflows/Stack-lts.yml) +[![Stack-nightly](https://github.com/stla/hspray/actions/workflows/Stack-nightly.yml/badge.svg)](https://github.com/stla/hspray/actions/workflows/Stack-nightly.yml) + + +Simple multivariate polynomials in Haskell. + +___ + + +```haskell +import Math.Algebra.Hspray +x = lone 1 :: Spray Double +y = lone 2 :: Spray Double +z = lone 3 :: Spray Double +poly = (2 *^ (x^**^3 ^*^ y ^*^ z) ^+^ x^**^2) ^*^ (4 *^ (x ^*^ y ^*^ z)) +prettySpray show "X" poly +-- "(4.0) * X^(3, 1, 1) + (8.0) * X^(4, 2, 2)" +``` + +More generally, one can use the type `Spray a` as long as the type `a` has +the instances `Eq` and `Algebra.Ring` (defined in the **numeric-prelude** +library). For example `a = Rational`: + +```haskell +import Math.Algebra.Hspray +import Data.Ratio +x = lone 1 :: Spray Rational +y = lone 2 :: Spray Rational +z = lone 3 :: Spray Rational +poly = ((2%3) *^ (x^**^3 ^*^ y ^*^ z) ^+^ x^**^2) ^*^ ((7%4) *^ (x ^*^ y ^*^ z)) +prettySpray show "X" poly +-- "(7 % 4) * X^(3, 1, 1) + (7 % 6) * X^(4, 2, 2)" +``` + +Or `a = Spray Double`: + +```haskell +import Math.Algebra.Hspray +p = lone 1 :: Spray Double +x = lone 1 :: Spray (Spray Double) +y = lone 2 :: Spray (Spray Double) +poly = ((p *^ x) ^+^ (p *^ y))^**^2 +prettySpray (prettySpray show "a") "X" poly +-- "((1.0) * a^(2)) * X^(0, 2) + ((2.0) * a^(2)) * X^(1, 1) + ((1.0) * a^(2)) * X^(2)" +``` + +Evaluation: + +```haskell +import Math.Algebra.Hspray +x = lone 1 :: Spray Double +y = lone 2 :: Spray Double +z = lone 3 :: Spray Double +poly = 2 *^ (x ^*^ y ^*^ z) +-- evaluate poly at x=2, y=1, z=2 +evalSpray poly [2, 1, 2] +-- 8.0 +``` + +Differentiation: + +```haskell +import Math.Algebra.Hspray +x = lone 1 :: Spray Double +y = lone 2 :: Spray Double +z = lone 3 :: Spray Double +poly = 2 *^ (x ^*^ y ^*^ z) ^+^ (3 *^ x^**^2) +-- derivate with respect to x +prettySpray show "X" $ derivSpray 1 poly +-- "(2.0) * X^(0, 1, 1) + (6.0) * X^(1)" +``` + +## Easier usage + +To construct a polynomial using the ordinary symbols `+`, `*` and `-`, +one can hide these operators from **Prelude** and import them from +the **numeric-prelude** library: + +```haskell +import Prelude hiding ((*), (+), (-)) +import qualified Prelude as P +import Algebra.Additive +import Algebra.Module +import Algebra.Ring +import Math.Algebra.Hspray +``` + +Or, maybe better (I didn't try yet), follow the "Usage" section on the +[Hackage page](https://hackage.haskell.org/package/numeric-prelude-0.4.4#usage) +of **numeric-prelude**. + +## Symbolic coefficients + +Assume you have the polynomial `a * (x² + y²) + 2b/3 * z`, +where `a` and `b` are symbolic coefficients. +You can define this polynomial as a `Spray` as follows: + +```haskell +import Prelude hiding ((*), (+), (-)) +import qualified Prelude as P +import Algebra.Additive +import Algebra.Module +import Algebra.Ring +import Math.Algebra.Hspray +import Data.Ratio + +x = lone 1 :: Spray (Spray Rational) +y = lone 2 :: Spray (Spray Rational) +z = lone 3 :: Spray (Spray Rational) +a = lone 1 :: Spray Rational +b = lone 2 :: Spray Rational + +poly = a *^ (x*x + y*y) + ((2%3) *^ b) *^ z +prettySpray (prettySpray show "a") "X" poly +-- "((2 % 3) * a^(0, 1)) * X^(0, 0, 1) + ((1 % 1) * a^(1)) * X^(0, 2) + ((1 % 1) * a^(1)) * X^(2)" +``` + +The `prettySpray` function shows the expansion of the polynomial. +You can extract the powers and the coefficients as follows: + +```haskell +l = toList poly +map fst l +-- [[0,0,1],[2],[0,2]] +map toList $ map snd l +-- [[([0,1],2 % 3)],[([1],1 % 1)],[([1],1 % 1)]] +``` diff --git a/Setup.hs b/Setup.hs --- a/Setup.hs +++ b/Setup.hs @@ -1,2 +1,2 @@ -import Distribution.Simple -main = defaultMain +import Distribution.Simple +main = defaultMain diff --git a/hspray.cabal b/hspray.cabal --- a/hspray.cabal +++ b/hspray.cabal @@ -1,52 +1,53 @@ -name: hspray -version: 0.1.2.0 -synopsis: Multivariate polynomials. -description: Manipulation of multivariate polynomials on a ring. -homepage: https://github.com/stla/hspray#readme -license: GPL-3 -license-file: LICENSE -author: Stéphane Laurent -maintainer: laurent_step@outlook.fr -copyright: 2022 Stéphane Laurent -category: Math, Algebra -build-type: Simple -extra-source-files: README.md - CHANGELOG.md -cabal-version: >=1.10 - -library - hs-source-dirs: src - exposed-modules: Math.Algebra.Hspray - build-depends: base >= 4.7 && < 5 - , containers >= 0.6.4.1 - , hashable >= 1.3.4.0 - , unordered-containers >= 0.2.17.0 - , numeric-prelude >= 0.4.4 - , text >= 1.2.5.0 - other-extensions: FlexibleInstances - , FlexibleContexts - , MultiParamTypeClasses - default-language: Haskell2010 - ghc-options: -Wall - -Wcompat - -Widentities - -Wincomplete-record-updates - -Wincomplete-uni-patterns - -Wmissing-export-lists - -Wmissing-home-modules - -Wpartial-fields - -Wredundant-constraints - -test-suite unit-tests - type: exitcode-stdio-1.0 - main-is: Main.hs - hs-source-dirs: tests/ - Build-Depends: base >= 4.7 && < 5 - , tasty - , tasty-hunit - , hspray - Default-Language: Haskell2010 - -source-repository head - type: git - location: https://github.com/stla/hspray +name: hspray +version: 0.1.3.0 +synopsis: Multivariate polynomials. +description: Manipulation of multivariate polynomials on a ring. +homepage: https://github.com/stla/hspray#readme +license: GPL-3 +license-file: LICENSE +author: Stéphane Laurent +maintainer: laurent_step@outlook.fr +copyright: 2022 Stéphane Laurent +category: Math, Algebra +build-type: Simple +extra-source-files: README.md + CHANGELOG.md +cabal-version: >=1.10 + +library + hs-source-dirs: src + exposed-modules: Math.Algebra.Hspray + build-depends: base >= 4.7 && < 5 + , containers >= 0.6.4.1 + , hashable >= 1.3.4.0 + , unordered-containers >= 0.2.17.0 + , numeric-prelude >= 0.4.4 + , text >= 1.2.5.0 + other-extensions: FlexibleInstances + , FlexibleContexts + , MultiParamTypeClasses + , InstanceSigs + default-language: Haskell2010 + ghc-options: -Wall + -Wcompat + -Widentities + -Wincomplete-record-updates + -Wincomplete-uni-patterns + -Wmissing-export-lists + -Wmissing-home-modules + -Wpartial-fields + -Wredundant-constraints + +test-suite unit-tests + type: exitcode-stdio-1.0 + main-is: Main.hs + hs-source-dirs: tests/ + Build-Depends: base >= 4.7 && < 5 + , tasty + , tasty-hunit + , hspray + Default-Language: Haskell2010 + +source-repository head + type: git + location: https://github.com/stla/hspray diff --git a/src/Math/Algebra/Hspray.hs b/src/Math/Algebra/Hspray.hs --- a/src/Math/Algebra/Hspray.hs +++ b/src/Math/Algebra/Hspray.hs @@ -1,252 +1,256 @@ -{-# LANGUAGE FlexibleInstances #-} -{-# LANGUAGE FlexibleContexts #-} -{-# LANGUAGE MultiParamTypeClasses #-} - -module Math.Algebra.Hspray - ( Spray - , lone - , unitSpray - , constantSpray - , fromList - , toList - , sprayTerms - , (*^) - , (.^) - , (^+^) - , (^-^) - , (^*^) - , (^**^) - , evalSpray - , prettySpray - , composeSpray - , bombieriSpray - , derivSpray - ) where -import qualified Algebra.Additive as AlgAdd -import qualified Algebra.Module as AlgMod -import qualified Algebra.Ring as AlgRing -import qualified Data.Foldable as DF -import Data.Function ( on ) -import Data.HashMap.Strict ( HashMap ) -import qualified Data.HashMap.Strict as HM -import Data.Hashable ( Hashable(hashWithSalt) ) -import Data.List ( sortBy ) -import qualified Data.Sequence as S -import Data.Sequence ( (><) - , Seq - , dropWhileR - , (|>) - , index - , adjust - ) -import Data.Text ( Text - , append - , cons - , intercalate - , pack - , snoc - , unpack - ) - - -infixr 7 *^, .^ - -infixl 6 ^+^, ^-^ - -infixl 7 ^*^ - -infixr 8 ^**^ - - -data Powers = Powers - { exponents :: Seq Int - , nvariables :: Int - } - deriving Show - -growSequence :: Seq Int -> Int -> Int -> Seq Int -growSequence s m n = s >< t where t = S.replicate (n - m) 0 - -harmonize :: (Powers, Powers) -> (Powers, Powers) -harmonize (pows1, pows2) = (Powers e1' n, Powers e2' n) - where - e1 = exponents pows1 - e2 = exponents pows2 - n1 = nvariables pows1 - n2 = nvariables pows2 - (e1', e2', n) = if n1 < n2 - then (growSequence e1 n1 n2, e2, n2) - else (e1, growSequence e2 n2 n1, n1) - -instance Eq Powers where - pows1 == pows2 = exponents pows1' == exponents pows2' - where (pows1', pows2') = harmonize (pows1, pows2) - -instance Hashable Powers where - hashWithSalt k pows = hashWithSalt k (exponents pows, nvariables pows) - -type Spray a = HashMap Powers a - -instance (AlgAdd.C a, Eq a) => AlgAdd.C (Spray a) where - p + q = addSprays p q - zero = HM.empty - negate = negateSpray - -instance (AlgRing.C a, Eq a) => AlgMod.C a (Spray a) where - lambda *> p = scaleSpray lambda p - -instance (AlgRing.C a, Eq a) => AlgRing.C (Spray a) where - p * q = multSprays p q - one = lone 0 - --- | Addition of two sprays -(^+^) :: (AlgAdd.C a, Eq a) => Spray a -> Spray a -> Spray a -(^+^) p q = p AlgAdd.+ q - --- | Substraction of two sprays -(^-^) :: (AlgAdd.C a, Eq a) => Spray a -> Spray a -> Spray a -(^-^) p q = p AlgAdd.- q - --- | Multiply two sprays -(^*^) :: (AlgRing.C a, Eq a) => Spray a -> Spray a -> Spray a -(^*^) p q = p AlgRing.* q - --- | Power of a spray -(^**^) :: (AlgRing.C a, Eq a) => Spray a -> Int -> Spray a -(^**^) p n = AlgRing.product (replicate n p) - --- | Scale spray by a scalar -(*^) :: (AlgRing.C a, Eq a) => a -> Spray a -> Spray a -(*^) lambda pol = lambda AlgMod.*> pol - --- | Scale spray by an integer -(.^) :: (AlgAdd.C a, Eq a) => Int -> Spray a -> Spray a -(.^) k pol = if k >= 0 - then AlgAdd.sum (replicate k pol) - else AlgAdd.negate $ AlgAdd.sum (replicate (-k) pol) - -simplifyPowers :: Powers -> Powers -simplifyPowers pows = Powers s (S.length s) - where s = dropWhileR (== 0) (exponents pows) - -simplifySpray :: Spray a -> Spray a -simplifySpray = HM.mapKeys simplifyPowers - -cleanSpray :: (AlgAdd.C a, Eq a) => Spray a -> Spray a -cleanSpray p = HM.filter (/= AlgAdd.zero) (simplifySpray p) - -addSprays :: (AlgAdd.C a, Eq a) => Spray a -> Spray a -> Spray a -addSprays p q = cleanSpray $ HM.foldlWithKey' f p q - where f s powers coef = HM.insertWith (AlgAdd.+) powers coef s - -negateSpray :: AlgAdd.C a => Spray a -> Spray a -negateSpray = HM.map AlgAdd.negate - -scaleSpray :: (AlgRing.C a, Eq a) => a -> Spray a -> Spray a -scaleSpray lambda p = cleanSpray $ HM.map (lambda AlgRing.*) p - -derivMonomial :: AlgRing.C a => Int -> (Powers, a) -> (Powers, a) -derivMonomial i (pows, coef) = if i' >= S.length expts - then (Powers S.empty 0, AlgAdd.zero) - else (pows', coef') - where - i' = i - 1 - expts = exponents pows - expt_i = expts `index` i' - expts' = adjust (subtract 1) i' expts - coef' = AlgAdd.sum (replicate expt_i coef) - pows' = Powers expts' (nvariables pows) - -derivSpray :: (AlgRing.C a, Eq a) => Int -> Spray a -> Spray a -derivSpray i p = cleanSpray $ HM.fromListWith (AlgAdd.+) monomials - where - p' = HM.toList p - monomials = [ derivMonomial i mp | mp <- p' ] - - -multMonomial :: AlgRing.C a => (Powers, a) -> (Powers, a) -> (Powers, a) -multMonomial (pows1, coef1) (pows2, coef2) = (pows, coef1 AlgRing.* coef2) - where - (pows1', pows2') = harmonize (pows1, pows2) - expts = S.zipWith (+) (exponents pows1') (exponents pows2') - pows = Powers expts (nvariables pows1') - -multSprays :: (AlgRing.C a, Eq a) => Spray a -> Spray a -> Spray a -multSprays p q = cleanSpray $ HM.fromListWith (AlgAdd.+) prods - where - p' = HM.toList p - q' = HM.toList q - prods = [ multMonomial mp mq | mp <- p', mq <- q' ] - --- | Spray corresponding to polynomial x_n -lone :: AlgRing.C a => Int -> Spray a -lone n = HM.singleton pows AlgRing.one - where - pows = if n == 0 - then Powers S.empty 0 - else Powers (S.replicate (n - 1) AlgAdd.zero |> AlgRing.one) n - --- | Unit spray -unitSpray :: AlgRing.C a => Spray a -unitSpray = lone 0 - --- | Constant spray -constantSpray :: (AlgRing.C a, Eq a) => a -> Spray a -constantSpray c = c *^ lone 0 - -evalMonomial :: AlgRing.C a => [a] -> (Powers, a) -> a -evalMonomial xyz (powers, coeff) = coeff - AlgRing.* AlgRing.product (zipWith (AlgRing.^) xyz pows) - where pows = DF.toList (fromIntegral <$> exponents powers) - --- | Evaluate a spray -evalSpray :: AlgRing.C a => Spray a -> [a] -> a -evalSpray p xyz = AlgAdd.sum $ map (evalMonomial xyz) (HM.toList p) - -identify :: (AlgRing.C a, Eq a) => Spray a -> Spray (Spray a) -identify = HM.map constantSpray - --- | Compose a spray with a change of variables -composeSpray :: (AlgRing.C a, Eq a) => Spray a -> [Spray a] -> Spray a -composeSpray p = evalSpray (identify p) - --- | Create a spray from list of terms -fromList :: (AlgRing.C a, Eq a) => [([Int], a)] -> Spray a -fromList x = cleanSpray $ HM.fromList $ map - (\(expts, coef) -> (Powers (S.fromList expts) (length expts), coef)) x - -prettyPowers :: String -> [Int] -> Text -prettyPowers var pows = append (pack x) (cons '(' $ snoc string ')') - where - x = " " ++ var ++ "^" - string = intercalate (pack ", ") (map (pack . show) pows) - --- | Pretty form of a spray -prettySpray :: (a -> String) -> String -> Spray a -> String -prettySpray prettyCoef var p = unpack $ intercalate (pack " + ") stringTerms - where - stringTerms = map stringTerm (sortBy (compare `on` fexpts) (HM.toList p)) - fexpts term = exponents $ fst term - stringTerm term = append - (snoc (snoc (cons '(' $ snoc stringCoef ')') ' ') '*') - (prettyPowers var pows) - where - pows = DF.toList $ exponents (fst term) - stringCoef = pack $ prettyCoef (snd term) - --- | Terms of a spray -sprayTerms :: Spray a -> HashMap (Seq Int) a -sprayTerms = HM.mapKeys exponents - --- | Spray as list -toList :: Spray a -> [([Int], a)] -toList p = HM.toList $ HM.mapKeys (DF.toList . exponents) p - --- | Bombieri spray -bombieriSpray :: AlgAdd.C a => Spray a -> Spray a -bombieriSpray = HM.mapWithKey f - where - f pows = times (pfactorial $ exponents pows) - pfactorial pows = product $ DF.toList $ factorial <$> S.filter (/= 0) pows - factorial n = product [1 .. n] - times k x = AlgAdd.sum (replicate k x) +{-# LANGUAGE FlexibleInstances #-} +{-# LANGUAGE FlexibleContexts #-} +{-# LANGUAGE MultiParamTypeClasses #-} +{-# LANGUAGE InstanceSigs #-} + +module Math.Algebra.Hspray + ( Powers (..) + , Spray + , lone + , unitSpray + , constantSpray + , fromList + , toList + , sprayTerms + , (*^) + , (.^) + , (^+^) + , (^-^) + , (^*^) + , (^**^) + , evalSpray + , prettySpray + , composeSpray + , bombieriSpray + , derivSpray + ) where +import qualified Algebra.Additive as AlgAdd +import qualified Algebra.Module as AlgMod +import qualified Algebra.Ring as AlgRing +import qualified Data.Foldable as DF +import Data.Function ( on ) +import Data.HashMap.Strict ( HashMap ) +import qualified Data.HashMap.Strict as HM +import Data.Hashable ( Hashable(hashWithSalt) ) +import Data.List ( sortBy ) +import qualified Data.Sequence as S +import Data.Sequence ( (><) + , Seq + , dropWhileR + , (|>) + , index + , adjust + ) +import Data.Text ( Text + , append + , cons + , intercalate + , pack + , snoc + , unpack + ) + + +infixr 7 *^, .^ + +infixl 6 ^+^, ^-^ + +infixl 7 ^*^ + +infixr 8 ^**^ + + +data Powers = Powers + { exponents :: Seq Int + , nvariables :: Int + } + deriving Show + +growSequence :: Seq Int -> Int -> Int -> Seq Int +growSequence s m n = s >< t where t = S.replicate (n - m) 0 + +harmonize :: (Powers, Powers) -> (Powers, Powers) +harmonize (pows1, pows2) = (Powers e1' n, Powers e2' n) + where + e1 = exponents pows1 + e2 = exponents pows2 + n1 = nvariables pows1 + n2 = nvariables pows2 + (e1', e2', n) = if n1 < n2 + then (growSequence e1 n1 n2, e2, n2) + else (e1, growSequence e2 n2 n1, n1) + +instance Eq Powers where + (==) :: Powers -> Powers -> Bool + pows1 == pows2 = exponents pows1' == exponents pows2' + where (pows1', pows2') = harmonize (pows1, pows2) + +instance Hashable Powers where + hashWithSalt :: Int -> Powers -> Int + hashWithSalt k pows = hashWithSalt k (exponents pows, nvariables pows) + +type Spray a = HashMap Powers a + +instance (AlgAdd.C a, Eq a) => AlgAdd.C (Spray a) where + p + q = addSprays p q + zero = HM.empty + negate = negateSpray + +instance (AlgRing.C a, Eq a) => AlgMod.C a (Spray a) where + lambda *> p = scaleSpray lambda p + +instance (AlgRing.C a, Eq a) => AlgRing.C (Spray a) where + p * q = multSprays p q + one = lone 0 + +-- | Addition of two sprays +(^+^) :: (AlgAdd.C a, Eq a) => Spray a -> Spray a -> Spray a +(^+^) p q = p AlgAdd.+ q + +-- | Substraction of two sprays +(^-^) :: (AlgAdd.C a, Eq a) => Spray a -> Spray a -> Spray a +(^-^) p q = p AlgAdd.- q + +-- | Multiply two sprays +(^*^) :: (AlgRing.C a, Eq a) => Spray a -> Spray a -> Spray a +(^*^) p q = p AlgRing.* q + +-- | Power of a spray +(^**^) :: (AlgRing.C a, Eq a) => Spray a -> Int -> Spray a +(^**^) p n = AlgRing.product (replicate n p) + +-- | Scale spray by a scalar +(*^) :: (AlgRing.C a, Eq a) => a -> Spray a -> Spray a +(*^) lambda pol = lambda AlgMod.*> pol + +-- | Scale spray by an integer +(.^) :: (AlgAdd.C a, Eq a) => Int -> Spray a -> Spray a +(.^) k pol = if k >= 0 + then AlgAdd.sum (replicate k pol) + else AlgAdd.negate $ AlgAdd.sum (replicate (-k) pol) + +simplifyPowers :: Powers -> Powers +simplifyPowers pows = Powers s (S.length s) + where s = dropWhileR (== 0) (exponents pows) + +simplifySpray :: Spray a -> Spray a +simplifySpray = HM.mapKeys simplifyPowers + +cleanSpray :: (AlgAdd.C a, Eq a) => Spray a -> Spray a +cleanSpray p = HM.filter (/= AlgAdd.zero) (simplifySpray p) + +addSprays :: (AlgAdd.C a, Eq a) => Spray a -> Spray a -> Spray a +addSprays p q = cleanSpray $ HM.foldlWithKey' f p q + where f s powers coef = HM.insertWith (AlgAdd.+) powers coef s + +negateSpray :: AlgAdd.C a => Spray a -> Spray a +negateSpray = HM.map AlgAdd.negate + +scaleSpray :: (AlgRing.C a, Eq a) => a -> Spray a -> Spray a +scaleSpray lambda p = cleanSpray $ HM.map (lambda AlgRing.*) p + +derivMonomial :: AlgRing.C a => Int -> (Powers, a) -> (Powers, a) +derivMonomial i (pows, coef) = if i' >= S.length expts + then (Powers S.empty 0, AlgAdd.zero) + else (pows', coef') + where + i' = i - 1 + expts = exponents pows + expt_i = expts `index` i' + expts' = adjust (subtract 1) i' expts + coef' = AlgAdd.sum (replicate expt_i coef) + pows' = Powers expts' (nvariables pows) + +derivSpray :: (AlgRing.C a, Eq a) => Int -> Spray a -> Spray a +derivSpray i p = cleanSpray $ HM.fromListWith (AlgAdd.+) monomials + where + p' = HM.toList p + monomials = [ derivMonomial i mp | mp <- p' ] + + +multMonomial :: AlgRing.C a => (Powers, a) -> (Powers, a) -> (Powers, a) +multMonomial (pows1, coef1) (pows2, coef2) = (pows, coef1 AlgRing.* coef2) + where + (pows1', pows2') = harmonize (pows1, pows2) + expts = S.zipWith (+) (exponents pows1') (exponents pows2') + pows = Powers expts (nvariables pows1') + +multSprays :: (AlgRing.C a, Eq a) => Spray a -> Spray a -> Spray a +multSprays p q = cleanSpray $ HM.fromListWith (AlgAdd.+) prods + where + p' = HM.toList p + q' = HM.toList q + prods = [ multMonomial mp mq | mp <- p', mq <- q' ] + +-- | Spray corresponding to polynomial x_n +lone :: AlgRing.C a => Int -> Spray a +lone n = HM.singleton pows AlgRing.one + where + pows = if n == 0 + then Powers S.empty 0 + else Powers (S.replicate (n - 1) AlgAdd.zero |> AlgRing.one) n + +-- | Unit spray +unitSpray :: AlgRing.C a => Spray a +unitSpray = lone 0 + +-- | Constant spray +constantSpray :: (AlgRing.C a, Eq a) => a -> Spray a +constantSpray c = c *^ lone 0 + +evalMonomial :: AlgRing.C a => [a] -> (Powers, a) -> a +evalMonomial xyz (powers, coeff) = coeff + AlgRing.* AlgRing.product (zipWith (AlgRing.^) xyz pows) + where pows = DF.toList (fromIntegral <$> exponents powers) + +-- | Evaluate a spray +evalSpray :: AlgRing.C a => Spray a -> [a] -> a +evalSpray p xyz = AlgAdd.sum $ map (evalMonomial xyz) (HM.toList p) + +identify :: (AlgRing.C a, Eq a) => Spray a -> Spray (Spray a) +identify = HM.map constantSpray + +-- | Compose a spray with a change of variables +composeSpray :: (AlgRing.C a, Eq a) => Spray a -> [Spray a] -> Spray a +composeSpray p = evalSpray (identify p) + +-- | Create a spray from list of terms +fromList :: (AlgRing.C a, Eq a) => [([Int], a)] -> Spray a +fromList x = cleanSpray $ HM.fromList $ map + (\(expts, coef) -> (Powers (S.fromList expts) (length expts), coef)) x + +prettyPowers :: String -> [Int] -> Text +prettyPowers var pows = append (pack x) (cons '(' $ snoc string ')') + where + x = " " ++ var ++ "^" + string = intercalate (pack ", ") (map (pack . show) pows) + +-- | Pretty form of a spray +prettySpray :: (a -> String) -> String -> Spray a -> String +prettySpray prettyCoef var p = unpack $ intercalate (pack " + ") stringTerms + where + stringTerms = map stringTerm (sortBy (compare `on` fexpts) (HM.toList p)) + fexpts term = exponents $ fst term + stringTerm term = append + (snoc (snoc (cons '(' $ snoc stringCoef ')') ' ') '*') + (prettyPowers var pows) + where + pows = DF.toList $ exponents (fst term) + stringCoef = pack $ prettyCoef (snd term) + +-- | Terms of a spray +sprayTerms :: Spray a -> HashMap (Seq Int) a +sprayTerms = HM.mapKeys exponents + +-- | Spray as list +toList :: Spray a -> [([Int], a)] +toList p = HM.toList $ HM.mapKeys (DF.toList . exponents) p + +-- | Bombieri spray +bombieriSpray :: AlgAdd.C a => Spray a -> Spray a +bombieriSpray = HM.mapWithKey f + where + f pows = times (pfactorial $ exponents pows) + pfactorial pows = product $ DF.toList $ factorial <$> S.filter (/= 0) pows + factorial n = product [1 .. n] + times k x = AlgAdd.sum (replicate k x) diff --git a/tests/Main.hs b/tests/Main.hs --- a/tests/Main.hs +++ b/tests/Main.hs @@ -1,73 +1,73 @@ -module Main where -import Data.Ratio ( (%) ) -import Math.Algebra.Hspray ( Spray, - (^+^), - (^*^), - (^**^), - (*^), - lone, - unitSpray, - evalSpray, - composeSpray, - fromList, - toList, - bombieriSpray, - derivSpray ) -import Test.Tasty ( defaultMain - , testGroup - ) -import Test.Tasty.HUnit ( assertEqual - , testCase - ) - -main :: IO () -main = defaultMain $ testGroup - "Testing hspray" - - [ testCase "bombieriSpray" $ do - let - x = lone 1 :: Spray Rational - y = lone 2 :: Spray Rational - z = lone 3 :: Spray Rational - poly = - (2 % 1) *^ ((2 % 1) *^ (x ^**^ 3 ^*^ y ^**^ 2)) ^+^ (4 % 1) *^ z ^+^ (5 % 1) *^ unitSpray - bpoly = - (24 % 1) *^ ((2 % 1) *^ (x ^**^ 3 ^*^ y ^**^ 2)) ^+^ (4 % 1) *^ z ^+^ (5 % 1) *^ unitSpray - assertEqual "" bpoly (bombieriSpray poly), - - testCase "composeSpray" $ do - let - x = lone 1 :: Spray Int - y = lone 2 :: Spray Int - z = lone 3 :: Spray Int - p = 2 *^ (2 *^ (x ^**^ 3 ^*^ y ^**^ 2)) ^+^ 4 *^ z ^+^ 5 *^ unitSpray - px = x ^+^ y ^+^ z - py = x ^*^ y ^*^ z - pz = y ^**^ 2 - q = composeSpray p [px, py, pz] - xyz = [2, 3, 4] - pxyz = map (`evalSpray` xyz) [px, py, pz] - assertEqual "" (evalSpray p pxyz) (evalSpray q xyz), - - testCase "fromList . toList = identity" $ do - let - x = lone 1 :: Spray Int - y = lone 2 :: Spray Int - z = lone 3 :: Spray Int - p = 2 *^ (2 *^ (x ^**^ 3 ^*^ y ^**^ 2)) ^+^ 4 *^ z ^+^ 5 *^ unitSpray - assertEqual "" p (fromList . toList $ p), - - testCase "derivSpray" $ do - let - x = lone 1 :: Spray Int - y = lone 2 :: Spray Int - z = lone 3 :: Spray Int - p1 = x ^+^ y ^*^ z ^**^ 3 - p2 = (x ^*^ y ^*^ z) ^+^ (2 *^ (x ^**^ 3 ^*^ y ^**^ 2)) - q = p1 ^*^ p2 - p1' = derivSpray 1 p1 - p2' = derivSpray 1 p2 - q' = derivSpray 1 q - assertEqual "" q' ((p1' ^*^ p2) ^+^ (p1 ^*^ p2')) - - ] +module Main where +import Data.Ratio ( (%) ) +import Math.Algebra.Hspray ( Spray, + (^+^), + (^*^), + (^**^), + (*^), + lone, + unitSpray, + evalSpray, + composeSpray, + fromList, + toList, + bombieriSpray, + derivSpray ) +import Test.Tasty ( defaultMain + , testGroup + ) +import Test.Tasty.HUnit ( assertEqual + , testCase + ) + +main :: IO () +main = defaultMain $ testGroup + "Testing hspray" + + [ testCase "bombieriSpray" $ do + let + x = lone 1 :: Spray Rational + y = lone 2 :: Spray Rational + z = lone 3 :: Spray Rational + poly = + (2 % 1) *^ ((2 % 1) *^ (x ^**^ 3 ^*^ y ^**^ 2)) ^+^ (4 % 1) *^ z ^+^ (5 % 1) *^ unitSpray + bpoly = + (24 % 1) *^ ((2 % 1) *^ (x ^**^ 3 ^*^ y ^**^ 2)) ^+^ (4 % 1) *^ z ^+^ (5 % 1) *^ unitSpray + assertEqual "" bpoly (bombieriSpray poly), + + testCase "composeSpray" $ do + let + x = lone 1 :: Spray Int + y = lone 2 :: Spray Int + z = lone 3 :: Spray Int + p = 2 *^ (2 *^ (x ^**^ 3 ^*^ y ^**^ 2)) ^+^ 4 *^ z ^+^ 5 *^ unitSpray + px = x ^+^ y ^+^ z + py = x ^*^ y ^*^ z + pz = y ^**^ 2 + q = composeSpray p [px, py, pz] + xyz = [2, 3, 4] + pxyz = map (`evalSpray` xyz) [px, py, pz] + assertEqual "" (evalSpray p pxyz) (evalSpray q xyz), + + testCase "fromList . toList = identity" $ do + let + x = lone 1 :: Spray Int + y = lone 2 :: Spray Int + z = lone 3 :: Spray Int + p = 2 *^ (2 *^ (x ^**^ 3 ^*^ y ^**^ 2)) ^+^ 4 *^ z ^+^ 5 *^ unitSpray + assertEqual "" p (fromList . toList $ p), + + testCase "derivSpray" $ do + let + x = lone 1 :: Spray Int + y = lone 2 :: Spray Int + z = lone 3 :: Spray Int + p1 = x ^+^ y ^*^ z ^**^ 3 + p2 = (x ^*^ y ^*^ z) ^+^ (2 *^ (x ^**^ 3 ^*^ y ^**^ 2)) + q = p1 ^*^ p2 + p1' = derivSpray 1 p1 + p2' = derivSpray 1 p2 + q' = derivSpray 1 q + assertEqual "" q' ((p1' ^*^ p2) ^+^ (p1 ^*^ p2')) + + ]