factory 0.2.0.3 → 0.2.0.4

raw patch · 29 files changed

+224/−75 lines, 29 files

Files

• changelog +4/−1
• factory.cabal +1/−1
• src/Factory/Data/Interval.hs +8/−3
• src/Factory/Data/MonicPolynomial.hs +17/−2
• src/Factory/Data/Monomial.hs +1/−1
• src/Factory/Data/Polynomial.hs +24/−14
• src/Factory/Data/Ring.hs +6/−1
• src/Factory/Math/Factorial.hs +1/−1
• src/Factory/Math/Hyperoperation.hs +4/−4
• src/Factory/Math/Implementations/Factorial.hs +3/−3
• src/Factory/Math/Implementations/Primality.hs +8/−3
• src/Factory/Math/Implementations/PrimeFactorisation.hs +6/−1
• src/Factory/Math/MultiplicativeOrder.hs +1/−1
• src/Factory/Math/Power.hs +1/−1
• src/Factory/Math/Primality.hs +14/−4
• src/Factory/Math/PrimeFactorisation.hs +6/−1
• src/Factory/Math/Primes.hs +6/−1
• src/Factory/Math/Probability.hs +19/−7
• src/Factory/Math/Radix.hs +25/−4
• src/Factory/Math/SquareRoot.hs +3/−3
• src/Factory/Math/Statistics.hs +33/−7
• src/Factory/Test/Performance/Factorial.hs +6/−1
• src/Factory/Test/Performance/Hyperoperation.hs +2/−2
• src/Factory/Test/Performance/Primality.hs +1/−1
• src/Factory/Test/Performance/Primes.hs +6/−1
• src/Factory/Test/Performance/SquareRoot.hs +7/−2
• src/Factory/Test/Performance/Statistics.hs +6/−1
• src/Factory/Test/QuickCheck/MonicPolynomial.hs +4/−2
• src/Factory/Test/QuickCheck/Primes.hs +1/−1

@@ -51,4 +51,7 @@ 	* Added 'Factory.Data.Interval.mkBounded'. 	* Generalised "Factory.Math.Statistics" to accept any 'Data.Foldable.Foldable' 'Functor', rather than merely lists. 0.2.0.3-	* Added "class Show" to several contexts, for migration to 'ghc-7.4'.+	* Added class 'Show' to some contexts in "Factory.Math.Radix", for migration to 'ghc-7.4'.+0.2.0.4+	* Added classes 'Eq' and 'Show' to many contexts, for migration to 'ghc-7.4'.+	* Minor re-formatting.

@@ -1,6 +1,6 @@ --Package-properties Name:			factory-Version:		0.2.0.3+Version:		0.2.0.4 Cabal-Version:		>= 1.6 Copyright:		(C) 2011 Dr. Alistair Ward License:		GPL

@@ -125,7 +125,12 @@ 	| otherwise	= b  -- | Bisect the /interval/ at the specified /end-point/; which should be between the two existing /end-points/.-splitAt' :: (Enum endPoint, Num endPoint, Ord endPoint) => endPoint -> Interval endPoint -> (Interval endPoint, Interval endPoint)+splitAt' :: (+	Enum	endPoint,+	Num	endPoint,+	Ord	endPoint,+	Show	endPoint+ ) => endPoint -> Interval endPoint -> (Interval endPoint, Interval endPoint) splitAt' i interval@(l, r) 	| any ($ i) [(< l), (>= r)]	= error $ "Factory.Data.Interval.splitAt':\tunsuitable index=" ++ show i ++ " for interval=" ++ show interval ++ "." 	| otherwise			= ((l, i), (succ i, r))@@ -168,7 +173,7 @@ 	(each leaf of which contains a list of up to 'minLength' integers, and each node of which contains an associative binary operator), 	and then collapsed to a scalar, by application of the operators. -}-divideAndConquer :: (Integral i, Data.Monoid.Monoid monoid)+divideAndConquer :: (Data.Monoid.Monoid monoid, Integral i, Show i) 	=> (i -> monoid)	-- ^ The monoid's constructor. 	-> Data.Ratio.Ratio i	-- ^ The ratio of the original span, at which to bisect the 'Interval'. 	-> i			-- ^ For efficiency, the /interval/ will not be bisected, when it's length has been reduced to this value.@@ -200,7 +205,7 @@ 	* Since the result can be large, 'divideAndConquer' is used to form operands of a similar order of magnitude, 	thus improving the efficiency of the big-number multiplication. -}-product' :: Integral i+product' :: (Integral i, Show i) 	=> Data.Ratio.Ratio i	-- ^ The ratio at which to bisect the 'Interval'. 	-> i			-- ^ For efficiency, the /interval/ will not be bisected, when it's length has been reduced to this value. 	-> Interval i

@@ -47,7 +47,13 @@ } deriving (Eq, Show)  -- | Smart constructor. Constructs an arbitrary /monic polynomial/.-mkMonicPolynomial :: (Num c, Ord e, Show e) => Data.Polynomial.Polynomial c e -> MonicPolynomial c e+mkMonicPolynomial :: (+	Eq	c,+	Num	c,+	Ord	e,+	Show	c,+	Show	e+ ) => Data.Polynomial.Polynomial c e -> MonicPolynomial c e mkMonicPolynomial polynomial 	| not $ Data.Polynomial.isMonic polynomial	= error $ "Factory.Data.MonicPolynomial.mkMonicPolynomial:\tnot monic; " ++ show polynomial 	| otherwise					= MkMonicPolynomial polynomial@@ -58,9 +64,11 @@ 	* CAVEAT: it's not strictly an instance of this class, since the result of some methods isn't /monic/. -} instance (+	Eq	c, 	Num	c, 	Num	e, 	Ord	e,+	Show	c, 	Show	e  ) => Data.Ring.Ring (MonicPolynomial c e)	where 	MkMonicPolynomial l =*= MkMonicPolynomial r	= MkMonicPolynomial $ l =*= r@@ -71,7 +79,14 @@ 	additiveIdentity				= MkMonicPolynomial Data.Ring.additiveIdentity	--CAVEAT: not monic !  -- Since the /leading term/ of the /denominator/ is one, the /coefficient/ isn't required to implement 'Fractional'.-instance (Num c, Num e, Ord e) => Data.QuotientRing.QuotientRing (MonicPolynomial c e)	where+instance (+	Eq	c,+	Num	c,+	Num	e,+	Ord	e,+	Show	c,+	Show	e+ ) => Data.QuotientRing.QuotientRing (MonicPolynomial c e)	where 	MkMonicPolynomial polynomialN `quotRem'` MkMonicPolynomial polynomialD	= ToolShed.Data.Pair.mirror MkMonicPolynomial $ longDivide polynomialN	where --		longDivide :: (Num c, Num e, Ord e) => Polynomial c e -> (Polynomial c e, Polynomial c e) 		longDivide numerator

@@ -95,7 +95,7 @@ (cL, eL) <*> (cR, eR)	= (cL * cR, eL + eR)  -- | Divide the two specified 'Monomial's.-(</>) :: (Fractional c, Num e)+(</>) :: (Eq c, Fractional c, Num e) 	=> Monomial c e	-- ^ Numerator. 	-> Monomial c e	-- ^ Denominator. 	-> Monomial c e

@@ -99,7 +99,12 @@ } deriving (Eq, Show)  -- | Makes /Polynomial/ a 'Data.Ring.Ring', over the /field/ composed from all possible /coefficients/; <http://en.wikipedia.org/wiki/Polynomial_ring>.-instance (Num c, Num e, Ord e) => Data.Ring.Ring (Polynomial c e) where+instance (+	Eq	c,+	Num	c,+	Num	e,+	Ord	e+ ) => Data.Ring.Ring (Polynomial c e) where 	MkPolynomial [] =*= _	= zero 	_ =*= MkPolynomial []	= zero 	polynomialL =*= polynomialR@@ -148,7 +153,12 @@ 		 ) $ map MkPolynomial . init {-remove terminal null-} . Data.List.tails . tail &&& map Data.Monomial.double $ getMonomialList p  -- | Defines the ability to divide /polynomials/.-instance (Fractional c, Num e, Ord e) => Data.QuotientRing.QuotientRing (Polynomial c e)	where+instance (+	Eq		c,+	Fractional	c,+	Num		e,+	Ord		e+ ) => Data.QuotientRing.QuotientRing (Polynomial c e)	where {- 	Uses /Euclidian division/. 	<http://en.wikipedia.org/wiki/Polynomial_long_division>.@@ -185,12 +195,12 @@ getLeadingTerm (MkPolynomial (m : _))	= m  -- | Removes terms with a /coefficient/ of zero.-pruneCoefficients :: Num c => Polynomial c e -> Polynomial c e+pruneCoefficients :: (Eq c, Num c) => Polynomial c e -> Polynomial c e pruneCoefficients (MkPolynomial [])	= zero pruneCoefficients p			= filter ((/= 0) . Data.Monomial.getCoefficient) `lift` p  -- | Sorts into /descending order/ of exponents, groups /like/ exponents, and calls 'pruneCoefficients'.-normalise :: (Num c, Ord e) => Polynomial c e -> Polynomial c e+normalise :: (Eq c, Num c, Ord e) => Polynomial c e -> Polynomial c e normalise	= pruneCoefficients . lift ( 	map ( 		foldr ((+) . Data.Monomial.getCoefficient) 0 &&& Data.Monomial.getExponent . head@@ -198,19 +208,19 @@  )  -- | Constructs an arbitrary /zeroeth-degree polynomial/, ie. independent of the /indeterminate/.-mkConstant :: (Num c, Num e) => c -> Polynomial c e+mkConstant :: (Eq c, Num c, Num e) => c -> Polynomial c e mkConstant 0	= zero mkConstant c	= MkPolynomial [(c, 0)]  -- | Constructs an arbitrary /first-degree polynomial/.-mkLinear :: (Num c, Num e)+mkLinear :: (Eq c, Num c, Num e) 	=> c	-- ^ Gradient. 	-> c	-- ^ Constant. 	-> Polynomial c e mkLinear m c	= pruneCoefficients $ MkPolynomial [(m, 1), (c, 0)]  -- | Smart constructor. Constructs an arbitrary /polynomial/.-mkPolynomial :: (Num c, Ord e) => MonomialList c e -> Polynomial c e+mkPolynomial :: (Eq c, Num c, Ord e) => MonomialList c e -> Polynomial c e mkPolynomial []	= zero mkPolynomial l	= normalise $ MkPolynomial l @@ -219,7 +229,7 @@ zero	= MkPolynomial []  -- | Constructs a constant /monomial/, independent of the /indeterminate/.-one :: (Num c, Num e) => Polynomial c e+one :: (Eq c, Num c, Num e) => Polynomial c e one	= mkConstant 1  -- | True if all /exponents/ are in the order defined by the specified comparator.@@ -237,11 +247,11 @@ inDescendingOrder	= inOrder (>=)  -- | True if no term has a /coefficient/ of zero.-isReduced :: Num c => Polynomial c e -> Bool+isReduced :: (Eq c, Num c) => Polynomial c e -> Bool isReduced	= all ((/= 0) . Data.Monomial.getCoefficient) . getMonomialList  -- | True if no term has a /coefficient/ of zero and the /exponents/ of successive terms are in /descending/ order.-isNormalised :: (Num c, Ord e) => Polynomial c e -> Bool+isNormalised :: (Eq c, Num c, Ord e) => Polynomial c e -> Bool isNormalised polynomial	= all ($ polynomial) [isReduced, inDescendingOrder]  {- |@@ -249,7 +259,7 @@  	* <http://en.wikipedia.org/wiki/Monic_polynomial#Classifications>. -}-isMonic :: Num c => Polynomial c e -> Bool+isMonic :: (Eq c, Num c) => Polynomial c e -> Bool isMonic (MkPolynomial [])	= False	--All coefficients are zero, and have therefore been removed. isMonic p			= (== 1) . Data.Monomial.getCoefficient $ getLeadingTerm p @@ -299,7 +309,7 @@  	* <http://en.wikipedia.org/wiki/Scalar_multiplication>. -}-(*=) :: (Num c, Num e) => Polynomial c e -> Data.Monomial.Monomial c e -> Polynomial c e+(*=) :: (Eq c, Num c, Num e) => Polynomial c e -> Data.Monomial.Monomial c e -> Polynomial c e polynomial *= monomial 	| Data.Monomial.getCoefficient monomial == 1	= map (`Data.Monomial.shiftExponent` Data.Monomial.getExponent monomial) `lift` polynomial 	| otherwise					= map (monomial <*>) `lift` polynomial@@ -309,7 +319,7 @@  	* Whilst one could naively implement this as @(x Data.Ring.=^ n) `mod` m@, this will result in arithmetic operatons on unnecessarily big integers. -}-raiseModulo :: (Integral c, Integral power, Num e, Ord e)+raiseModulo :: (Integral c, Integral power, Num e, Ord e, Show power) 	=> Polynomial c e	-- ^ The base. 	-> power		-- ^ The exponent to which the base should be raised. 	-> c			-- ^ The modulus.@@ -345,7 +355,7 @@ 	* If the /polynomial/ is very sparse, this may be inefficient, 	since it /memoizes/ the complete sequence of powers up to the polynomial's /degree/. -}-evaluate :: (Num n, Integral e)+evaluate :: (Num n, Integral e, Show e) 	=> n	-- ^ The /indeterminate/. 	-> Polynomial n e 	-> n	-- ^ The Result.

@@ -72,7 +72,12 @@ 	* Exponentiation is implemented as a sequence of either squares of, or multiplications by, the /ring/-member; 	<http://en.wikipedia.org/wiki/Exponentiation_by_squaring>. -}-(=^) :: (Ring r, Eq r, Integral power) => r -> power -> r+(=^) :: (+	Eq		r,+	Integral	power,+	Ring		r,+	Show		power+ ) => r -> power -> r _ =^ 0	= multiplicativeIdentity ring =^ power 	| power < 0							= error $ "Factory.Data.Ring.(=^):\tthe result isn't guaranteed to be a ring-member, for power=" ++ show power

@@ -33,5 +33,5 @@  -- | Defines the methods expected of a /factorial/-algorithm. class Algorithmic algorithm	where-	factorial	:: Integral i => algorithm -> i -> i+	factorial	:: (Integral i, Show i) => algorithm -> i -> i 

@@ -67,7 +67,7 @@ 		<http://en.wikipedia.org/wiki/Tetration>, 		<http://www.tetration.org/Fractals/Atlas/index.html>. -}-powerTower :: (Integral base, Integral hyperExponent) => base -> hyperExponent -> base+powerTower :: (Integral base, Integral hyperExponent, Show base) => base -> hyperExponent -> base powerTower 0 hyperExponent 	| even hyperExponent	= 1 	| otherwise		= 0@@ -77,7 +77,7 @@ 	| otherwise			= Data.List.genericIndex (iterate (base ^) 1) hyperExponent  -- | The /hyperoperation/-sequence; <http://en.wikipedia.org/wiki/Hyperoperation>.-hyperoperation :: Integral rank => rank -> Base -> HyperExponent -> Base+hyperoperation :: (Integral rank, Show rank) => rank -> Base -> HyperExponent -> Base hyperoperation rank base hyperExponent 	| rank < fromIntegral succession	= error $ "Factory.Math.Hyperoperation.hyperoperation:\tundefined for rank; " ++ show rank 	| hyperExponent < 0			= error $ "Factory.Math.Hyperoperation.hyperoperation:\tundefined for hyper-exponent; " ++ show hyperExponent@@ -101,11 +101,11 @@ 					e'	= {-fromIntegral $-} r ^# pred e  -- | The /Ackermann-Peter/-function; <http://en.wikipedia.org/wiki/Ackermann_function#Ackermann_numbers>.-ackermannPeter :: Integral rank => rank -> HyperExponent -> Base+ackermannPeter :: (Integral rank, Show rank) => rank -> HyperExponent -> Base ackermannPeter rank	= (+ negate 3) . hyperoperation rank 2 {-base-} . (+ 3)  -- | True if @hyperoperation base hyperExponent@ has the same value for each specified 'rank'.-areCoincidental :: Integral rank => Base -> HyperExponent -> [rank] -> Bool+areCoincidental :: (Integral rank, Show rank) => Base -> HyperExponent -> [rank] -> Bool areCoincidental _ _ []				= True areCoincidental _ _ [_]				= True areCoincidental base hyperExponent ranks	= all (== h) hs	where

@@ -98,7 +98,7 @@ primeMultiplicity prime	= sum . takeWhile (> 0) . tail . iterate (`div` prime)  -- | Returns the /rising factorial/; <http://mathworld.wolfram.com/RisingFactorial.html>-risingFactorial :: Integral i+risingFactorial :: (Integral i, Show i) 	=> i	-- ^ The lower bound of the integer-range, whose product is returned. 	-> i	-- ^ The number of integers in the range above. 	-> i	-- ^ The result.@@ -107,7 +107,7 @@ risingFactorial x n	= Data.Interval.product' (recip 2) 64 $ Data.Interval.normalise (x, pred $ x + n)  -- | Returns the /falling factorial/; <http://mathworld.wolfram.com/FallingFactorial.html>-fallingFactorial :: Integral i+fallingFactorial :: (Integral i, Show i) 	=> i	-- ^ The upper bound of the integer-range, whose product is returned. 	-> i	-- ^ The number of integers in the range beneath. 	-> i	-- ^ The result.@@ -125,7 +125,7 @@ 	then manipulate them using the module "Data.PrimeFactors", 	and evaluate it using by /Data.PrimeFactors.product'/. -}-(!/!) :: (Integral i, Fractional f)+(!/!) :: (Integral i, Fractional f, Show i) 	=> i	-- ^ The /numerator/. 	-> i	-- ^ The /denominator/. 	-> f	-- ^ The resulting fraction.

@@ -107,7 +107,12 @@  	[@Vibhor Bhatt and G. K. Patra@]			<http://www.cmmacs.ernet.in/cmmacs/Publications/resch_rep/rrcm0307.pdf>, -}-isPrimeByAKS :: (Math.PrimeFactorisation.Algorithmic factorisationAlgorithm, Control.DeepSeq.NFData i, Integral i) => factorisationAlgorithm -> i -> Bool+isPrimeByAKS :: (+	Control.DeepSeq.NFData			i,+	Integral				i,+	Math.PrimeFactorisation.Algorithmic	factorisationAlgorithm,+	Show					i+ ) => factorisationAlgorithm -> i -> Bool isPrimeByAKS factorisationAlgorithm n	= and [ 	not $ Math.PerfectPower.isPerfectPower n,	--Step 1. 	Math.Primality.areCoprime n `all` filter (/= n) [2 .. r],	--Step 3.@@ -155,7 +160,7 @@ 	the remainder belong to the subset of /liars/. 	In consequence, many false results must be accumulated for different bases, to convincingly identify a prime. -}-witnessesCompositeness :: Integral i+witnessesCompositeness :: (Integral i, Show i) 	=> i	-- ^ Candidate integer. 	-> i 	-> Int@@ -186,7 +191,7 @@  	* <http://oeis.org/A014233>, <http://oeis.org/A006945>. -}-isPrimeByMillerRabin :: Integral i => i -> Bool+isPrimeByMillerRabin :: (Integral i, Show i) => i -> Bool isPrimeByMillerRabin primeCandidate	= not $ witnessesCompositeness primeCandidate ( 	fst $ last binaryFactors	--Odd-remainder.  ) (

@@ -98,7 +98,12 @@  	* This algorithm works best when there's a factor close to the /square-root/. -}-factoriseByFermatsMethod :: (Control.DeepSeq.NFData base, Integral base, Control.DeepSeq.NFData exponent, Num exponent) => base -> Data.PrimeFactors.Factors base exponent+factoriseByFermatsMethod :: (+	Control.DeepSeq.NFData	base,+	Control.DeepSeq.NFData	exponent,+	Integral		base,+	Num			exponent+ ) => base -> Data.PrimeFactors.Factors base exponent factoriseByFermatsMethod i 	| i <= 3				= [Data.Exponential.rightIdentity i] 	| even i				= Data.Exponential.rightIdentity 2 : factoriseByFermatsMethod (i `div` 2) {-recurse-}

@@ -42,7 +42,7 @@  	* <http://mathworld.wolfram.com/MultiplicativeOrder.html>. -}-multiplicativeOrder :: (Math.PrimeFactorisation.Algorithmic primeFactorisationAlgorithm, Control.DeepSeq.NFData i, Integral i)+multiplicativeOrder :: (Math.PrimeFactorisation.Algorithmic primeFactorisationAlgorithm, Control.DeepSeq.NFData i, Integral i, Show i) 	=> primeFactorisationAlgorithm 	-> i	-- ^ Base. 	-> i	-- ^ Modulus.

@@ -62,7 +62,7 @@  	* <http://en.wikipedia.org/wiki/Modular_exponentiation>. -}-raiseModulo :: (Integral i, Integral power)+raiseModulo :: (Integral i, Integral power, Show power) 	=> i	-- ^ Base. 	-> power 	-> i	-- ^ Modulus.

@@ -40,7 +40,7 @@  -- | Defines the methods expected of a primality-testing algorithm. class Algorithmic algorithm	where-	isPrime	:: (Control.DeepSeq.NFData i, Integral i) => algorithm -> i -> Bool+	isPrime	:: (Control.DeepSeq.NFData i, Integral i, Show i) => algorithm -> i -> Bool  {- | 	'True' if the two specified integers are /relatively prime/,@@ -68,7 +68,7 @@  	* TODO: confirm that all values must be tested. -}-isFermatWitness :: Integral i => i -> Bool+isFermatWitness :: (Integral i, Show i) => i -> Bool isFermatWitness i	= not . all isFermatPseudoPrime $ filter (areCoprime i) [2 .. pred i]	where 	isFermatPseudoPrime base	= Math.Power.raiseModulo base (pred i) i == 1	--CAVEAT: a /Fermat Pseudo-prime/ must also be a /composite/ number. @@ -79,7 +79,12 @@  	* <http://mathworld.wolfram.com/CarmichaelNumber.html>. -}-isCarmichaelNumber :: (Algorithmic algorithm, Control.DeepSeq.NFData i, Integral i) => algorithm -> i -> Bool+isCarmichaelNumber :: (+	Algorithmic		algorithm,+	Control.DeepSeq.NFData	i,+	Integral		i,+	Show			i+ ) => algorithm -> i -> Bool isCarmichaelNumber algorithm i	= not $ or [ 	i <= 2, 	even i,@@ -88,5 +93,10 @@  ]  -- | An ordered list of the /Carmichael/ numbers; <http://en.wikipedia.org/wiki/Carmichael_number>.-carmichaelNumbers :: (Algorithmic algorithm, Control.DeepSeq.NFData i, Integral i) => algorithm -> [i]+carmichaelNumbers :: (+	Algorithmic		algorithm,+	Control.DeepSeq.NFData	i,+	Integral		i,+	Show			i+ ) => algorithm -> [i] carmichaelNumbers algorithm	= isCarmichaelNumber algorithm `filter` [3, 5 ..]

@@ -118,7 +118,12 @@  	* AKA /EulerPhi/. -}-eulersTotient :: (Algorithmic algorithm, Control.DeepSeq.NFData i, Integral i) => algorithm -> i -> i+eulersTotient :: (+	Algorithmic		algorithm,+	Control.DeepSeq.NFData	i,+	Integral		i,+	Show			i+ ) => algorithm -> i -> i eulersTotient _ 1	= 1 eulersTotient algorithm i 	| i <= 0	= error $ "Factory.Math.PrimeFactorisation.eulersTotient:\tundefined for; " ++ show i

@@ -41,5 +41,10 @@  	* <http://mathworld.wolfram.com/Primorial.html>. -}-primorial :: (Algorithmic algorithm, Control.DeepSeq.NFData i, Data.Array.IArray.Ix i, Integral i) => algorithm -> [i]+primorial :: (+	Algorithmic		algorithm,+	Control.DeepSeq.NFData	i,+	Data.Array.IArray.Ix	i,+	Integral		i+ ) => algorithm -> [i] primorial	= scanl (*) 1 . primes

@@ -49,7 +49,7 @@ 	| NormalDistribution f f				-- ^ Defines a /Normal/-distribution with a particular /mean/ and /variance/; <http://en.wikipedia.org/wiki/Normal_distribution>. 	deriving (Eq, Read, Show) -instance (Num a, Ord a) => ToolShed.SelfValidate.SelfValidator (ContinuousDistribution a)	where+instance (Num a, Ord a, Show a) => ToolShed.SelfValidate.SelfValidator (ContinuousDistribution a)	where 	getErrors distribution	= ToolShed.SelfValidate.extractErrors $ case distribution of 		UniformDistribution interval	-> [(Data.Interval.isReversed interval, "Reversed interval='" ++ show interval ++ "'.")] 		NormalDistribution _ v		-> [(v < 0, "Negative variance=" ++ show v ++ ".")]@@ -57,7 +57,7 @@ -- | Describes a /discrete probability-distribution/; <http://en.wikipedia.org/wiki/List_of_probability_distributions#Discrete_distributions>. data DiscreteDistribution f	= PoissonDistribution f	deriving (Eq, Read, Show) -instance (Num f, Ord f) => ToolShed.SelfValidate.SelfValidator (DiscreteDistribution f)	where+instance (Num f, Ord f, Show f) => ToolShed.SelfValidate.SelfValidator (DiscreteDistribution f)	where 	getErrors (PoissonDistribution lambda)	= ToolShed.SelfValidate.extractErrors [(lambda < 0, "Negative lambda=" ++ show lambda ++ ".")]  {- |@@ -66,7 +66,7 @@  	* <http://en.wikipedia.org/wiki/Box%E2%80%93Muller_transform>. -}-boxMullerTransform :: (Floating f, Ord f)+boxMullerTransform :: (Floating f, Ord f, Show f) 	=> (f, f)	-- ^ Independent, /uniformly distributed/ random numbers, which must be within the /semi-closed unit interval/, /(0, 1]/. 	-> (f, f)	-- ^ Independent, /normally distributed/ random numbers, with standardized /mean/=0 and /variance/=1. boxMullerTransform cartesian@@ -93,7 +93,12 @@  	* <http://en.wikipedia.org/wiki/Normal_distribution>, <http://mathworld.wolfram.com/NormalDistribution.html>. -}-generateStandardizedNormalDistribution :: (System.Random.RandomGen randomGen, RealFloat f, System.Random.Random f) => randomGen -> [f]+generateStandardizedNormalDistribution :: (+	RealFloat		f,+	Show			f,+	System.Random.Random	f,+	System.Random.RandomGen	randomGen+ ) => randomGen -> [f] generateStandardizedNormalDistribution	= ToolShed.Data.List.linearise . uncurry (zipWith $ curry boxMullerTransform) . ToolShed.Data.Pair.mirror ( 	System.Random.randomRs (minPositiveFloat undefined, 1)  ) . System.Random.split@@ -109,7 +114,12 @@ 	the generated population will only tend towards the requested /mean/ and /variance/ of, as the sample-size tends towards infinity. 	Whilst one could arrange for these criteria to be precisely met for any sample-size, the sample would lose a degree of randomness as a result. -}-generateContinuousPopulation :: (RealFloat f, System.Random.Random f, System.Random.RandomGen randomGen)+generateContinuousPopulation :: (+	RealFloat		f,+	Show			f,+	System.Random.Random	f,+	System.Random.RandomGen	randomGen+ ) 	=> Int	-- ^ number of items. 	-> ContinuousDistribution f 	-> randomGen	-- ^ A generator of /uniformly distributed/ random numbers.@@ -136,10 +146,11 @@ 		so for large /lambda/, this implementation returns the appropriate 'NormalDistribution', which is similar for large /lambda/. -} generatePoissonDistribution :: (+	Integral		events, 	RealFloat		lambda,+	Show			lambda, 	System.Random.Random	lambda,-	System.Random.RandomGen	randomGen,-	Integral		events+	System.Random.RandomGen	randomGen  ) 	=> lambda	-- ^ Defines the required approximate value of both /mean/ and /variance/. 	-> randomGen@@ -163,6 +174,7 @@ generateDiscretePopulation :: ( 	Ord			f, 	RealFloat		f,+	Show			f, 	System.Random.Random	f, 	System.Random.RandomGen	randomGen, 	Integral		events

@@ -58,7 +58,13 @@ 	* The conversion to 'Char' can only succeed where printable and intelligible characters exist to represent all digits in the chosen base; 	which in practice means @(-36 <= base <= 36)@. -}-toBase :: (Data.Array.IArray.Ix decimal, Integral base, Integral decimal, Show decimal, Show base) => base -> decimal -> String+toBase :: (+	Data.Array.IArray.Ix	decimal,+	Integral		base,+	Integral		decimal,+	Show			base,+	Show			decimal+ ) => base -> decimal -> String toBase 10 decimal	= show decimal	--Base unchanged. toBase _ 0		= "0"		--Zero has the same representation in any base. toBase base decimal@@ -88,7 +94,12 @@  	* Both negative numbers and negative bases are permissible. -}-fromBase :: (Integral base, Integral decimal, Read decimal, Show base) => base -> String -> decimal+fromBase :: (+	Integral	base,+	Integral	decimal,+	Read		decimal,+	Show		base+ ) => base -> String -> decimal fromBase 10 s	= read s	--Base unchanged. fromBase _ "0"	= 0		--Zero has the same representation in any base. fromBase base s@@ -108,11 +119,21 @@  	* <http://en.wikipedia.org/wiki/Digit_sum>. -}-digitSum :: (Data.Array.IArray.Ix decimal, Integral base, Integral decimal, Show decimal) => base -> decimal -> decimal+digitSum :: (+	Data.Array.IArray.Ix	decimal,+	Integral		base,+	Integral		decimal,+	Show			base,+	Show			decimal+ ) => base -> decimal -> decimal digitSum 10	= fromIntegral . foldr ((+) . Data.Char.digitToInt) 0 . show digitSum base	= sum . Data.Maybe.mapMaybe (`lookup` decodes) . toBase base  -- | <http://en.wikipedia.org/wiki/Digital_root>.-digitalRoot :: (Data.Array.IArray.Ix decimal, Integral decimal) => decimal -> decimal+digitalRoot :: (+	Data.Array.IArray.Ix	decimal,+	Integral		decimal,+	Show			decimal+ ) => decimal -> decimal digitalRoot	= until (<= 9) (digitSum (10 :: Int)) 

@@ -53,14 +53,14 @@  -- | Defines the methods expected of a /square-root/ algorithm. class Algorithmic algorithm	where-	squareRootFrom	:: Real operand+	squareRootFrom	:: (Real operand, Show operand) 		=> algorithm 		-> Estimate			-- ^ An initial estimate from which to start. 		-> Math.Precision.DecimalDigits	-- ^ The required precision. 		-> operand			-- ^ The value for which to find the /square-root/. 		-> Result			-- ^ Returns an improved estimate of the /square-root/, found using the specified algorithm, accurate to at least the required number of decimal digits. -	squareRoot	:: Real operand+	squareRoot	:: (Real operand, Show operand) 		=> algorithm 		-> Math.Precision.DecimalDigits	-- ^ The required precision. 		-> operand			-- ^ The value for which to find the /square-root/.@@ -82,7 +82,7 @@ rSqrt	= sqrt . realToFrac  -- | Uses 'Double'-precision floating-point arithmetic, to obtain an initial estimate for the /square-root/, and its accuracy.-getEstimate :: Real operand => operand -> Estimate+getEstimate :: (Real operand, Show operand) => operand -> Estimate getEstimate y 	| y < 0		= error $ "Factory.Math.SquareRoot.getEstimate:\tthere's no real square-root of " ++ show y 	| otherwise	= (Math.Precision.simplify decimalDigits {-doubles performance by roughly length of the Rational representation-} . toRational $ rSqrt y, decimalDigits)

@@ -58,7 +58,12 @@  	* Should the caller define the result-type as 'Data.Ratio.Rational', then it will be free from rounding-errors. -}-getDispersionFromMean :: (Data.Foldable.Foldable f, Functor f, Real r, Fractional result) => (Data.Ratio.Rational -> Data.Ratio.Rational) -> f r -> result+getDispersionFromMean :: (+	Data.Foldable.Foldable	f,+	Fractional		result,+	Functor			f,+	Real			r+ ) => (Data.Ratio.Rational -> Data.Ratio.Rational) -> f r -> result getDispersionFromMean weight x	= getMean $ fmap (weight . (+ negate mean) . realToFrac) x	where 	mean :: Data.Ratio.Rational 	mean	= getMean x@@ -68,11 +73,21 @@  	* Should the caller define the result-type as 'Data.Ratio.Rational', then it will be free from rounding-errors. -}-getVariance :: (Data.Foldable.Foldable f, Functor f, Real r, Fractional variance) => f r -> variance+getVariance :: (+	Data.Foldable.Foldable	f,+	Fractional		variance,+	Functor			f,+	Real			r+ ) => f r -> variance getVariance	= getDispersionFromMean Math.Power.square  -- | Determines the /standard-deviation/ of the specified numbers; <http://en.wikipedia.org/wiki/Standard_deviation>.-getStandardDeviation :: (Data.Foldable.Foldable f, Functor f, Real r, Floating result) => f r -> result+getStandardDeviation :: (+	Data.Foldable.Foldable	f,+	Floating		result,+	Functor			f,+	Real			r+ ) => f r -> result getStandardDeviation	= sqrt . getVariance  {- |@@ -80,11 +95,22 @@  	* Should the caller define the result-type as 'Data.Ratio.Rational', then it will be free from rounding-errors. -}-getAverageAbsoluteDeviation :: (Data.Foldable.Foldable f, Functor f, Real r, Fractional result) => f r -> result+getAverageAbsoluteDeviation :: (+	Data.Foldable.Foldable	f,+	Fractional		result,+	Functor			f,+	Real			r+ ) => f r -> result getAverageAbsoluteDeviation	= getDispersionFromMean abs  -- | Determines the /coefficient-of-variance/ of the specified numbers; <http://en.wikipedia.org/wiki/Coefficient_of_variation>.-getCoefficientOfVariance :: (Data.Foldable.Foldable f, Functor f, Real r, Floating result) => f r -> result+getCoefficientOfVariance :: (+	Data.Foldable.Foldable	f,+	Eq			result,+	Floating		result,+	Functor			f,+	Real			r+ ) => f r -> result getCoefficientOfVariance l 	| mean == 0	= error "Factory.Math.Statistics.getCoefficientOfVariance:\tundefined if mean is zero." 	| otherwise	= getStandardDeviation l / abs mean@@ -92,7 +118,7 @@ 		mean	= getMean l  -- | The number of unordered /combinations/ of /r/ objects taken from /n/; <http://en.wikipedia.org/wiki/Combination>.-nCr :: (Math.Factorial.Algorithmic factorialAlgorithm, Integral i)+nCr :: (Math.Factorial.Algorithmic factorialAlgorithm, Integral i, Show i) 	=> factorialAlgorithm 	-> i	-- ^ The total number of items from which to select. 	-> i	-- ^ The number of items in a sample.@@ -110,7 +136,7 @@ 		denominator		= Math.Factorial.factorial factorialAlgorithm smaller  -- | The number of /permutations/ of /r/ objects taken from /n/; <http://en.wikipedia.org/wiki/Permutations>.-nPr :: Integral i+nPr :: (Integral i, Show i) 	=> i	-- ^ The total number of items from which to select. 	-> i	-- ^ The number of items in a sample. 	-> i	-- ^ The number of permutations.

@@ -34,7 +34,12 @@ import qualified	ToolShed.System.TimePure  -- | Measures the CPU-time required by 'Math.Factorial.factorial'.-factorialPerformance :: (Math.Factorial.Algorithmic algorithm, Control.DeepSeq.NFData i, Integral i) => algorithm -> i -> IO (Double, i)+factorialPerformance :: (+	Control.DeepSeq.NFData		i,+	Integral			i,+	Math.Factorial.Algorithmic	algorithm,+	Show				i+ ) => algorithm -> i -> IO (Double, i) factorialPerformance algorithm	= ToolShed.System.TimePure.getCPUSeconds . Math.Factorial.factorial algorithm  -- | Measures the CPU-time required by a naive implementation.

@@ -31,7 +31,7 @@ import qualified	ToolShed.System.TimePure  -- | Measures the CPU-time required by 'Math.Hyperoperation.hyperoperation'.-hyperoperationPerformance :: Integral rank => rank -> Math.Hyperoperation.Base -> Math.Hyperoperation.HyperExponent -> IO (Double, Integer)+hyperoperationPerformance :: (Integral rank, Show rank) => rank -> Math.Hyperoperation.Base -> Math.Hyperoperation.HyperExponent -> IO (Double, Integer) hyperoperationPerformance rank base	= ToolShed.System.TimePure.getCPUSeconds . Math.Hyperoperation.hyperoperation rank base  {- |@@ -57,7 +57,7 @@  	* CAVEAT: nothing is returned, since the result is printed ... and it never terminates. -}-hyperoperationPerformanceGraphExponent :: Integral rank+hyperoperationPerformanceGraphExponent :: (Integral rank, Show rank) 	=> Bool	-- ^ Verbose. 	-> rank 	-> Math.Hyperoperation.Base

@@ -39,7 +39,7 @@ 	| otherwise	= ToolShed.System.TimePure.getCPUSeconds . take i $ Math.Primality.carmichaelNumbers primalityAlgorithm  -- | Measures the CPU-time required to determine whether the specified integer is prime, which is returned together with the Boolean result.-isPrimePerformance :: (Control.DeepSeq.NFData i, Integral i) => Math.Primality.Algorithmic primalityAlgorithm => primalityAlgorithm -> i -> IO (Double, Bool)+isPrimePerformance :: (Control.DeepSeq.NFData i, Integral i, Show i) => Math.Primality.Algorithmic primalityAlgorithm => primalityAlgorithm -> i -> IO (Double, Bool) isPrimePerformance primalityAlgorithm	= ToolShed.System.TimePure.getCPUSeconds . Math.Primality.isPrime primalityAlgorithm  {- |

@@ -31,5 +31,10 @@ import qualified	ToolShed.System.TimePure  -- | Measures the CPU-time required by 'Math.Primes.primes', to find the specified prime.-primesPerformance :: (Math.Primes.Algorithmic algorithm, Control.DeepSeq.NFData i, Data.Array.IArray.Ix i, Integral i) => algorithm -> Int -> IO (Double, i)+primesPerformance :: (+	Control.DeepSeq.NFData	i,+	Data.Array.IArray.Ix	i,+	Math.Primes.Algorithmic	algorithm,+	Integral		i+ ) => algorithm -> Int -> IO (Double, i) primesPerformance algorithm	= ToolShed.System.TimePure.getCPUSeconds . (Math.Primes.primes algorithm !!)

@@ -32,7 +32,11 @@ import qualified	ToolShed.System.TimePure  -- | Measures the CPU-time required by 'Math.SquareRoot.squareRootFrom', which is returned together with the approximate rational result.-squareRootPerformance :: (Math.SquareRoot.Algorithmic algorithm, Real operand) => algorithm -> operand -> Math.Precision.DecimalDigits -> IO (Double, Math.SquareRoot.Result)+squareRootPerformance :: (+	Math.SquareRoot.Algorithmic	algorithm,+	Real				operand,+	Show				operand+ ) => algorithm -> operand -> Math.Precision.DecimalDigits -> IO (Double, Math.SquareRoot.Result) squareRootPerformance algorithm operand requiredDecimalDigits = ToolShed.System.TimePure.getCPUSeconds $ Math.SquareRoot.squareRoot algorithm requiredDecimalDigits operand  {- |@@ -44,8 +48,9 @@ squareRootPerformanceGraph :: ( 	Math.SquareRoot.Algorithmic	algorithm, 	Math.SquareRoot.Iterator	algorithm,+	Real				operand, 	Show				algorithm,-	Real				operand+	Show				operand  ) => algorithm -> operand -> IO () squareRootPerformanceGraph algorithm operand	= mapM_ ( 	\requiredDecimalDigits	-> putStrLn . (

@@ -31,7 +31,12 @@ import qualified	ToolShed.System.TimePure  -- | Measures the CPU-time required by 'Math.Statistics.nCr'.-nCrPerformance :: (Math.Factorial.Algorithmic factorialAlgorithm, Control.DeepSeq.NFData i, Integral i)+nCrPerformance :: (+	Control.DeepSeq.NFData		i,+	Integral			i,+	Math.Factorial.Algorithmic	factorialAlgorithm,+	Show				i+ ) 	=> factorialAlgorithm 	-> i	-- ^ The total number from which to select. 	-> i	-- ^ The number of items in a sample.

@@ -39,10 +39,12 @@ import qualified	Test.QuickCheck  instance (-	Test.QuickCheck.Arbitrary	c, 	Integral			c,+	Integral			e,+	Test.QuickCheck.Arbitrary	c, 	Test.QuickCheck.Arbitrary	e,-	Integral			e+	Show				c,+	Show				e  ) => Test.QuickCheck.Arbitrary (Data.MonicPolynomial.MonicPolynomial c e)	where 	arbitrary	= do 		polynomial	<- Test.QuickCheck.arbitrary

@@ -54,7 +54,7 @@ 	coarbitrary	= undefined	--CAVEAT: stops warnings from ghc. #endif -isPrime :: (Control.DeepSeq.NFData i, Integral i) => i -> Bool+isPrime :: (Control.DeepSeq.NFData i, Integral i, Show i) => i -> Bool isPrime	= Math.Primality.isPrime primalityAlgorithm	where 	primalityAlgorithm :: Math.Implementations.Primality.Algorithm Math.Implementations.PrimeFactorisation.Algorithm 	primalityAlgorithm	= ToolShed.Defaultable.defaultValue