Copyright	(c) 2014 Bryan O'Sullivan
License	BSD3
Maintainer	[email protected]
Stability	experimental
Portability	portable
Safe Haskell	None
Language	Haskell2010

Numeric.Sum

Contents

Summation type class
- Usage
Kahan-Babuška-Neumaier summation
Order-2 Kahan-Babuška summation
Less desirable approaches
- Kahan summation
- Pairwise summation
References

Description

Functions for summing floating point numbers more accurately than the naive sum function and its counterparts in the vector package and elsewhere.

When used with floating point numbers, in the worst case, the sum function accumulates numeric error at a rate proportional to the number of values being summed. The algorithms in this module implement different methods of /compensated summation/, which reduce the accumulation of numeric error so that it either grows much more slowly than the number of inputs (e.g. logarithmically), or remains constant.

Synopsis

class Summation s where
- zero :: s
- add :: s -> Double -> s
- sum :: Foldable f => (s -> Double) -> f Double -> Double
sumVector :: (Vector v Double, Summation s) => (s -> Double) -> v Double -> Double
data KBNSum = KBNSum !Double !Double
kbn :: KBNSum -> Double
data KB2Sum = KB2Sum !Double !Double !Double
kb2 :: KB2Sum -> Double
data KahanSum = KahanSum !Double !Double
kahan :: KahanSum -> Double
pairwiseSum :: Vector v Double => v Double -> Double

Summation type class

class Summation s where Source #

A class for summation of floating point numbers.

Minimal complete definition

zero, add

Methods

zero :: s Source #

The identity for summation.

add :: s -> Double -> s Source #

Add a value to a sum.

sum :: Foldable f => (s -> Double) -> f Double -> Double Source #

Sum a collection of values.

Example: foo = sum kbn [1,2,3]

Instances

Instances details

Summation KB2Sum Source #
Instance details Defined in Numeric.Sum Methods zero :: KB2Sum Source # add :: KB2Sum -> Double -> KB2Sum Source # sum :: Foldable f => (KB2Sum -> Double) -> f Double -> Double Source #
Summation KBNSum Source #
Instance details Defined in Numeric.Sum Methods zero :: KBNSum Source # add :: KBNSum -> Double -> KBNSum Source # sum :: Foldable f => (KBNSum -> Double) -> f Double -> Double Source #
Summation KahanSum Source #
Instance details Defined in Numeric.Sum Methods zero :: KahanSum Source # add :: KahanSum -> Double -> KahanSum Source # sum :: Foldable f => (KahanSum -> Double) -> f Double -> Double Source #
Summation Double Source #
Instance details Defined in Numeric.Sum Methods zero :: Double Source # add :: Double -> Double -> Double Source # sum :: Foldable f => (Double -> Double) -> f Double -> Double Source #

sumVector :: (Vector v Double, Summation s) => (s -> Double) -> v Double -> Double Source #

O(n) Sum a vector of values.

Usage

Most of these summation algorithms are intended to be used via the Summation typeclass interface. Explicit type annotations should not be necessary, as the use of a function such as kbn or kb2 to extract the final sum out of a Summation instance gives the compiler enough information to determine the precise type of summation algorithm to use.

As an example, here is a (somewhat silly) function that manually computes the sum of elements in a list.

sillySumList :: [Double] -> Double
sillySumList = loop zero
  where loop s []     = kbn s
        loop s (x:xs) = seq s' loop s' xs
          where s'    = add s x

In most instances, you can simply use the much more general sum function instead of writing a summation function by hand.

-- Avoid ambiguity around which sum function we are using.
import Prelude hiding (sum)
--
betterSumList :: [Double] -> Double
betterSumList xs = sum kbn xs

Kahan-Babuška-Neumaier summation

data KBNSum Source #

Kahan-Babuška-Neumaier summation. This is a little more computationally costly than plain Kahan summation, but is always at least as accurate.

Constructors

KBNSum !Double !Double

Instances

Instances details

Data KBNSum Source #
Instance details Defined in Numeric.Sum Methods gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> KBNSum -> c KBNSum # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c KBNSum # toConstr :: KBNSum -> Constr # dataTypeOf :: KBNSum -> DataType # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c KBNSum) # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c KBNSum) # gmapT :: (forall b. Data b => b -> b) -> KBNSum -> KBNSum # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> KBNSum -> r # gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> KBNSum -> r # gmapQ :: (forall d. Data d => d -> u) -> KBNSum -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> KBNSum -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> KBNSum -> m KBNSum # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> KBNSum -> m KBNSum # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> KBNSum -> m KBNSum #
Monoid KBNSum Source #	Since: 0.3.0.0
Instance details Defined in Numeric.Sum Methods mempty :: KBNSum # mappend :: KBNSum -> KBNSum -> KBNSum # mconcat :: [KBNSum] -> KBNSum #
Semigroup KBNSum Source #	Since: 0.3.0.0
Instance details Defined in Numeric.Sum Methods (<>) :: KBNSum -> KBNSum -> KBNSum # sconcat :: NonEmpty KBNSum -> KBNSum # stimes :: Integral b => b -> KBNSum -> KBNSum #
Show KBNSum Source #
Instance details Defined in Numeric.Sum Methods showsPrec :: Int -> KBNSum -> ShowS # show :: KBNSum -> String # showList :: [KBNSum] -> ShowS #
NFData KBNSum Source #
Instance details Defined in Numeric.Sum Methods rnf :: KBNSum -> () #
Eq KBNSum Source #
Instance details Defined in Numeric.Sum Methods (==) :: KBNSum -> KBNSum -> Bool # (/=) :: KBNSum -> KBNSum -> Bool #
Summation KBNSum Source #
Instance details Defined in Numeric.Sum Methods zero :: KBNSum Source # add :: KBNSum -> Double -> KBNSum Source # sum :: Foldable f => (KBNSum -> Double) -> f Double -> Double Source #
Unbox KBNSum Source #
Instance details Defined in Numeric.Sum
Vector Vector KBNSum Source #
Instance details Defined in Numeric.Sum Methods basicUnsafeFreeze :: Mutable Vector s KBNSum -> ST s (Vector KBNSum) basicUnsafeThaw :: Vector KBNSum -> ST s (Mutable Vector s KBNSum) basicLength :: Vector KBNSum -> Int basicUnsafeSlice :: Int -> Int -> Vector KBNSum -> Vector KBNSum basicUnsafeIndexM :: Vector KBNSum -> Int -> Box KBNSum basicUnsafeCopy :: Mutable Vector s KBNSum -> Vector KBNSum -> ST s () elemseq :: Vector KBNSum -> KBNSum -> b -> b
MVector MVector KBNSum Source #
Instance details Defined in Numeric.Sum Methods basicLength :: MVector s KBNSum -> Int basicUnsafeSlice :: Int -> Int -> MVector s KBNSum -> MVector s KBNSum basicOverlaps :: MVector s KBNSum -> MVector s KBNSum -> Bool basicUnsafeNew :: Int -> ST s (MVector s KBNSum) basicInitialize :: MVector s KBNSum -> ST s () basicUnsafeReplicate :: Int -> KBNSum -> ST s (MVector s KBNSum) basicUnsafeRead :: MVector s KBNSum -> Int -> ST s KBNSum basicUnsafeWrite :: MVector s KBNSum -> Int -> KBNSum -> ST s () basicClear :: MVector s KBNSum -> ST s () basicSet :: MVector s KBNSum -> KBNSum -> ST s () basicUnsafeCopy :: MVector s KBNSum -> MVector s KBNSum -> ST s () basicUnsafeMove :: MVector s KBNSum -> MVector s KBNSum -> ST s () basicUnsafeGrow :: MVector s KBNSum -> Int -> ST s (MVector s KBNSum)
newtype Vector KBNSum Source #
Instance details Defined in Numeric.Sum newtype Vector KBNSum = V_KBNSum (Vector (Double, Double))
newtype MVector s KBNSum Source #
Instance details Defined in Numeric.Sum newtype MVector s KBNSum = MV_KBNSum (MVector s (Double, Double))

kbn :: KBNSum -> Double Source #

Return the result of a Kahan-Babuška-Neumaier sum.

Order-2 Kahan-Babuška summation

data KB2Sum Source #

Second-order Kahan-Babuška summation. This is more computationally costly than Kahan-Babuška-Neumaier summation, running at about a third the speed. Its advantage is that it can lose less precision (in admittedly obscure cases).

This method compensates for error in both the sum and the first-order compensation term, hence the use of "second order" in the name.

Constructors

KB2Sum !Double !Double !Double

Instances

Instances details

Data KB2Sum Source #
Instance details Defined in Numeric.Sum Methods gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> KB2Sum -> c KB2Sum # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c KB2Sum # toConstr :: KB2Sum -> Constr # dataTypeOf :: KB2Sum -> DataType # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c KB2Sum) # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c KB2Sum) # gmapT :: (forall b. Data b => b -> b) -> KB2Sum -> KB2Sum # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> KB2Sum -> r # gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> KB2Sum -> r # gmapQ :: (forall d. Data d => d -> u) -> KB2Sum -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> KB2Sum -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> KB2Sum -> m KB2Sum # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> KB2Sum -> m KB2Sum # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> KB2Sum -> m KB2Sum #
Monoid KB2Sum Source #	Since: 0.3.0.0
Instance details Defined in Numeric.Sum Methods mempty :: KB2Sum # mappend :: KB2Sum -> KB2Sum -> KB2Sum # mconcat :: [KB2Sum] -> KB2Sum #
Semigroup KB2Sum Source #	Since: 0.3.0.0
Instance details Defined in Numeric.Sum Methods (<>) :: KB2Sum -> KB2Sum -> KB2Sum # sconcat :: NonEmpty KB2Sum -> KB2Sum # stimes :: Integral b => b -> KB2Sum -> KB2Sum #
Show KB2Sum Source #
Instance details Defined in Numeric.Sum Methods showsPrec :: Int -> KB2Sum -> ShowS # show :: KB2Sum -> String # showList :: [KB2Sum] -> ShowS #
NFData KB2Sum Source #
Instance details Defined in Numeric.Sum Methods rnf :: KB2Sum -> () #
Eq KB2Sum Source #
Instance details Defined in Numeric.Sum Methods (==) :: KB2Sum -> KB2Sum -> Bool # (/=) :: KB2Sum -> KB2Sum -> Bool #
Summation KB2Sum Source #
Instance details Defined in Numeric.Sum Methods zero :: KB2Sum Source # add :: KB2Sum -> Double -> KB2Sum Source # sum :: Foldable f => (KB2Sum -> Double) -> f Double -> Double Source #
Unbox KB2Sum Source #
Instance details Defined in Numeric.Sum
Vector Vector KB2Sum Source #
Instance details Defined in Numeric.Sum Methods basicUnsafeFreeze :: Mutable Vector s KB2Sum -> ST s (Vector KB2Sum) basicUnsafeThaw :: Vector KB2Sum -> ST s (Mutable Vector s KB2Sum) basicLength :: Vector KB2Sum -> Int basicUnsafeSlice :: Int -> Int -> Vector KB2Sum -> Vector KB2Sum basicUnsafeIndexM :: Vector KB2Sum -> Int -> Box KB2Sum basicUnsafeCopy :: Mutable Vector s KB2Sum -> Vector KB2Sum -> ST s () elemseq :: Vector KB2Sum -> KB2Sum -> b -> b
MVector MVector KB2Sum Source #
Instance details Defined in Numeric.Sum Methods basicLength :: MVector s KB2Sum -> Int basicUnsafeSlice :: Int -> Int -> MVector s KB2Sum -> MVector s KB2Sum basicOverlaps :: MVector s KB2Sum -> MVector s KB2Sum -> Bool basicUnsafeNew :: Int -> ST s (MVector s KB2Sum) basicInitialize :: MVector s KB2Sum -> ST s () basicUnsafeReplicate :: Int -> KB2Sum -> ST s (MVector s KB2Sum) basicUnsafeRead :: MVector s KB2Sum -> Int -> ST s KB2Sum basicUnsafeWrite :: MVector s KB2Sum -> Int -> KB2Sum -> ST s () basicClear :: MVector s KB2Sum -> ST s () basicSet :: MVector s KB2Sum -> KB2Sum -> ST s () basicUnsafeCopy :: MVector s KB2Sum -> MVector s KB2Sum -> ST s () basicUnsafeMove :: MVector s KB2Sum -> MVector s KB2Sum -> ST s () basicUnsafeGrow :: MVector s KB2Sum -> Int -> ST s (MVector s KB2Sum)
newtype Vector KB2Sum Source #
Instance details Defined in Numeric.Sum newtype Vector KB2Sum = V_KB2Sum (Vector (Double, Double, Double))
newtype MVector s KB2Sum Source #
Instance details Defined in Numeric.Sum newtype MVector s KB2Sum = MV_KB2Sum (MVector s (Double, Double, Double))

kb2 :: KB2Sum -> Double Source #

Return the result of an order-2 Kahan-Babuška sum.

Less desirable approaches

Kahan summation

data KahanSum Source #

Kahan summation. This is the least accurate of the compensated summation methods. In practice, it only beats naive summation for inputs with large magnitude. Kahan summation can be less accurate than naive summation for small-magnitude inputs.

This summation method is included for completeness. Its use is not recommended. In practice, KBNSum is both 30% faster and more accurate.

Constructors

KahanSum !Double !Double

Instances

Instances details

Data KahanSum Source #
Instance details Defined in Numeric.Sum Methods gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> KahanSum -> c KahanSum # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c KahanSum # toConstr :: KahanSum -> Constr # dataTypeOf :: KahanSum -> DataType # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c KahanSum) # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c KahanSum) # gmapT :: (forall b. Data b => b -> b) -> KahanSum -> KahanSum # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> KahanSum -> r # gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> KahanSum -> r # gmapQ :: (forall d. Data d => d -> u) -> KahanSum -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> KahanSum -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> KahanSum -> m KahanSum # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> KahanSum -> m KahanSum # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> KahanSum -> m KahanSum #
Monoid KahanSum Source #	Since: 0.3.0.0
Instance details Defined in Numeric.Sum Methods mempty :: KahanSum # mappend :: KahanSum -> KahanSum -> KahanSum # mconcat :: [KahanSum] -> KahanSum #
Semigroup KahanSum Source #	Since: 0.3.0.0
Instance details Defined in Numeric.Sum Methods (<>) :: KahanSum -> KahanSum -> KahanSum # sconcat :: NonEmpty KahanSum -> KahanSum # stimes :: Integral b => b -> KahanSum -> KahanSum #
Show KahanSum Source #
Instance details Defined in Numeric.Sum Methods showsPrec :: Int -> KahanSum -> ShowS # show :: KahanSum -> String # showList :: [KahanSum] -> ShowS #
NFData KahanSum Source #
Instance details Defined in Numeric.Sum Methods rnf :: KahanSum -> () #
Eq KahanSum Source #
Instance details Defined in Numeric.Sum Methods (==) :: KahanSum -> KahanSum -> Bool # (/=) :: KahanSum -> KahanSum -> Bool #
Summation KahanSum Source #
Instance details Defined in Numeric.Sum Methods zero :: KahanSum Source # add :: KahanSum -> Double -> KahanSum Source # sum :: Foldable f => (KahanSum -> Double) -> f Double -> Double Source #
Unbox KahanSum Source #
Instance details Defined in Numeric.Sum
Vector Vector KahanSum Source #
Instance details Defined in Numeric.Sum Methods basicUnsafeFreeze :: Mutable Vector s KahanSum -> ST s (Vector KahanSum) basicUnsafeThaw :: Vector KahanSum -> ST s (Mutable Vector s KahanSum) basicLength :: Vector KahanSum -> Int basicUnsafeSlice :: Int -> Int -> Vector KahanSum -> Vector KahanSum basicUnsafeIndexM :: Vector KahanSum -> Int -> Box KahanSum basicUnsafeCopy :: Mutable Vector s KahanSum -> Vector KahanSum -> ST s () elemseq :: Vector KahanSum -> KahanSum -> b -> b
MVector MVector KahanSum Source #
Instance details Defined in Numeric.Sum Methods basicLength :: MVector s KahanSum -> Int basicUnsafeSlice :: Int -> Int -> MVector s KahanSum -> MVector s KahanSum basicOverlaps :: MVector s KahanSum -> MVector s KahanSum -> Bool basicUnsafeNew :: Int -> ST s (MVector s KahanSum) basicInitialize :: MVector s KahanSum -> ST s () basicUnsafeReplicate :: Int -> KahanSum -> ST s (MVector s KahanSum) basicUnsafeRead :: MVector s KahanSum -> Int -> ST s KahanSum basicUnsafeWrite :: MVector s KahanSum -> Int -> KahanSum -> ST s () basicClear :: MVector s KahanSum -> ST s () basicSet :: MVector s KahanSum -> KahanSum -> ST s () basicUnsafeCopy :: MVector s KahanSum -> MVector s KahanSum -> ST s () basicUnsafeMove :: MVector s KahanSum -> MVector s KahanSum -> ST s () basicUnsafeGrow :: MVector s KahanSum -> Int -> ST s (MVector s KahanSum)
newtype Vector KahanSum Source #
Instance details Defined in Numeric.Sum newtype Vector KahanSum = V_KahanSum (Vector (Double, Double))
newtype MVector s KahanSum Source #
Instance details Defined in Numeric.Sum newtype MVector s KahanSum = MV_KahanSum (MVector s (Double, Double))

kahan :: KahanSum -> Double Source #

Return the result of a Kahan sum.

Pairwise summation

pairwiseSum :: Vector v Double => v Double -> Double Source #

O(n) Sum a vector of values using pairwise summation.

This approach is perhaps 10% faster than KBNSum, but has poorer bounds on its error growth. Instead of having roughly constant error regardless of the size of the input vector, in the worst case its accumulated error grows with O(log n).

References

Kahan, W. (1965), Further remarks on reducing truncation errors. Communications of the ACM 8(1):40.
Neumaier, A. (1974), Rundungsfehleranalyse einiger Verfahren zur Summation endlicher Summen. Zeitschrift für Angewandte Mathematik und Mechanik 54:39–51.
Klein, A. (2006), A Generalized Kahan-Babuška-Summation-Algorithm. Computing 76(3):279-293.
Higham, N.J. (1993), The accuracy of floating point summation. SIAM Journal on Scientific Computing 14(4):783–799.