HERA-0.2ContentsIndex
Data.Number.Ball
Synopsis
data Ball = Ball {
center :: !Dyadic
radius :: !Dyadic
}
makeA :: Precision -> Dyadic -> Dyadic -> Ball
make :: Dyadic -> Dyadic -> Ball
normalizeBall :: Precision -> Ball -> Ball
lower :: Precision -> Ball -> Dyadic
upper :: Precision -> Ball -> Dyadic
lower_ :: Ball -> Dyadic
upper_ :: Ball -> Dyadic
sgnLower :: Ball -> Int
sgnUpper :: Ball -> Int
width :: Ball -> Dyadic
compareB :: Ball -> Ball -> POrdering
below :: Ball -> Ball -> Bool
contains :: Ball -> Dyadic -> Bool
intersectA :: Monad m => Precision -> Ball -> Ball -> m Ball
intersect :: Monad m => Ball -> Ball -> m Ball
add :: Precision -> Ball -> Ball -> Ball
sub :: Precision -> Ball -> Ball -> Ball
neg :: Precision -> Ball -> Ball
absB :: Precision -> Ball -> Ball
mul :: Precision -> Ball -> Ball -> Ball
div :: Monad m => Precision -> Ball -> Ball -> m Ball
sqrt :: Monad m => Precision -> Ball -> m Ball
exp :: Precision -> Ball -> Ball
log :: Monad m => Precision -> Ball -> m Ball
maxB :: Precision -> Ball -> Ball -> Ball
minB :: Precision -> Ball -> Ball -> Ball
fromDyadic :: Precision -> Dyadic -> Ball
fromString :: Precision -> String -> Ball
fromInt :: Precision -> Int -> Ball
fromWord :: Precision -> Word -> Ball
Documentation
data Ball
Ball represents a closed interval [center-radius, center+radius]
Constructors
Ball
center :: !Dyadiccenter of the ball
radius :: !Dyadicradius of the ball
show/hide Instances
Show Ball
makeA
:: Precisiondesired precision of the center
-> Dyadicleft endpoint
-> Dyadicright endpoint
-> Ball
Make a ball from endpoints
make
:: Dyadicleft endpoint
-> Dyadicright endpoint
-> Ball
Make a ball from endpoints so that no precision is lost.
normalizeBall :: Precision -> Ball -> Ball
Normalize the given ball's center to the specified precision. Resulting ball might be larger.
lower :: Precision -> Ball -> Dyadic
Lower endpoint of the ball rounded down to specified precision.
upper :: Precision -> Ball -> Dyadic
Upper endpoint of the ball rounded up to specified precision.
lower_ :: Ball -> Dyadic
Lower endpoint with precision of the center
upper_ :: Ball -> Dyadic
Upper endpoint with precision of the center
sgnLower :: Ball -> Int
Sign of lower endpoint of the ball. This should be faster than using signum (center b - radius b)
sgnUpper :: Ball -> Int
Analogous to sgnLower.
width :: Ball -> Dyadic
Upper bound on the width of the ball. 2 * radius b rounded up.
compareB :: Ball -> Ball -> POrdering

Compare two balls.

  • if upper a < lower b then Less
  • if upper b < lower a then Greater
  • otherwise balls are incomparable.
below :: Ball -> Ball -> Bool
Check if second ball is included in the first
contains :: Ball -> Dyadic -> Bool
Check if dyadic is element of the ball.
intersectA
:: Monad m
=> Precisionprecision of the resulting ball's center
-> Ball
-> Ball
-> m Ball
Returns an intersection of two balls. If balls are disjoint then computation fails with fail.
intersect :: Monad m => Ball -> Ball -> m Ball
Intersection of two balls exactly (no precision is lost).
add :: Precision -> Ball -> Ball -> Ball

Addition of two balls. 

  •  center = center a + center b
  •  radius = radius a + radius b

Rounding errors are added to the radius.

sub :: Precision -> Ball -> Ball -> Ball

Subtraction of two balls. 

  •  center = center a - center b
  •  radius = radius a + radius b

Rounding errors are added to the radius.

neg :: Precision -> Ball -> Ball

Negation of the ball.

  • center = - center b rounded to specified precision.
  • radius is only modified for the rounding error.
absB :: Precision -> Ball -> Ball
mul :: Precision -> Ball -> Ball -> Ball

Multiplication of two balls. (centers of both balls are assumed positive)

  • If none of the balls contains 0 then 
 center = center a * center b + radius a * radius b
 radius = center a * radius b + radius a * center b
  • If one of the operands (left) contains 0 
 center = center a * upper b
 radius = radius a * upper b
  • If both of the balls contain 0 
 lower =  min ((lower a) * (upper b)) ((lower b) * (upper a))
 upper =  max ((lower a) * (lower b)) ((upper b) * (upper a))

Rounding errors are added to the radius.

div :: Monad m => Precision -> Ball -> Ball -> m Ball

Division of two balls

  • If radius is "large" then divide endpoints and makeA a ball from them.
  • If radius is "small" then division can be optimized 
  •  center = center a / center b
  • (radius = radius a * center b + center a * radius b) / (center b * center b) + 2 * 2 ^ (e1 - e2 - p) where p is precision of the result, e1 = getExp c1, e2 = getExp c2 . This way the resulting interval is guaranteed to be correct.

Rounding errors are added to the radius.

If divisor ball contains zero compuatation fails with fail.

sqrt :: Monad m => Precision -> Ball -> m Ball
Square root of a ball. If interval contains 0 then computation fails.
exp :: Precision -> Ball -> Ball
 e ^ b
log :: Monad m => Precision -> Ball -> m Ball
Natural logarithm of a ball. If interval contains 0 then computation fails.
maxB :: Precision -> Ball -> Ball -> Ball

Maximum of two balls, meaning:

  • lower = max (lower a) (lower b) rounded down
  • upper = max (upper a) (upper b) rounded up
minB :: Precision -> Ball -> Ball -> Ball
Analogous to maxB.
fromDyadic :: Precision -> Dyadic -> Ball
MakeA a ball from dyadic. Radius is 0 if desired precision is not smaller than precision of dyadic.
fromString :: Precision -> String -> Ball
Similar to fromDyadic.
fromInt :: Precision -> Int -> Ball
Similar to fromDyadic.
fromWord :: Precision -> Word -> Ball
Similar to fromInt.
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