Data.Number.Ball

Synopsis

Documentation

Ball represents a closed interval [center-radius, center+radius]

Constructors

Ball
Fields center :: !Dyadic center of the ball radius :: !Dyadic radius of the ball

Instances

Show Ball

makeA Source

Arguments

:: Precision	desired precision of the center
-> Dyadic	left endpoint
-> Dyadic	right endpoint
-> Ball

Make a ball from endpoints

make Source

Arguments

:: Dyadic	left endpoint
-> Dyadic	right endpoint
-> Ball

Make a ball from endpoints so that no precision is lost.

normalizeBall :: Precision -> Ball -> Ball Source

Normalize the given ball's center to the specified precision. Resulting ball might be larger.

lower :: Precision -> Ball -> Dyadic Source

Lower endpoint of the ball rounded down to specified precision.

upper :: Precision -> Ball -> Dyadic Source

Upper endpoint of the ball rounded up to specified precision.

lower_ :: Ball -> Dyadic Source

Lower endpoint with precision of the center

upper_ :: Ball -> Dyadic Source

Upper endpoint with precision of the center

sgnLower :: Ball -> Int Source

Sign of lower endpoint of the ball. This should be faster than using signum (center b - radius b)

sgnUpper :: Ball -> Int Source

Analogous to sgnLower.

width :: Ball -> Dyadic Source

Upper bound on the width of the ball. 2 * radius b rounded up.

compareB :: Ball -> Ball -> POrdering Source

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 Source

Check if second ball is included in the first

contains :: Ball -> Dyadic -> Bool Source

Check if dyadic is element of the ball.

intersectA Source

Arguments

:: Monad m
=> Precision	precision 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 Source

Intersection of two balls exactly (no precision is lost).

add :: Precision -> Ball -> Ball -> Ball Source

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 Source

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 Source

Negation of the ball.

center = - center b rounded to specified precision.
radius is only modified for the rounding error.

absB :: Precision -> Ball -> Ball Source

mul :: Precision -> Ball -> Ball -> Ball Source

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 Source

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 Source

Square root of a ball. If interval contains 0 then computation fails.

exp :: Precision -> Ball -> Ball Source