Merge d0707a8 into 24b7e2f

docs/decorations.md

@@ -0,0 +1,47 @@
+# Decorations:
+
+Decorations are flags or labels attached to intervals to indicate the status of a given interval as the product of a function evaluation on a given initial interval.
+
+The possible decorations and their ordering are as follows (pg. 46):
+`com` > `dac` > `def` > `trv` > `ill`.
+
+
+- `com` ("common"): The meanings are (pg. 44): x is a bounded, nonempty subset of Dom(f); f is
+continuous at each point of x; and the computed interval f(x) is bounded.
+x is a nonempty subset of Dom(f), and the restriction of f to x is continuous;
+
+- `dac` ("defined & continuous"): x is a nonempty subset of Dom(f), and the
+restriction of f to x is continuous;
+
+- `def` ("defined"): x is a nonempty subset of Dom(f), i.e. f is defined at each point of x
+
+- `trv` ("trivial"): always true; gives no information
+
+- `ill` ("ill-formed"): Not an Interval (an error occurred), e.g. Dom(f) = empty
+
+## Initialisation
+Each decorated interval is initialised with a decoration according to the status of its interval `x` (pg. 46 of the standard):
+
+- `com`: if `x` is nonempty and bounded;
+- `dac` if `x` is unbounded;
+- `trv` if `x` is empty.
+
+
+## Action of functions
+
+A decoration is the combination of an interval together with the sequence of functions that it has passed through.
+
+As an example, consider the `sqrt` function acting on an interval `x`.
+
+If `x` is empty, then `sqrt` will return the, empty interval, together with the trivial decoration.
+
+If `x` is [a, Inf] with a > 0 (`dac`), then `sqrt(x)` will return `[sqrt(a), Inf]`, so is also unbounded, and so will return (`[sqrt(a), Inf], dac`).
+
+The domain of `sqrt` is `D := Dom(sqrt) = [0, Inf]`. We must first check if
+`x ⊆ D`. If so, then the `sqrt` function is continuous and bounded on that interval, so we return `min(`com`, decoration(x))`,  where `decoration(x)` is the decoration of `x`.
+
+E.g. consider the `sign` function:
+sign(x) = 1, if x > 0; sign(x) = 0 if x = 0; sign(x) = -1 if x < 0
+
+This function is defined for all x in R, but is not continuous at 0. Thus if the interval contains 0, the decoration that results cannot be better than `def`.
+Thus `sqrt(sign([0,10]))` will return `([0,1], def)`, where the decoration is only `def`, indicating that we cannot guarantee that the function `sqrt ∘ sign` is continuous on the interval. (In fact, we know that it is not continuous, but this is not guaranteed by the decoration.)

src/ValidatedNumerics.jl

@@ -25,11 +25,10 @@ import Base:
     precision,
     isfinite, isnan
 
-
 export
-    Interval,
+    Interval, AbstractInterval,
     @interval, @biginterval, @floatinterval, @make_interval,
-    diam, radius, mid, mag, mig, hull, isinside,
+    diam, radius, mid, mag, mig, hull,
     emptyinterval, ∅, ∞, isempty, interior, isdisjoint, ⪽,
     precedes, strictprecedes, ≺,
     entireinterval, isentire, nai, isnai, isthin, iscommon,
@@ -55,6 +54,12 @@ end
 export
     IntervalBox, @intervalbox
 
+## Decorations
+export
+    @decorated,
+    interval_part, decoration, DecoratedInterval,
+    com, dac, def, trv, ill
+
 ## Root finding
 export
     newton, krawczyk,
@@ -63,6 +68,7 @@ export
     find_roots,
     find_roots_midpoint
 
+
 function __init__()
     setrounding(BigFloat, RoundNearest)
     setrounding(Float64, RoundNearest)
@@ -76,6 +82,7 @@ end
 
 include("intervals/intervals.jl")
 include("multidim/multidim.jl")
+include("decorations/decorations.jl")
 
 include("root_finding/root_finding.jl")
 

src/decorations/decorations.jl

@@ -0,0 +1,2 @@
+include("intervals.jl")
+include("functions.jl")