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Decorations

Decorations are flags, or labels, attached to intervals to indicate the status of a given interval as the result of evaluating a function on an initial interval. The combination of an interval $X$ and a decoration $d$ is called a decorated interval.

The allowed decorations and their ordering are as follows: com > dac > def > trv > ill.

Suppose that a decorated interval $(X, d)$ is the result of evaluating a function $f$, or the composition of a sequence of functions, on an initial decorated interval $(X_0, d_0)$. The meaning of the resulting decoration $d$ is as follows:

com ("common"): $X$ is a closed, bounded, nonempty subset of the domain of $f$; $f$ is continuous on the interval $X$; and the resulting interval $f(X)$ is bounded.
dac ("defined & continuous"): $X$ is a nonempty subset of $\mathrm{Dom}(f)$, and $f$ is continuous on $X$.
def ("defined"): $X$ is a nonempty subset of $\mathrm{Dom}(f)$, 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. $\mathrm{Dom}(f) = \emptyset$.

An example will be given at the end of this section.

Initialisation

The simplest way to create a DecoratedInterval is with the @decorated macro, which does correct rounding:

julia> @decorated(0.1, 0.3)
[0.0999999, 0.300001]

The DecoratedInterval constructor may also be used if necessary:

julia> X = DecoratedInterval(3, 4)
[3, 4]

By default, decorations are not displayed. The following turns on display of decorations:

julia> displaymode(decorations=true)

julia> X
[3, 4]_com

If no decoration is explicitly specified when a DecoratedInterval is created, then it is initialised with a decoration according to its interval X:

com: if X is nonempty and bounded;
dac if X is unbounded;
trv if X is empty.

An explicit decoration may be provided for advanced use:

julia> DecoratedInterval(3, 4, dac)
[3, 4]_dac

julia> DecoratedInterval(X, def)
[3, 4]_def

Here, a new DecoratedInterval was created by extracting the interval from another one and appending a different decoration.

Action of functions

A decoration is the combination of an interval together with the sequence of functions that it has passed through. Here are some examples:

julia> X1 = @decorated(0.5, 3)
[0.5, 3]_com

julia> sqrt(X1)
[0.707106, 1.73206]_com

In this case, both input and output are "common" intervals, meaning that they are closed and bounded, and that the resulting function is continuous over the input interval, so that fixed-point theorems may be applied. Since sqrt(X1) ⊆ X1, we know that there must be a fixed point of the function inside the interval X1 (in this case, sqrt(1) == 1).

julia> X2 = DecoratedInterval(3, ∞)
[3, ∞]_dac

julia> sqrt(X2)
[1.73205, ∞]_dac

Since the intervals are unbounded here, the maximum decoration possible is dac.

julia> X3 = @decorated(-3, 4)
[-3, 4]_com

julia> sign(X3)
[-1, 1]_def

The sign function is discontinuous at 0, but is defined everywhere on the input interval, so the decoration is def.

julia> X4 = @decorated(-3.5, 4.1)
[-3.5, 4.10001]_com

julia> sqrt(X4)
[0, 2.02485]_trv

The negative part of X is discarded by the sqrt function, since its domain is [0,∞]. (This process of discarding parts of the input interval that are not in the domain is called "loose evaluation".) The fact that this occurred is, however, recorded by the resulting decoration, trv, indicating a loss of information: "nothing is known" about the relationship between the output interval and the input.

In this case, we know why the decoration was reduced to trv. But if this were just a single step in a longer calculation, a resulting trv decoration shows only that something like this happened at some step. For example:

julia> X5 = @decorated(-3, 3)
[-3, 3]_com

julia> asin(sqrt(X5))
[0, 1.5708]_trv

julia> X6 = @decorated(0, 3)
[0, 3]_com

julia> asin(sqrt(X6))
[0, 1.5708]_trv

In both cases, asin(sqrt(X)) gives a result with a trv decoration, but we do not know at which step this happened, unless we break down the function into its constituent parts:

julia> sqrt(X5)
[0, 1.73206]_trv

julia> sqrt(X6)
[0, 1.73206]_com

This shows that loose evaluation occurred in different parts of the expression in the two different cases.

In general, the trv decoration is thus used only to signal that "something unexpected" happened during the calculation. Often this is later used to split up the original interval into pieces and reevaluate the function on each piece to refine the information that is obtained about the function.