AngouriMath

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AngouriMath

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StepPieces

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Description

Summary

An expression with floors and ceilings of x in it, written over a bounded range as the piecewise it is
there: each floor and ceiling replaced by its value on each piece, the pieces' ends exact, and each end owned by
the piece whose value it has. floor(x^2) over [0, 2] is 0 on [0, 1), 1 on
[1, sqrt(2)), 2 on [sqrt(2), sqrt(3)), 3 on [sqrt(3), 2) and 4 at 2.
https://github.com/asc-community/AngouriMath/issues/1807

Remarks

The ends are where a step's argument g is a whole number: the roots of g(x) = k for each whole
k in g's range there, as the solver gives them, each checked to be one exactly. What makes the
answer a partition rather than a list of the roots the solver happened to find is the proof that between two
consecutive ends g reaches no whole number: by bisection, each part settled by intervals that hold every
value there (see Over(AngouriMath.Entity,AngouriMath.Entity.Variable,AngouriMath.Numerics.Interval)) -- g inside one (k, k + 1), or g strictly monotone with no whole number between its values at the part's ends, or, next to an end where
g = k, the first derivative of g that does not vanish there of one sign throughout, so that by
Taylor's theorem g - k has one sign beside the end. A root the solver missed is a part none of these
settles, and the whole is declined: null, not an answer with a piece missing.
 The last criterion is for the ends where g only touches a whole number, which are most of them for a
sine: floor(sin(pi x (4 - x)/2)) over [0, 4] reaches 1, -1 and 0 at the turning points of its
argument, where g' = 0 and no monotone part has the end in it.

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