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Composition

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Composition[f1, f2, f3, ...]
    represents a composition of the functions f1, f2, f3, ....

Composition allows you to build up compositions of functions which can
later be applied to specific arguments. Applied to arguments, the
composition acts innermost-first:
    Composition[f, g, h][x, y]  ->  f[g[h[x, y]]].

Composition has the attributes Flat and OneIdentity.
Composition can be entered in the form f1 @* f2 @* ....

Composition objects containing Identity or InverseFunction[f] are
automatically simplified when possible:
    Composition[]                       ->  Identity
    Composition[f]                      ->  f
    Composition[f, Identity, g]         ->  Composition[f, g]
    Composition[f, InverseFunction[f]]  ->  Identity.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= Composition[f, g, h][x, y]
Out[1]= f[g[h[x, y]]]

In[2]:= f @* g @* h @ x
Out[2]= f[g[h[x]]]

In[3]:= Composition[f, Identity, g]
Out[3]= Composition[f, g]

In[4]:= Composition[f, InverseFunction[f]][x]
Out[4]= x

In[5]:= Composition[f, g] @* Composition[a, b]
Out[5]= Composition[f, g, a, b]

Implementation notes

builtin_composition (src/core.c) handles only the algebraic simplifications of Composition[f1,...,fn]: Composition[] -> Identity, Composition[f] -> f, dropping Identity arguments, and cancelling adjacent f/InverseFunction[f] pairs (in either order, iterated to fixed point). The actual application Composition[f1,...,fn][args] -> f1[f2[...fn[args]...]] is performed in the evaluator (eval.c). The symbol carries ATTR_FLAT | ATTR_ONEIDENTITY.

Attributes: Flat, OneIdentity, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= Composition[f, g, h][x]
Out[1]= f[g[h[x]]]

In[1]:= Composition[Sqrt, Abs][-16]
Out[1]= 4

In[1]:= Composition[f, InverseFunction[f]]
Out[1]= Identity

Notes

Composition[f1, f2, ...] represents a function that acts innermost-first when applied. It has attributes Flat and OneIdentity, and can be entered as f1 @* f2 @* .... Compositions containing Identity or InverseFunction[f] automatically simplify.