RatCanonPrototype¶

Status: Stable

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

Description¶

RatCanonPrototype[expr] (Phase-1 prototype) reduces a rational function over the differential/algebraic tower of expr via one FLINT reduction.

Examples (4)¶

Every input below was run against the current Mathilda build and its output recorded.

Applications (4)¶

Cancels the common factor over Q

In[1]:= RatCanonPrototype[(x^2 - 1)/(x - 1)]
Out[1]= 1 + x

E^x is treated as one tower generator

In[2]:= RatCanonPrototype[(E^(2 x) - 1)/(E^x - 1)]
Out[2]= 1 + E^x

A Log tower kernel

In[3]:= RatCanonPrototype[(Log[x]^2 - 1)/(Log[x] + 1)]
Out[3]= -1 + Log[x]

Combines to a single reduced fraction

In[4]:= RatCanonPrototype[x/(x + 1) + 1/(x + 1)]
Out[4]= 1

Implementation notes¶

Algorithm. RatCanonPrototype[expr] is a Phase-1 prototype of rational-function canonicalisation over the differential/algebraic tower of expr. builtin_ratcanon_prototype (src/poly/ratcanon.c) performs a kernel-substitution round trip: rcp_forward walks the expression and replaces every non-rational kernel (each Exp[...], Log[...], Sqrt[...], Sin[...], etc.) with a fresh free symbol, recording the map; flint_rational_together reduces the resulting plain rational function over Q in one FLINT reduction; rcp_backward substitutes the kernels back; and eval_and_free applies the algebraic relations that the free-symbol abstraction hid (I^2 -> -1, Sqrt[k]^2 -> k, and so on). If the reduction declines, the builtin returns NULL and the call is left unevaluated.

Data structures. An RcpMap holds the kernel↔symbol correspondence; everything else is ordinary Expr passed into the FLINT rational engine (flint_rational_together, the same kernel Together/Cancel use). It is registered Protected in ratcanon_init.

Complexity / limits. Dominated by the single FLINT reduction over Q, which is why it cancels common factors cheaply — (x^2-1)/(x-1) collapses to 1 + x, and the exponential tower (E^(2x)-1)/(E^x-1) to 1 + E^x by treating E^x as one generator. It is a prototype: it reduces over Q with each kernel as an independent generator, so it does not know the algebraic relations between kernels (e.g. it declines a mixed Sqrt-in-denominator case and the Pythagorean Sin/Cos identity). It is a building block for a fuller tower-canonicalisation, not a general simplifier — use Together, Cancel or Simplify for production work.

Attributes: Protected.

References¶

Notes & additional examples¶

Notes¶

RatCanonPrototype[expr] is a Phase-1 prototype that reduces a rational function over the differential/algebraic tower of expr via a single FLINT reduction. It abstracts each non-rational kernel (E^x, Log[x], …) to a fresh generator, reduces over Q, substitutes the kernels back, and re-applies the algebraic relations — so it cancels common factors and combines fractions cheaply, as in the examples above.

It is a prototype, not a general simplifier: it treats each kernel as an independent generator, so it does not know relations between kernels and will leave an expression unevaluated when its heuristic declines (for instance a Sqrt in the denominator, or the Pythagorean Sin/Cos identity). For production use prefer Together, Cancel or Simplify.