Work with exact rational functions

Load the focused plugin. Its .rf and .rationalFunction aliases are the same callable value, and .poly loads as its exact dependency.

Runnable RiXRun cell

.Plugin.Load("ratfun"); R := .rf`(x^2 - 1)/(x - 1)`; {: R(3), R.Numerator().Coefficients(), R.Denominator().Coefficients() };

Convert a symbolic or structural form

.R() mirrors Polynomial .P(). An explicit symbolic input is the clearest choice when other names are coefficients.

Runnable RiXRun cell

.Plugin.Load("ratfun"); y := 2; R := {#x# (x + y)/(x - 1)}.R(); S := (`(t^2 - 4)/(t - 2)`).R(:t); {: R(3), S(5) };

The named backtick header is useful when bare free symbols would be ambiguous.

Runnable RiXRun cell

.Plugin.Load("ratfun"); y := 2; R := `.rf.Var(x):(x + y)/(x - 1)`; R(3);

Polynomial promotion and field operations

After .ratfun is loaded, / between Polynomials creates a RationalFunction. The usual field operations stay in that semantic type. //, %, and /% remain the distinct quotient/remainder operations supplied by .algebra.

Runnable RiXRun cell

.Plugin.Load("algebra"); ## loads ratfun, then poly P := .p`x^2 - 1`; Q := .p`x - 1`; R := P/Q; S := 1/Q; T := (R + S)^2; {: R(2), S(2), T(2), P // Q, P % Q };

Composition

Calling with a Polynomial or RationalFunction substitutes it and returns a canonical RationalFunction, or the narrower Polynomial type if the resulting denominator is 1.

Runnable RiXRun cell

.Plugin.Load("ratfun"); R := .rf`(x + 1)/(x - 1)`; P := .p`x^2`; Q := .rf`1/x`; {: R(P)(2), R.Compose(Q)(2) };

Reactive coefficients

Canonicalization evaluates coefficients exactly when a value is built. Reactive definitions rebuild the RationalFunction whenever a read dependency changes; ordinary definitions remain snapshots.

Runnable RiXRun cell

.Plugin.Load("ratfun"); $$y := 2; $$P := `.p.Var(x):x + @($y)`; Q := .p`x - 1`; $$R := $P/Q; $$S := $R + 1/Q; $y := 4; {: $R(3), $S(3) }; ## (7/2, 4)

Cancellation and domains

RationalFunctions use reduced fraction-field semantics. Cancelled source restrictions are not remembered.

Runnable RiXRun cell

.Plugin.Load("ratfun"); R := .rf`(x^2 - 1)/(x - 1)`; {: R(1), R.Domain(), R == .rf`x + 1` };

Here R(1) is 2. Use .fracfun when the removable hole at 1 must remain part of the source-domain meaning.

Choose and verify an explicit rational form

Canonical equality remains independent of display. These presentations retain their exact transformation data and reconstruct the source before reporting a verified result.

Runnable RiXRun cell

.Plugin.Load("ratfun"); R := .rf`(x^5+x^3+2*x+1)/((x-1)^2*(x^2+1))`; partial := R.PartialFractions(); factored := R.Factored(); .Table( ["property", "exact value"], [ ["polynomial part", partial.PolynomialPart().Coefficients()], ["rational linear terms", partial.Terms()], ["proper residual", partial.Residual()], ["all denominator factors linear over Q", partial[:linearComplete]], ["partial round trip", partial.Expand() == R], ["factored round trip", factored.Expand() == R] ] );

Inspect reduced zeros and poles

Runnable RiXRun cell

.Plugin.Load("ratfun"); R := .rf`6*(x-2)^3*(x+1)/((x-3)^2*(x^2+1))`; evidence := R.PoleZeroEvidence(); .Table( ["property", "exact evidence"], [ ["rational zeros", evidence.Zeros()[:entries]], ["rational poles", evidence.Poles()[:entries]], ["unfactored pole residual", evidence.Poles()[:residual].Coefficients()], ["verified coprime pair", evidence[:coprime]] ] );