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Add TaylorSeriesRationalFunction and improve Derivative for univariate rational functions #4104

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Sep 4, 2020
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Add TaylorSeriesRationalFunction and improve Derivative for univariate rational functions #4104

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508c4d9
ENHANCE: Added Taylor series for rational functions
hulpke Aug 20, 2020
8519e20
Fixed the suggested simplification (list/pol confusion)
hulpke Sep 4, 2020
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2 changes: 2 additions & 0 deletions doc/ref/ratfun.xml
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Original file line number Diff line number Diff line change
Expand Up @@ -430,6 +430,8 @@ extended versions for Laurent polynomials exist).

<#Include Label="UnivariateRationalFunctionByCoefficients">

<#Include Label="TaylorSeriesRationalFunction">

</Section>


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21 changes: 21 additions & 0 deletions lib/ratfun.gd
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Expand Up @@ -1340,6 +1340,27 @@ DeclareOperation("DegreeIndeterminate",[IsPolynomial,IsPosInt]);
DeclareAttribute("Derivative",IsUnivariateRationalFunction);
DeclareOperation("Derivative",[IsPolynomialFunction,IsPosInt]);

#############################################################################
##
#A TaylorSeriesRationalFunction( <ratfun>, <at>, <deg> )
##
## <#GAPDoc Label="TaylorSeriesRationalFunction">
## <ManSection>
## <Attr Name="TaylorSeriesRationalFunction" Arg='ratfun, at, deg]'/>
##
## <Description>
## Computes the taylor series up to degree <A>deg</A> of <A>ratfun</A> at
## <A>at</A>.
## <Example><![CDATA[
## gap> TaylorSeriesRationalFunction((x^5+3*x+7)/(x^5+x+1),0,11);
## -50*x^11+36*x^10-26*x^9+22*x^8-18*x^7+14*x^6-10*x^5+4*x^4-4*x^3+4*x^2-4*x+7
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("TaylorSeriesRationalFunction");


#############################################################################
##
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50 changes: 47 additions & 3 deletions lib/ratfunul.gi
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Expand Up @@ -1069,18 +1069,62 @@ end);
RedispatchOnCondition(Derivative,true,
[IsPolynomial],[IsLaurentPolynomial],0);

InstallOtherMethod(Derivative,"uratfun,ind",true,
InstallOtherMethod(Derivative,"uratfun",true,
[IsUnivariateRationalFunction],0,
function(ratfun)
local num,den;
local num,den,R,cf,pow,dr,a;
# try to be good for the case of iterated derivatives by using if the
# denominator is a power
num:=NumeratorOfRationalFunction(ratfun);
den:=DenominatorOfRationalFunction(ratfun);
return (Derivative(num)*den-num*Derivative(den))/(den^2);
R:=CoefficientsRing(DefaultRing([den]));
cf:=Collected(Factors(den));
pow:=Gcd(List(cf,x->x[2]));
if pow>1 then
dr:=Product(List(cf,x->x[1]^(x[2]/pow))); # root
else
dr:=den;
fi;
#cf:=(Derivative(num)*dr-num*pow*Derivative(dr))/dr^(pow+1);
num:=Derivative(num)*dr-num*pow*Derivative(dr);
den:=dr^(pow+1);
if IsOne(Gcd(num,dr)) then
# avoid cancellation
num:=CoefficientsOfUnivariateLaurentPolynomial(num);
den:=CoefficientsOfUnivariateLaurentPolynomial(den);
cf:=UnivariateRationalFunctionByExtRepNC(FamilyObj(ratfun),
num[1],den[1],num[2]-den[2],
IndeterminateNumberOfUnivariateRationalFunction(ratfun));
# maintain factorization of denominator
StoreFactorsPol(R,DenominatorOfRationalFunction(cf),
ListWithIdenticalEntries(pow+1,dr));
else
cf:=num/den; # cancellation
fi;
return cf;
#return (Derivative(num)*den-num*Derivative(den))/(den^2);
end);

RedispatchOnCondition(Derivative,true,
[IsPolynomial],[IsUnivariateRationalFunction],0);


InstallGlobalFunction(TaylorSeriesRationalFunction,function(f,at,deg)
local t,i,x;
if not (IsUnivariateRationalFunction(f) and deg>=0) then
Error("function is not univariate pol, or degree negative");
fi;
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x:=IndeterminateOfUnivariateRationalFunction(f);
t:=Zero(f);
for i in [0..deg] do
if i>0 then
f:=Derivative(f);
fi;
t:=t+Value(f,at)*(x-at)^i/Factorial(i);
od;
Comment on lines +1118 to +1124
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Assuming we require deg >= 0 (which is not clear to me from the description of the function), could also do this

Suggested change
t:=Zero(f);
for i in [0..deg] do
if i>0 then
f:=Derivative(f);
fi;
t:=t+Value(f,at)*(x-at)^i/Factorial(i);
od;
t:=Value(f,at)*x^0;
for i in [1..deg] do
f:=Derivative(f);
t:=t+Value(f,at)*(x-at)^i/Factorial(i);
od;

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I think this is a wash in terms of cost or code readability, so I left it out. (Added test for deg\ge0)

return t;
end);

#############################################################################
##
#F Discriminant( <f> ) . . . . . . . . . . . . discriminant of polynomial f
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