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spad
)abbrev package MYEXP MyExp
MyExp(F: Algebra(Fraction(Integer))): with myexp: F -> F myexp: (F, NonNegativeInteger) -> F == add Map2 ==> ListFunctions2(NonNegativeInteger, F) myexp(x, n) == a(i : NonNegativeInteger) : F == (1/(factorial i)) * x^i reduce(_+@((F, F) -> F), map(a, [i for i in 0..n])$Map2)
myexp(x: F): F == myexp(x, 32)
spad
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/8403278387071313463-25px001.spad
      using old system compiler.
   MYEXP abbreviates package MyExp 
------------------------------------------------------------------------
   initializing NRLIB MYEXP for MyExp 
   compiling into NRLIB MYEXP 
   processing macro definition Map2 ==> ListFunctions2(NonNegativeInteger,F) 
   compiling exported myexp : (F,NonNegativeInteger) -> F
Time: 0.04 SEC.
compiling exported myexp : F -> F Time: 0 SEC.
(time taken in buildFunctor: 0)
;;; *** |MyExp| REDEFINED
;;; *** |MyExp| REDEFINED Time: 0 SEC.
Cumulative Statistics for Constructor MyExp Time: 0.04 seconds
finalizing NRLIB MYEXP Processing MyExp for Browser database: --->-->MyExp(constructor): Not documented!!!! --->-->MyExp((myexp (F F))): Not documented!!!! --->-->MyExp((myexp (F F (NonNegativeInteger)))): Not documented!!!! --->-->MyExp(): Missing Description ; compiling file "/var/aw/var/LatexWiki/MYEXP.NRLIB/MYEXP.lsp" (written 13 JUL 2015 02:24:20 PM):
; /var/aw/var/LatexWiki/MYEXP.NRLIB/MYEXP.fasl written ; compilation finished in 0:00:00.029 ------------------------------------------------------------------------ MyExp is now explicitly exposed in frame initial MyExp will be automatically loaded when needed from /var/aw/var/LatexWiki/MYEXP.NRLIB/MYEXP

That's generic code and has to be compiled. Look at the types of the result.

fricas
myexp(1.0)

\label{eq1}2.7182818284 \<u> 590452354(1)
Type: Float
fricas
myexp(1.0,4)

\label{eq2}2.7083333333 \<u> 333333333(2)
Type: Float
fricas
myexp(1)

\label{eq3}{28610550901763172837819811744646057}\over{105252334773477412
06688720486400000}(3)
Type: Fraction(Integer)
fricas
myexp(matrix[[1,0],[0,2]] :: SQMATRIX(2,FRAC(INT))) :: SQMATRIX(2,Float)

\label{eq4}\left[ 
\begin{array}{cc}
{2.7182818284 \<u> 590452354}&{0.0}
\
{0.0}&{7.3890560989 \</u> 306502273}
(4)
Type: SquareMatrix?(2,Float)

And now we create a domain of functions and let it behave like an algebra over a field of rational numbers where multiplication is concatenation of functions and addition is... well, see below.

spad
)abbrev domain MYFUN MyFun
MyFun(F: Field): Algebra(Fraction(Integer)) with apply: (%, F) -> F coerce: (F -> F) -> % == add Rep := F -> F X ==> x pretend (F->F) Y ==> y pretend (F->F) zr(f: F): F == 0 id(f: F): F == f plus(x: %, y: %): % == (f: F): F +-> X(f) + Y(f) times(x: %, y: %): % == (f: F): F +-> X(Y f) ltimes(l: Fraction(Integer), y: %): % == (f: F): F +-> l*Y(f)
0: % == zr 1: % == id (x: %) + (y: %) == plus(x,y) (x: %) * (y: %) == times(x,y) (l: Fraction(Integer)) * (y: %) == ltimes(l,y) coerce(x: %): OutputForm == "myfun-object" apply(x: %, f: F): F == X f coerce(fun: F -> F): % == fun
spad
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/1886816482829968029-25px003.spad
      using old system compiler.
   MYFUN abbreviates domain MyFun 
------------------------------------------------------------------------
   initializing NRLIB MYFUN for MyFun 
   compiling into NRLIB MYFUN 
   processing macro definition X ==> pretend(x,F -> F) 
   processing macro definition Y ==> pretend(y,F -> F) 
   compiling local zr : F -> F
Time: 0.01 SEC.
compiling local id : F -> F MYFUN;id is replaced by f Time: 0 SEC.
compiling local plus : ($,$) -> $ Time: 0 SEC.
compiling local times : ($,$) -> $ Time: 0 SEC.
compiling local ltimes : (Fraction Integer,$) -> $ Time: 0.01 SEC.
compiling exported Zero : () -> $ Time: 0 SEC.
compiling exported One : () -> $ Time: 0 SEC.
compiling exported + : ($,$) -> $ Time: 0 SEC.
compiling exported * : ($,$) -> $ Time: 0 SEC.
compiling exported * : (Fraction Integer,$) -> $ Time: 0 SEC.
compiling exported coerce : $ -> OutputForm MYFUN;coerce;$Of;11 is replaced by myfun-object Time: 0 SEC.
compiling exported apply : ($,F) -> F MYFUN;apply;$2F;12 is replaced by SPADCALLfx Time: 0 SEC.
compiling exported coerce : F -> F -> $ MYFUN;coerce;M$;13 is replaced by fun Time: 0 SEC.
(time taken in buildFunctor: 0)
;;; *** |MyFun| REDEFINED
;;; *** |MyFun| REDEFINED Time: 0 SEC.
Warnings: [1] plus: elt has no value [2] times: elt has no value [3] apply: elt has no value
Cumulative Statistics for Constructor MyFun Time: 0.02 seconds
finalizing NRLIB MYFUN Processing MyFun for Browser database: --->-->MyFun(constructor): Not documented!!!! --->-->MyFun((apply (F % F))): Not documented!!!! --->-->MyFun((coerce (% (Mapping F F)))): Not documented!!!! --->-->MyFun(): Missing Description ; compiling file "/var/aw/var/LatexWiki/MYFUN.NRLIB/MYFUN.lsp" (written 13 JUL 2015 02:24:20 PM):
; /var/aw/var/LatexWiki/MYFUN.NRLIB/MYFUN.fasl written ; compilation finished in 0:00:00.032 ------------------------------------------------------------------------ MyFun is now explicitly exposed in frame initial MyFun will be automatically loaded when needed from /var/aw/var/LatexWiki/MYFUN.NRLIB/MYFUN

fricas
F := Fraction Integer

\label{eq5}\hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Integer}\ })(5)
Type: Type
fricas
H := MyFun(F)

\label{eq6}\hbox{\axiomType{MyFun}\ } (\hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Integer}\ }))(6)
Type: Type
fricas
phi(f: F): F == f/2
Function declaration phi : Fraction(Integer) -> Fraction(Integer) has been added to workspace.
Type: Void
fricas
hphi:H := phi::H
fricas
Compiling function phi with type Fraction(Integer) -> Fraction(
      Integer)

\label{eq7}myfun - object(7)
Type: MyFun?(Fraction(Integer))
fricas
apply(hphi, 1)

\label{eq8}1 \over 2(8)
Type: Fraction(Integer)
fricas
a==>apply
Type: Void

Computing the first 3 terms of 1+\varphi(1)+\frac{1}{2}\varphi(\varphi(1)).

fricas
1+a(hphi,1)+1/2 * a(hphi,a(hphi,1))

\label{eq9}{13}\over 8(9)
Type: Fraction(Integer)

And here we first compute 1+\varphi+\frac{1}{2}\varphi^2 and then apply this function to 1.

fricas
hexp:=myexp(hphi,2)

\label{eq10}myfun - object(10)
Type: MyFun?(Fraction(Integer))
fricas
apply(hexp,1)

\label{eq11}{13}\over 8(11)
Type: Fraction(Integer)

It also works with finite fields.

fricas
P := PrimeField 17

\label{eq12}\hbox{\axiomType{PrimeField}\ } (17)(12)
Type: Type
fricas
Q := MyFun(P)

\label{eq13}\hbox{\axiomType{MyFun}\ } (\hbox{\axiomType{PrimeField}\ } (1
7))(13)
Type: Type
fricas
psi(f: P): P == f+1
Function declaration psi : PrimeField(17) -> PrimeField(17) has been added to workspace.
Type: Void
fricas
qpsi:Q := psi::Q
fricas
Compiling function psi with type PrimeField(17) -> PrimeField(17)

\label{eq14}myfun - object(14)
Type: MyFun?(PrimeField?(17))
fricas
apply(qpsi, 1)

\label{eq15}2(15)
Type: PrimeField?(17)
fricas
qexp:=myexp(qpsi,2)

\label{eq16}myfun - object(16)
Type: MyFun?(PrimeField?(17))
fricas
apply(qexp,1)

\label{eq17}13(17)
Type: PrimeField?(17)




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