(1) -> <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: 0.03 SEC.
compiling exported myexp : F -> F Time: 0 SEC.
(time taken in buildFunctor: 0) Time: 0 SEC.
Cumulative Statistics for Constructor MyExp Time: 0.03 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 10 AUG 2025 08:13:29 AM):
; wrote /var/aw/var/LatexWiki/MYEXP.NRLIB/MYEXP.fasl ; compilation finished in 0:00:00.008 ------------------------------------------------------------------------ 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.
myexp(1.0)
myexp(1.0,4)
myexp(1)
myexp(matrix[[1,0], [0, 2]] :: SQMATRIX(2, FRAC(INT))) :: SQMATRIX(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.
)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 == message("myfun-object")$OutputForm apply(x: %, f: F): F == X f coerce(fun: F -> F): % == fun
Compiling FriCAS source code from file /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/4752705461069213214-25px003.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 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 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 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: 1086) 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: 0.01 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 10 AUG 2025 08:13:30 AM):
; wrote /var/aw/var/LatexWiki/MYFUN.NRLIB/MYFUN.fasl ; compilation finished in 0:00:00.012 ------------------------------------------------------------------------ MyFun is now explicitly exposed in frame initial MyFun will be automatically loaded when needed from /var/aw/var/LatexWiki/MYFUN.NRLIB/MYFUN
F := Fraction Integer
H := MyFun(F)
phi(f: F): F == f/2
Function declaration phi : Fraction(Integer) -> Fraction(Integer) has been added to workspace.
hphi:H := phi::H
Compiling function phi with type Fraction(Integer) -> Fraction( Integer)
apply(hphi,1)
a==>apply
Computing the first 3 terms of 1+\varphi(1)+\frac{1}{2}\varphi(\varphi(1)).
1+a(hphi,1)+1/2 * a(hphi, a(hphi, 1))
And here we first compute 1+\varphi+\frac{1}{2}\varphi^2 and then apply this function to 1.
hexp:=myexp(hphi,2)
apply(hexp,1)
It also works with finite fields.
P := PrimeField 17
Q := MyFun(P)
psi(f: P): P == f+1
Function declaration psi : PrimeField(17) -> PrimeField(17) has been added to workspace.
qpsi:Q := psi::Q
Compiling function psi with type PrimeField(17) -> PrimeField(17)
apply(qpsi,1)
qexp:=myexp(qpsi,2)
apply(qexp,1)