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FunWithFunctions

Edit detail for FunWithFunctions revision 4 of 5

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Editor: Ralf Hemmecke Time: 2011/04/05 06:34:52 GMT-7
Note:

added:
It also works with finite fields.
\begin{axiom}
P := PrimeField 17
Q := MyFun(P)
psi(f: P): P == f+1
qpsi:Q := psi::Q
apply(qpsi, 1)
qexp:=myexp(qpsi,2)
apply(qexp,1)
\end{axiom}

fricas

(1) -> <spad>

fricas

)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>

fricas

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: 0.02 SEC.

   compiling exported myexp : F -> F
Time: 0.00 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |MyExp| REDEFINED

;;;     ***       |MyExp| REDEFINED
Time: 0.00 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 17 SEP 2023 07:35:40 PM):

; 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.

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myexp(1.0)

$\label{eq1}2.7182818284 \<u> 590452354$

(1)

Type: Float

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myexp(1.0,4)

$\label{eq2}2.7083333333 \<u> 333333333$

(2)

Type: Float

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myexp(1)

$\label{eq3}\frac{28610550901763172837819811744646057}{105252334773477412 06688720486400000}$

(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: 0.00 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: 0.00 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
****** comp fails at level 1 with expression: ******
error in function coerce 

("myfun-object")
****** level 1  ******
$x:= myfun-object
$m:= (OutputForm)
$f:=
((((|x| # #) (|ltimes| #)) ((|ltimes| #) (|times| #)) ((|times| #) (|plus| #))
  ((|plus| #) (|id| #)) ...))

   >> Apparent user error:
   Cannot coerce myfun-object 
      of mode myfun-object 
      to mode (OutputForm)

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F := Fraction Integer

$\label{eq5}\hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Integer}\ })$

(5)

Type: Type

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H := MyFun(F)

   MyFun is an unknown constructor and so is unavailable. Did you mean 
      to use -> but type something different instead?

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

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

   There are no library operations named a 
      Use HyperDoc Browse or issue
                                 )what op a
      to learn if there is any operation containing " a " in its name.

   Cannot find a definition or applicable library operation named a 
      with argument type(s) 
                               Variable(hphi)
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

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

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hexp:=myexp(hphi,2)

$\label{eq6}{{\frac{1}{2}}\ {{hphi}^{2}}}+ hphi + 1$

(6)

Type: Polynomial(Fraction(Integer))

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apply(hexp,1)

   There are 2 exposed and 2 unexposed library operations named apply 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op apply
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named apply
      with argument type(s) 
                        Polynomial(Fraction(Integer))
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

It also works with finite fields.

fricas

P := PrimeField 17

$\label{eq7}\hbox{\axiomType{PrimeField}\ } (17)$

(7)

Type: Type

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Q := MyFun(P)

   MyFun is an unknown constructor and so is unavailable. Did you mean 
      to use -> but type something different instead?