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FunWithFunctions

Edit detail for FunWithFunctions revision 3 of 5

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Editor: Yrogirg Time: 2011/04/05 06:19:53 GMT-7
Note:

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

changed:
-And here we first compute $1+\varphi+\varphi^2$ and then apply this function to 1.
And here we first compute $1+\varphi+\frac{1}{2}\varphi^2$ and then apply this function to 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/zope2/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: 0.55 SEC.

   compiling exported myexp : F -> F
Time: 0 SEC.

(time taken in buildFunctor:  10)

;;;     ***       |MyExp| REDEFINED

;;;     ***       |MyExp| REDEFINED
Time: 0.01 SEC.

   Cumulative Statistics for Constructor MyExp
      Time: 0.56 seconds

   finalizing NRLIB MYEXP 
   Processing MyExp for Browser database:
--->-->MyExp((myexp (F F))): Not documented!!!!
--->-->MyExp((myexp (F F (NonNegativeInteger)))): Not documented!!!!
--->-->MyExp(constructor): Not documented!!!!
--->-->MyExp(): Missing Description
; compiling file "/var/zope2/var/LatexWiki/MYEXP.NRLIB/MYEXP.lsp" (written 05 APR 2011 06:19:28 AM):

; /var/zope2/var/LatexWiki/MYEXP.NRLIB/MYEXP.fasl written
; compilation finished in 0:00:00.151
------------------------------------------------------------------------
   MyExp is now explicitly exposed in frame initial 
   MyExp will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/MYEXP.NRLIB/MYEXP

   >> System error:
   The bounding indices 163 and 162 are bad for a sequence of length 162.
See also:
  The ANSI Standard, Glossary entry for "bounding index designator"
  The ANSI Standard, writeup for Issue SUBSEQ-OUT-OF-BOUNDS:IS-AN-ERROR

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

axiom

myexp(1.0)

$\label{eq1}2.7182818284590452354$

(1)

Type: Float

axiom

myexp(1.0,4)

$\label{eq2}2.7083333333333333333$

(2)

Type: Float

axiom

myexp(1)

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

(3)

Type: Fraction(Integer)

axiom

myexp(matrix[[1,0],[0,2]] :: SQMATRIX(2,FRAC(INT))) :: SQMATRIX(2,Float)

$\label{eq4}\left[ \begin{array}{cc} {2.7182818284590452354}&{0.0} \ {0.0}&{7.3890560989306502273}$

(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/zope2/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: 0.52 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: 0.01 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: 0.01 SEC.

(time taken in buildFunctor:  10)

;;;     ***       |MyFun| REDEFINED

;;;     ***       |MyFun| REDEFINED
Time: 0.01 SEC.

   Cumulative Statistics for Constructor MyFun
      Time: 0.56 seconds

   finalizing NRLIB MYFUN 
   Processing MyFun for Browser database:
--->-->MyFun((apply (F % F))): Not documented!!!!
--->-->MyFun((coerce (% (Mapping F F)))): Not documented!!!!
--->-->MyFun(constructor): Not documented!!!!
--->-->MyFun(): Missing Description
; compiling file "/var/zope2/var/LatexWiki/MYFUN.NRLIB/MYFUN.lsp" (written 05 APR 2011 06:19:30 AM):

; /var/zope2/var/LatexWiki/MYFUN.NRLIB/MYFUN.fasl written
; compilation finished in 0:00:00.133
------------------------------------------------------------------------
   MyFun is now explicitly exposed in frame initial 
   MyFun will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/MYFUN.NRLIB/MYFUN

   >> System error:
   The bounding indices 163 and 162 are bad for a sequence of length 162.
See also:
  The ANSI Standard, Glossary entry for "bounding index designator"
  The ANSI Standard, writeup for Issue SUBSEQ-OUT-OF-BOUNDS:IS-AN-ERROR

axiom

F := Fraction Integer

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

(5)

Type: Type

axiom

H := MyFun(F)

$\label{eq6}\hbox{\axiomType{MyFun}\ } (\hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Integer}\ }))$

(6)

Type: Type

axiom

phi(f: F): F == f/2

   Function declaration phi : Fraction(Integer) -> Fraction(Integer) 
      has been added to workspace.

Type: Void

axiom

hphi:H := phi::H

axiom

Compiling function phi with type Fraction(Integer) -> Fraction(
      Integer)

$\label{eq7}myfun - object$

(7)

Type: MyFun?(Fraction(Integer))

axiom

apply(hphi, 1)

$\label{eq8}1 \over 2$

(8)

Type: Fraction(Integer)

axiom

a==>apply

Type: Void

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

axiom

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.

axiom

hexp:=myexp(hphi,2)

$\label{eq10}myfun - object$

(10)

Type: MyFun?(Fraction(Integer))

axiom

apply(hexp,1)

$\label{eq11}{13}\over 8$

(11)

Type: Fraction(Integer)