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SandBoxComplexManifold

Edit detail for SandBoxComplexManifold revision 4 of 7

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Editor: Bill Page Time: 2009/06/20 21:30:55 GMT-7
Note: sqrt

changed:
-ComplexManifold(R:Join(Field,RadicalCategory,TranscendentalFunctionCategory)): Join(RadicalCategory,DirectProductCategory(2,R)) with
ComplexManifold(R:Join(Field,RadicalCategory,TranscendentalFunctionCategory,OrderedSet)): Join(RadicalCategory,DirectProductCategory(2,R)) with

added:
    coerce: % -> Complex R

changed:
-    arg(x:%):R == (2::R)*atan(imag(x)/(norm(x)+real(x)))
-    sqrt(x:%):% == per( sqrt(sqrt(norm(x)))*directProduct(vector [cos(pi()*arg(x)) + sin(pi()*arg(x)), cos(pi()*arg(x)) - sin(pi()*arg(x))] ) )
    arg(x:%):R ==
      real(x)<0 and imag(x)=0 => pi()
      (2::R)*atan(imag(x)/(sqrt(norm(x))+real(x)))
    sqrt(x:%):% == per( sqrt(sqrt(norm(x)))*directProduct(vector [ _
      cos(arg(x)/(2::R)) + sin(arg(x)/(2::R)), _
      cos(arg(x)/(2::R)) - sin(arg(x)/(2::R))] ) )

changed:
-
    coerce(x:%):Complex(R) == complex(real x,imag x)

added:
s3:=sqrt(I)
s3*s3

changed:
-s3:=sqrt(c1)
-real s3
-imag s3
c1::Complex(EXPR INT)
sqrt %
s4:=sqrt(c1)
real s4
imag s4
s4*s4
%::Complex(EXPR INT)
normalize %
%::Complex(EXPR INT)::Complex(INT)

spad

)abbrev domain CM ComplexManifold
ComplexManifold(R:Join(Field,RadicalCategory,TranscendentalFunctionCategory,OrderedSet)): Join(RadicalCategory,DirectProductCategory(2,R)) with
    elt:(%,"pos") -> R
    elt:(%,"neg") -> R
    imaginary: () -> %
    real: % -> R
    imag: % -> R
    conj: % -> %
    norm: % -> R
    arg:  % -> R
    coerce: % -> Complex R
  == DirectProduct(2,R) add
    -- represent as conjugate pair
    Rep == DirectProduct(2,R)
    elt(x,"pos") == rep(x).1
    elt(x,"neg") == rep(x).2

    imaginary():% == per directProduct vector [1,-1]
    real(x:%):R == (rep(x).1 + rep(x).2)/(2::R)
    imag(x:%):R == (rep(x).1 - rep(x).2)/(2::R)
    -- just swap
    conj(x:%):% == per directProduct vector [rep(x).2,rep(x).1]
    -- multiplication is interesting
    (x:% * y:%):% == per directProduct vector [ _
      real(x)*rep(y).1 + imag(x)*rep(y).2,      _
      real(x)*rep(y).2 - imag(x)*rep(y).1]
    norm(x:%):R == retract(x*conj(x))
    arg(x:%):R ==
      real(x)<0 and imag(x)=0 => pi()
      (2::R)*atan(imag(x)/(sqrt(norm(x))+real(x)))
    sqrt(x:%):% == per( sqrt(sqrt(norm(x)))*directProduct(vector [ _
      cos(arg(x)/(2::R)) + sin(arg(x)/(2::R)), _
      cos(arg(x)/(2::R)) - sin(arg(x)/(2::R))] ) )

    -- [sqrt(abs real x)+sqrt(abs imag x).
    -- sign(real x)*sqrt(abs real x)-sign(imag x)*sqrt(abs imag x)]
    -- too complicated?
    --sqrt(x:%):% == per directProduct vector [       _
    --  (if real(x)>=0 then sqrt real(x) else sqrt(-real(x))) +    _
    --  (if imag(x)<=0 then sqrt(-imag(x)) else sqrt imag(x)),     _
    --  (if real(x)>=0 then sqrt real(x) else -sqrt(-real(x))) +   _
    --  (if imag(x)<=0 then sqrt(-imag(x)) else -sqrt imag(x)) ]

    coerce(x:%):OutputForm == complex(real x,imag x)$Complex(R)::OutputForm
    coerce(x:%):Complex(R) == complex(real x,imag x)

spad

   Compiling OpenAxiom source code from file 
      /var/zope2/var/LatexWiki/7243446078452659057-25px001.spad using 
      Spad compiler.
   CM abbreviates domain ComplexManifold 
------------------------------------------------------------------------
   initializing NRLIB CM for ComplexManifold 
   compiling into NRLIB CM 
   Adding Integer modemaps
   Adding NonNegativeInteger modemaps
   Adding PositiveInteger modemaps
   Adding $ modemaps
   Adding R modemaps
   Adding R modemaps
   Adding R modemaps
   Adding R modemaps
   Adding Integer modemaps
   Adding NonNegativeInteger modemaps
   Adding PositiveInteger modemaps
   Adding Rep modemaps
   compiling exported elt : (%,pos) -> R
   Adding DirectProduct(2,R) modemaps
Time: 0.14 SEC.

   compiling exported elt : (%,neg) -> R
   Adding DirectProduct(2,R) modemaps
Time: 0.02 SEC.

   compiling exported imaginary : () -> %
   Adding DirectProduct(2,R) modemaps
   Adding Vector R modemaps
   Adding List R modemaps
Time: 0.05 SEC.

   compiling exported real : % -> R
   Adding DirectProduct(2,R) modemaps
Time: 0.02 SEC.

   compiling exported imag : % -> R
   Adding DirectProduct(2,R) modemaps
Time: 0.01 SEC.

   compiling exported conj : % -> %
   Adding DirectProduct(2,R) modemaps
   Adding Vector R modemaps
   Adding List R modemaps
Time: 0.03 SEC.

   compiling exported * : (%,%) -> %
   Adding DirectProduct(2,R) modemaps
   Adding Vector R modemaps
   Adding List R modemaps
Time: 0.09 SEC.

   compiling exported norm : % -> R
Time: 0 SEC.

   compiling exported arg : % -> R
   Adding Boolean modemaps
Time: 0 SEC.

   compiling exported sqrt : % -> %
   Adding DirectProduct(2,R) modemaps
   Adding Vector R modemaps
   Adding List R modemaps
Time: 0.03 SEC.

   Adding OutputForm modemaps
   compiling exported coerce : % -> OutputForm
   Adding Complex R modemaps
Time: 0.02 SEC.

   Adding Complex R modemaps
   compiling exported coerce : % -> Complex R
Time: 0.01 SEC.

   Adding R modemaps
   augmenting R: DifferentialRing
   Adding R modemaps
   augmenting R: Evalable R
   Adding R modemaps
   augmenting R: LinearlyExplicitRingOver Integer
   Adding R modemaps
   augmenting R: PartialDifferentialRing Symbol
   Adding R modemaps
   augmenting R: RetractableTo Fraction Integer
   Adding R modemaps
   augmenting R: RetractableTo Integer
   Adding R modemaps
   augmenting R: DifferentialRing
   Adding R modemaps
   augmenting R: DifferentialRing
   Adding R modemaps
   augmenting R: RetractableTo Integer
   Adding R modemaps
   augmenting R: OrderedAbelianMonoidSup
   augmenting $: shallowlyMutable
   Adding R modemaps
   augmenting R: Finite
   Adding R modemaps
   augmenting R: OrderedAbelianMonoidSup
   Adding R modemaps
   augmenting R: OrderedRing
Predicate: 
(|has| $ (ATTRIBUTE |finiteAggregate|))   replaced by: 
T Predicate: 
(|has| $ (ATTRIBUTE |shallowlyMutable|))   replaced by: 
NIL Predicate: 
(|has| $ (ATTRIBUTE |finiteAggregate|))   replaced by: 
T (time taken in buildFunctor:  33)
Time: 0.53 SEC.

   Cumulative Statistics for Constructor ComplexManifold
      Time: 0.95 seconds

--------------non extending category----------------------
   ComplexManifold #1 of category Join(RadicalCategory,
      DirectProductCategory(2,#1)) with 
         ?.pos : (%,pos) -> #1
         ?.neg : (%,neg) -> #1
         imaginary : () -> %
         real : % -> #1
         imag : % -> #1
         conj : % -> %
         norm : % -> #1
         arg : % -> #1
         coerce : % -> Complex #1
    has no DirectProductCategory(NIL,#1)
   finalizing NRLIB CM 
   Processing ComplexManifold for Browser database:
--->-->ComplexManifold((elt (R % pos))): Not documented!!!!
--->-->ComplexManifold((elt (R % neg))): Not documented!!!!
--->-->ComplexManifold((imaginary (%))): Not documented!!!!
--->-->ComplexManifold((real (R %))): Not documented!!!!
--->-->ComplexManifold((imag (R %))): Not documented!!!!
--->-->ComplexManifold((conj (% %))): Not documented!!!!
--->-->ComplexManifold((norm (R %))): Not documented!!!!
--->-->ComplexManifold((arg (R %))): Not documented!!!!
--->-->ComplexManifold((coerce ((Complex R) %))): Not documented!!!!
--->-->ComplexManifold(constructor): Not documented!!!!
--->-->ComplexManifold: Missing Description
------------------------------------------------------------------------
   ComplexManifold is now explicitly exposed in frame initial 
   ComplexManifold will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/CM.NRLIB/code.o

axiom

a:CM(EXPR INT) := 3

(1)

Type: ComplexManifold? Expression Integer

axiom

b:CM(EXPR INT) := -5

(2)

Type: ComplexManifold? Expression Integer

axiom

real(a)

(3)

Type: Expression Integer

axiom

norm a

(4)

Type: Expression Integer

axiom

norm b

(5)

Type: Expression Integer

axiom

ab:=a*b

(6)

Type: ComplexManifold? Expression Integer

axiom

real ab

(7)

Type: Expression Integer

axiom

imag ab

(8)

Type: Expression Integer

axiom

s1:=sqrt(a)

(9)

Type: ComplexManifold? Expression Integer

axiom

real s1

(10)

Type: Expression Integer

axiom

imag s1

(11)

Type: Expression Integer

axiom

s2:=sqrt(b)

(12)

Type: ComplexManifold? Expression Integer

axiom

real s2

(13)

Type: Expression Integer

axiom

imag s2

(14)

Type: Expression Integer

axiom

I:CM(EXPR INT) := imaginary()

(15)

Type: ComplexManifold? Expression Integer

axiom

real I

(16)

Type: Expression Integer

axiom

norm I

(17)

Type: Expression Integer

axiom

imag I

(18)

Type: Expression Integer

axiom

imag conj I

(19)

Type: Expression Integer

axiom

s3:=sqrt(I)

(20)

Type: ComplexManifold? Expression Integer

axiom

s3*s3

(21)

Type: ComplexManifold? Expression Integer

axiom

c1:=conj(a+b*I)

(22)

Type: ComplexManifold? Expression Integer

axiom

real c1

(23)

Type: Expression Integer

axiom

imag c1

(24)

Type: Expression Integer

axiom

norm c1

(25)

Type: Expression Integer

axiom

c1::Complex(EXPR INT)

(26)

Type: Complex Expression Integer

axiom

sqrt %

(27)

Type: Complex Expression Integer

axiom

s4:=sqrt(c1)

(28)

Type: ComplexManifold? Expression Integer

axiom

real s4

(29)

Type: Expression Integer

axiom

imag s4

(30)

Type: Expression Integer

axiom

s4*s4

(31)

Type: ComplexManifold? Expression Integer

axiom

%::Complex(EXPR INT)

(32)

Type: Complex Expression Integer

axiom

normalize %

(33)

Type: Expression Integer

axiom

%::Complex(EXPR INT)::Complex(INT)

   Cannot convert from type Complex Expression Integer to Complex
      Integer for value
     +---+
   5\|- 1  + 3