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SandBoxComplexManifold
  
last edited 12 years ago by test1

Edit detail for SandBoxComplexManifold revision 3 of 7
1 2 3 4 5 6 7
Editor: Bill Page
Time: 2009/06/20 20:37:50 GMT-7
Note: sqrt 
changed:
-ComplexManifold(R:Join(Field,RadicalCategory,OrderedSet)): Join(RadicalCategory,DirectProductCategory(2,R)) with
ComplexManifold(R:Join(Field,RadicalCategory,TranscendentalFunctionCategory)): Join(RadicalCategory,DirectProductCategory(2,R)) with

added:
    norm: % -> R
    arg:  % -> R

added:
    -- represent as conjugate pair

added:
    -- just swap

changed:
-
-    -- sqrt(abs real x)+sqrt(abs imag x)
-    -- sign(real x)*sqrt(abs real x)-sign(imag x)*sqrt(abs imag x)
-    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)) ]
    -- 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 == (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))] ) )

    -- [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


added:
norm a
norm b
ab:=a*b
real ab
imag ab

added:
real I
norm I

added:
norm c1
s3:=sqrt(c1)
real s3
imag s3

spad
)abbrev domain CM ComplexManifold
ComplexManifold(R:Join(Field,RadicalCategory,TranscendentalFunctionCategory)): Join(RadicalCategory,DirectProductCategory(2,R)) with
    elt:(%,"pos") -> R
    elt:(%,"neg") -> R
    imaginary: () -> %
    real: % -> R
    imag: % -> R
    conj: % -> %
    norm: % -> R
    arg:  % -> 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 == (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))] ) )

    -- [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
spad
   Compiling OpenAxiom source code from file 
      /var/zope2/var/LatexWiki/7433806807856594830-25px001.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 Integer modemaps
   Adding NonNegativeInteger modemaps
   Adding PositiveInteger modemaps
   Adding Rep modemaps
   compiling exported elt : (%,pos) -> R
   Adding DirectProduct(2,R) modemaps
Time: 0.12 SEC.

   compiling exported elt : (%,neg) -> R
   Adding DirectProduct(2,R) modemaps
Time: 0.01 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 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: 0.02 SEC.

   compiling exported * : (%,%) -> %
   Adding DirectProduct(2,R) modemaps
   Adding Vector R modemaps
   Adding List R modemaps
Time: 0.06 SEC.

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

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

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

   Adding OutputForm modemaps
   compiling exported coerce : % -> OutputForm
   Adding Complex R modemaps
Time: 0.05 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:  75)
Time: 0.87 SEC.

   Cumulative Statistics for Constructor ComplexManifold
      Time: 1.21 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
    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(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
LatexWiki Image(1)
Type: ComplexManifold? Expression Integer
axiom
b:CM(EXPR INT) := -5
LatexWiki Image(2)
Type: ComplexManifold? Expression Integer
axiom
real(a)
LatexWiki Image(3)
Type: Expression Integer
axiom
norm a
LatexWiki Image(4)
Type: Expression Integer
axiom
norm b
LatexWiki Image(5)
Type: Expression Integer
axiom
ab:=a*b
LatexWiki Image(6)
Type: ComplexManifold? Expression Integer
axiom
real ab
LatexWiki Image(7)
Type: Expression Integer
axiom
imag ab
LatexWiki Image(8)
Type: Expression Integer
axiom
s1:=sqrt(a)
LatexWiki Image(9)
Type: ComplexManifold? Expression Integer
axiom
real s1
LatexWiki Image(10)
Type: Expression Integer
axiom
imag s1
LatexWiki Image(11)
Type: Expression Integer
axiom
s2:=sqrt(b)
LatexWiki Image(12)
Type: ComplexManifold? Expression Integer
axiom
real s2
LatexWiki Image(13)
Type: Expression Integer
axiom
imag s2
LatexWiki Image(14)
Type: Expression Integer
axiom
I:CM(EXPR INT) := imaginary()
LatexWiki Image(15)
Type: ComplexManifold? Expression Integer
axiom
real I
LatexWiki Image(16)
Type: Expression Integer
axiom
norm I
LatexWiki Image(17)
Type: Expression Integer
axiom
imag I
LatexWiki Image(18)
Type: Expression Integer
axiom
imag conj I
LatexWiki Image(19)
Type: Expression Integer
axiom
c1:=conj(a+b*I)
LatexWiki Image(20)
Type: ComplexManifold? Expression Integer
axiom
real c1
LatexWiki Image(21)
Type: Expression Integer
axiom
imag c1
LatexWiki Image(22)
Type: Expression Integer
axiom
norm c1
LatexWiki Image(23)
Type: Expression Integer
axiom
s3:=sqrt(c1)
LatexWiki Image(24)
Type: ComplexManifold? Expression Integer
axiom
real s3
LatexWiki Image(25)
Type: Expression Integer
axiom
imag s3
LatexWiki Image(26)
Type: Expression Integer