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SandBoxBiQuaternions3
  
last edited 16 years ago

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Biquaternion Calculus Domain

**D. Cyganski and Bill Page - July 2007**

This version is implemented as a new domain in SPAD.

\begin{spad}
)abbrev domain BIQ BiQuaternion
BiQuaternion(R:Join(OrderedSet,CommutativeRing)): Exports == Implementation where
  C==>Complex Expression R
  Exports ==> QuaternionCategory(C) with
    qlist: List C -> %
    -- takes a complex list (parameter l) into a quaternion
    listq: % -> List C
    -- takes a quaternion into a list
    matrixq: % -> SquareMatrix(2,C)
    -- takes a quaternion into a matrix
    sig0: ()->%
    sig1: ()->%
    sig2: ()->%
    sig3: ()->%
    siglist: % -> List C
    -- Pauli basis representation of the biquaternion
    if C has PartialDifferentialRing(Symbol) then
      D: (%,Symbol,Symbol,Symbol) -> %
    -- quaternion derivative
    if C has RadicalCategory and C has Field and C has TranscendentalFunctionCategory then
      abs: % -> C
      rot: (C,%) -> %
      exp: % -> %
    -- biquaternion rotation
      _/: (%,%) -> %
      _/: (C,%) -> %
      _/: (%,C) -> %
    coerce: Complex R -> %

  Implementation ==> Quaternion C add
    import C

    coerce(z:Complex R):% ==
      import Expression(R),ComplexFunctions2(R,Expression R)
      map(coerce,z)::%

    -- Define a function that takes a complex list (parameter l) into a quaternion
    qlist(l:List C):%==
      import Integer
      quatern(l 1,l 2,l 3,l 4)
    -- Define a function that takes a quaternion into a list
    listq(x:%):List C == [real x, imagI x, imagJ x, imagK x]
    -- Define a function that takes a biquat into a matrix
    matrixq(x:%):SquareMatrix(2,C) ==
       import List List C
       matrix [[real x + imaginary()*imagI(x), imagJ x + imaginary()*imagK(x)],
               [-imagJ(x) + imaginary()*imagK(x), real x - imaginary()*imagI(x)]]
    -- Define a function that produces the Pauli basis representation of the biquaternion
    siglist(x:%):List C == [real x, -imaginary()*imagK(x),-imaginary()*imagJ(x),imaginary()*imagI(x)]
    sig0():% == quatern(1,0,0,0)
    sig1():% == imaginary() * quatern(0,0,0,1)
    sig2():% == imaginary() * quatern(0,0,1,0)
    sig3():% == -imaginary() * quatern(0,1,0,0)
    -- Define the quaternion derivative (Morgan, 2001, Eq. 2)
    if C has PartialDifferentialRing(Symbol) then
      D(q:%,x:Symbol,y:Symbol,z:Symbol):% == sig1()*D(q,x)+sig2()*D(q,y)+sig3()*D(q,z)
    -- Define a biquaternion rotation operator that takes a biquat through a rotation
    -- of theta radians about the axis defined by the unit q biquat (Morgan 2001, Eq 3).
    if C has RadicalCategory and C has Field and C has TranscendentalFunctionCategory then
      rot(theta:C,q:%):% ==
        import Integer, SparseMultivariatePolynomial(Integer, Kernel(C))
        cos(theta/2::C)::% - imaginary()*q*sin(theta/2::C)
      ((x:%) / (y:%)):% == x*inv(y)$%
      ((x:C) / (y:%)):% == (x::%)*inv(y)
      ((x:%) / (y:C)):% == x*inv(y::%)
      abs(q:%):C ==
        sqrt(retract(q*conjugate(q)))
      exp(q:%):% ==
        import Integer, SparseMultivariatePolynomial(Integer, Kernel(C))
        q-conjugate(q)=0 => exp(retract(q+conjugate(q))/2::C)*sig0()
        exp(retract(q+conjugate(q))/2::C) * (sig0()*cos(abs(q)) + (q-conjugate(q))/abs(q-conjugate(q)) * sin(abs(q)))
\end{spad}

\begin{axiom}
)show BiQuaternion
\end{axiom}

\begin{axiom}
Q := BiQuaternion Integer
q:Q := quatern(q0,q1,q2,q3) 
\end{axiom}

For testing the derivative we define this set of operators
\begin{axiom}
Ft:=operator 'Ft; Fx:=operator 'Fx; Fy:=operator 'Fy; Fz:=operator 'Fz;
\end{axiom}

Now form a general quaternion which is a function of x,y,z
\begin{axiom}
F:Q:=Ft(x,y,z)*sig0()+Fx(x,y,z)*sig1()+Fy(x,y,z)*sig2()+Fz(x,y,z)*sig3()
\end{axiom}


In the Pauli basis the derivative of this biquat should produce (Morgan 2001, eq 1)::

  D(Ft+F.sigma)=div(F)+(grad(Ft)+%i*curl(F)).sigma

which it does
\begin{axiom}
siglist(D(F,x,y,z))
\end{axiom}


Test

 (comment out this test later)

\begin{axiom}
%i::Q
abs(%i::Q)
abs(q)
cos(abs(%i::Q))
\end{axiom}

If I've defined these correctly, then the rotation about the x axis defined by qx below by 2 radians
should give the same answer as exponentiation to '-%i*qx' (not a very complete test)
\begin{axiom}
qx:Q:=sig1()
siglist(rot(2,qx))
siglist(exp(-%i::Q*qx))
\end{axiom}

which it does
\begin{axiom}
(%%(-1)=%%(-2))@Boolean
\end{axiom}

I would love to express a proof of equality such as::

   rot(theta,q) = exp((-theta/2)*%i*q)

for arbitrary real $\theta$ and biquaternion q as I would in Maple.

\begin{axiom}
theta:Complex Expression Integer := _\theta
map(simplify, siglist( rot(theta,q) - exp((-%i*theta/2) * q)))::List Expression Complex Integer
\end{axiom}

\begin{axiom}
map(simplify,siglist(rot(2,qx)))
\end{axiom}




Biquaternion Calculus Domain


D. Cyganski and Bill Page - July 2007


This version is implemented as a new domain in SPAD.


\begin{spad}
)abbrev domain BIQ BiQuaternion
BiQuaternion(R:Join(OrderedSet,CommutativeRing)): Exports == Implementation where
  C==>Complex Expression R
  Exports ==> QuaternionCategory(C) with
    qlist: List C -> %
    -- takes a complex list (parameter l) into a quaternion
    listq: % -> List C
    -- takes a quaternion into a list
    matrixq: % -> SquareMatrix(2,C)
    -- takes a quaternion into a matrix
    sig0: ()->%
    sig1: ()->%
    sig2: ()->%
    sig3: ()->%
    siglist: % -> List C
    -- Pauli basis representation of the biquaternion
    if C has PartialDifferentialRing(Symbol) then
      D: (%,Symbol,Symbol,Symbol) -> %
    -- quaternion derivative
    if C has RadicalCategory and C has Field and C has TranscendentalFunctionCategory then
      abs: % -> C
      rot: (C,%) -> %
      exp: % -> %
    -- biquaternion rotation
      _/: (%,%) -> %
      _/: (C,%) -> %
      _/: (%,C) -> %
    coerce: Complex R -> %
  Implementation ==> Quaternion C add
    import C
    coerce(z:Complex R):% ==
      import Expression(R),ComplexFunctions2(R,Expression R)
      map(coerce,z)::%


    -- Define a function that takes a complex list (parameter l) into a quaternion
    qlist(l:List C):%==
      import Integer
      quatern(l 1,l 2,l 3,l 4)
    -- Define a function that takes a quaternion into a list
    listq(x:%):List C == [real x, imagI x, imagJ x, imagK x]
    -- Define a function that takes a biquat into a matrix
    matrixq(x:%):SquareMatrix(2,C) ==
       import List List C
       matrix [[real x + imaginary()imagI(x), imagJ x + imaginary()imagK(x)],
               [-imagJ(x) + imaginary()imagK(x), real x - imaginary()imagI(x)]]
    -- Define a function that produces the Pauli basis representation of the biquaternion
    siglist(x:%):List C == [real x, -imaginary()imagK(x),-imaginary()imagJ(x),imaginary()imagI(x)]
    sig0():% == quatern(1,0,0,0)
    sig1():% == imaginary()  quatern(0,0,0,1)
    sig2():% == imaginary()  quatern(0,0,1,0)
    sig3():% == -imaginary()  quatern(0,1,0,0)
    -- Define the quaternion derivative (Morgan, 2001, Eq. 2)
    if C has PartialDifferentialRing(Symbol) then
      D(q:%,x:Symbol,y:Symbol,z:Symbol):% == sig1()D(q,x)+sig2()D(q,y)+sig3()D(q,z)
    -- Define a biquaternion rotation operator that takes a biquat through a rotation
    -- of theta radians about the axis defined by the unit q biquat (Morgan 2001, Eq 3).
    if C has RadicalCategory and C has Field and C has TranscendentalFunctionCategory then
      rot(theta:C,q:%):% ==
        import Integer, SparseMultivariatePolynomial(Integer, Kernel(C))
        cos(theta/2::C)::% - imaginary()qsin(theta/2::C)
      ((x:%) / (y:%)):% == xinv(y)$%
      ((x:C) / (y:%)):% == (x::%)inv(y)
      ((x:%) / (y:C)):% == xinv(y::%)
      abs(q:%):C ==
        sqrt(retract(qconjugate(q)))
      exp(q:%):% ==
        import Integer, SparseMultivariatePolynomial(Integer, Kernel(C))
        q-conjugate(q)=0 => exp(retract(q+conjugate(q))/2::C)sig0()
        exp(retract(q+conjugate(q))/2::C)  (sig0()cos(abs(q)) + (q-conjugate(q))/abs(q-conjugate(q)) * sin(abs(q)))
\end{spad}






\begin{axiom}
)show BiQuaternion
\end{axiom}


\begin{axiom}
Q := BiQuaternion Integer
q:Q := quatern(q0,q1,q2,q3) 
\end{axiom}


For testing the derivative we define this set of operators
\begin{axiom}
Ft:=operator 'Ft; Fx:=operator 'Fx; Fy:=operator 'Fy; Fz:=operator 'Fz;
\end{axiom}


Now form a general quaternion which is a function of x,y,z
\begin{axiom}
F:Q:=Ft(x,y,z)sig0()+Fx(x,y,z)sig1()+Fy(x,y,z)sig2()+Fz(x,y,z)sig3()
\end{axiom}


In the Pauli basis the derivative of this biquat should produce (Morgan 2001, eq 1):

  D(Ft+F.sigma)=div(F)+(grad(Ft)+%i*curl(F)).sigma





which it does
\begin{axiom}
siglist(D(F,x,y,z))
\end{axiom}


Test


 (comment out this test later)


\begin{axiom}
%i::Q
abs(%i::Q)
abs(q)
cos(abs(%i::Q))
\end{axiom}


If I've defined these correctly, then the rotation about the x axis defined by qx below by 2 radians
should give the same answer as exponentiation to -%i*qx (not a very complete test)
\begin{axiom}
qx:Q:=sig1()
siglist(rot(2,qx))
siglist(exp(-%i::Q*qx))
\end{axiom}


which it does
\begin{axiom}
(%%(-1)=%%(-2))@Boolean
\end{axiom}


I would love to express a proof of equality such as:

   rot(theta,q) = exp((-theta/2)*%i*q)





for arbitrary real $\theta$ and biquaternion q as I would in Maple.


\begin{axiom}
theta:Complex Expression Integer := _\theta
map(simplify, siglist( rot(theta,q) - exp((-%itheta/2)  q)))::List Expression Complex Integer
\end{axiom}


\begin{axiom}
map(simplify,siglist(rot(2,qx)))
\end{axiom}





    Some or all expressions may not have rendered properly,
    because Axiom returned the following error:
Error: export HOME=/var/zope2/var/LatexWiki; ulimit -t 600; export LD_LIBRARY_PATH=/usr/local/lib/fricas/target/x86_64-linux-gnu/lib; LANG=en_US.UTF-8 /usr/local/lib/fricas/target/x86_64-linux-gnu/bin/fricas -nosman < /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/405825799807838506-25px.axm
Killed


Checking for foreign routines
FRICAS="/usr/local/lib/fricas/target/x86_64-linux-gnu"
spad-lib="/usr/local/lib/fricas/target/x86_64-linux-gnu/lib/libspad.so"
foreign routines found
openServer result -2
                       FriCAS Computer Algebra System 
                            Version: FriCAS 1.3.9
                 Timestamp: Sun Sep 17 04:15:23 PM CEST 2023
-----------------------------------------------------------------------------
   Issue )copyright to view copyright notices.
   Issue )summary for a summary of useful system commands.
   Issue )quit to leave FriCAS and return to shell.
-----------------------------------------------------------------------------


(1) -> (1) -> (1) -> (1) -> (1) -> (1) -> <spad>
)abbrev domain BIQ BiQuaternion
BiQuaternion(R:Join(OrderedSet,CommutativeRing)): Exports == Implementation where
  C==>Complex Expression R
  Exports ==> QuaternionCategory(C) with
    qlist: List C -> %
    -- takes a complex list (parameter l) into a quaternion
    listq: % -> List C
    -- takes a quaternion into a list
    matrixq: % -> SquareMatrix(2,C)
    -- takes a quaternion into a matrix
    sig0: ()->%
    sig1: ()->%
    sig2: ()->%
    sig3: ()->%
    siglist: % -> List C
    -- Pauli basis representation of the biquaternion
    if C has PartialDifferentialRing(Symbol) then
      D: (%,Symbol,Symbol,Symbol) -> %
    -- quaternion derivative
    if C has RadicalCategory and C has Field and C has TranscendentalFunctionCategory then
      abs: % -> C
      rot: (C,%) -> %
      exp: % -> %
    -- biquaternion rotation
      /: (%,%) -> %
      /: (C,%) -> %
      _/: (%,C) -> %
    coerce: Complex R -> %
  Implementation ==> Quaternion C add
    import C
    coerce(z:Complex R):% ==
      import Expression(R),ComplexFunctions2(R,Expression R)
      map(coerce,z)::%


    -- Define a function that takes a complex list (parameter l) into a quaternion
    qlist(l:List C):%==
      import Integer
      quatern(l 1,l 2,l 3,l 4)
    -- Define a function that takes a quaternion into a list
    listq(x:%):List C == [real x, imagI x, imagJ x, imagK x]
    -- Define a function that takes a biquat into a matrix
    matrixq(x:%):SquareMatrix(2,C) ==
       import List List C
       matrix [[real x + imaginary()imagI(x), imagJ x + imaginary()imagK(x)],
               [-imagJ(x) + imaginary()imagK(x), real x - imaginary()imagI(x)]]
    -- Define a function that produces the Pauli basis representation of the biquaternion
    siglist(x:%):List C == [real x, -imaginary()imagK(x),-imaginary()imagJ(x),imaginary()imagI(x)]
    sig0():% == quatern(1,0,0,0)
    sig1():% == imaginary()  quatern(0,0,0,1)
    sig2():% == imaginary()  quatern(0,0,1,0)
    sig3():% == -imaginary()  quatern(0,1,0,0)
    -- Define the quaternion derivative (Morgan, 2001, Eq. 2)
    if C has PartialDifferentialRing(Symbol) then
      D(q:%,x:Symbol,y:Symbol,z:Symbol):% == sig1()D(q,x)+sig2()D(q,y)+sig3()D(q,z)
    -- Define a biquaternion rotation operator that takes a biquat through a rotation
    -- of theta radians about the axis defined by the unit q biquat (Morgan 2001, Eq 3).
    if C has RadicalCategory and C has Field and C has TranscendentalFunctionCategory then
      rot(theta:C,q:%):% ==
        import Integer, SparseMultivariatePolynomial(Integer, Kernel(C))
        cos(theta/2::C)::% - imaginary()qsin(theta/2::C)
      ((x:%) / (y:%)):% == xinv(y)$%
      ((x:C) / (y:%)):% == (x::%)inv(y)
      ((x:%) / (y:C)):% == xinv(y::%)
      abs(q:%):C ==
        sqrt(retract(qconjugate(q)))
      exp(q:%):% ==
        import Integer, SparseMultivariatePolynomial(Integer, Kernel(C))
        q-conjugate(q)=0 => exp(retract(q+conjugate(q))/2::C)sig0()
        exp(retract(q+conjugate(q))/2::C)  (sig0()cos(abs(q)) + (q-conjugate(q))/abs(q-conjugate(q))  sin(abs(q)))</spad>
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/8384816125117043612-25px001.spad
      using old system compiler.
   BIQ abbreviates domain BiQuaternion 
------------------------------------------------------------------------
   initializing NRLIB BIQ for BiQuaternion 
   compiling into NRLIB BIQ 
***** Domain: R already in scope
   importing Complex Expression R
   compiling exported coerce : Complex R -> %
Time: 0.05 SEC.


   compiling exported qlist : List Complex Expression R -> %
Time: 0.01 SEC.


   compiling exported listq : % -> List Complex Expression R
Time: 0.00 SEC.


   compiling exported matrixq : % -> SquareMatrix(2,Complex Expression R)
Time: 0.01 SEC.


   compiling exported siglist : % -> List Complex Expression R
Time: 0.00 SEC.


   compiling exported sig0 : () -> %
Time: 0.00 SEC.


   compiling exported sig1 : () -> %
Time: 0 SEC.


   compiling exported sig2 : () -> %
Time: 0.00 SEC.


   compiling exported sig3 : () -> %
Time: 0.00 SEC.


   compiling exported D : (%,Symbol,Symbol,Symbol) -> %
Time: 0.02 SEC.






**** Domain: (Complex (Expression R)) already in scope
augmenting (Complex (Expression R)): (RadicalCategory)
** Domain: (Complex (Expression R)) already in scope
augmenting (Complex (Expression R)): (Field)
**** Domain: (Complex (Expression R)) already in scope
augmenting (Complex (Expression R)): (TranscendentalFunctionCategory)
   compiling exported rot : (Complex Expression R,%) -> %
Time: 0.04 SEC.
   compiling exported / : (%,%) -> %
Time: 0.00 SEC.


   compiling exported / : (Complex Expression R,%) -> %
Time: 0 SEC.


   compiling exported / : (%,Complex Expression R) -> %
Time: 0 SEC.


   compiling exported abs : % -> Complex Expression R
Time: 0 SEC.


   compiling exported exp : % -> %
Time: 0.07 SEC.




**** Domain: (Complex (Expression R)) already in scope
augmenting (Complex (Expression R)): (Field)
** Domain: (Complex (Expression R)) already in scope
augmenting (Complex (Expression R)): (RadicalCategory)
** Domain: (Complex (Expression R)) already in scope
augmenting (Complex (Expression R)): (TranscendentalFunctionCategory)
** Domain: (Complex (Expression R)) already in scope
augmenting (Complex (Expression R)): (CharacteristicNonZero)
** Domain: (Complex (Expression R)) already in scope
augmenting (Complex (Expression R)): (ConvertibleTo (InputForm))
** Domain: (Complex (Expression R)) already in scope
augmenting (Complex (Expression R)): (DifferentialRing)
** Domain: (Complex (Expression R)) already in scope
augmenting (Complex (Expression R)): (Eltable (Complex (Expression R)) (Complex (Expression R)))
** Domain: (Complex (Expression R)) already in scope
augmenting (Complex (Expression R)): (EntireRing)
** Domain: (Complex (Expression R)) already in scope
augmenting (Complex (Expression R)): (Evalable (Complex (Expression R)))
** Domain: (Complex (Expression R)) already in scope
augmenting (Complex (Expression R)): (Field)
** Domain: (Complex (Expression R)) already in scope
augmenting (Complex (Expression R)): (InnerEvalable (Symbol) (Complex (Expression R)))
** Domain: (Complex (Expression R)) already in scope
augmenting (Complex (Expression R)): (IntegerNumberSystem)
** Domain: (Complex (Expression R)) already in scope
augmenting (Complex (Expression R)): (LinearlyExplicitOver (Integer))
** Domain: (Complex (Expression R)) already in scope
augmenting (Complex (Expression R)): (OrderedSet)
** Domain: (Complex (Expression R)) already in scope
augmenting (Complex (Expression R)): (RetractableTo (Fraction (Integer)))
**** Domain: (Complex (Expression R)) already in scope
augmenting (Complex (Expression R)): (RetractableTo (Integer))
(time taken in buildFunctor:  30710)


;;;     ***       |BiQuaternion| REDEFINED


;;;     ***       |BiQuaternion| REDEFINED
Time: 0.19 SEC.
   Cumulative Statistics for Constructor BiQuaternion
      Time: 0.40 seconds


   finalizing NRLIB BIQ 
   Processing BiQuaternion for Browser database:
--->-->BiQuaternion(constructor): Not documented!!!!
--->-->BiQuaternion((qlist (% (List (Complex (Expression R)))))): Not documented!!!!
--->-->BiQuaternion((listq ((List (Complex (Expression R))) %))): Not documented!!!!
--->-->BiQuaternion((matrixq ((SquareMatrix 2 (Complex (Expression R))) %))): Not documented!!!!
--->-->BiQuaternion((sig0 (%))): Not documented!!!!
--->-->BiQuaternion((sig1 (%))): Not documented!!!!
--->-->BiQuaternion((sig2 (%))): Not documented!!!!
--->-->BiQuaternion((sig3 (%))): Not documented!!!!
--->-->BiQuaternion((siglist ((List (Complex (Expression R))) %))): Not documented!!!!
--->-->BiQuaternion((D (% % (Symbol) (Symbol) (Symbol)))): Not documented!!!!
--->-->BiQuaternion((abs ((Complex (Expression R)) %))): Not documented!!!!
--->-->BiQuaternion((rot (% (Complex (Expression R)) %))): Not documented!!!!
--->-->BiQuaternion((exp (% %))): Not documented!!!!
--->-->BiQuaternion((/ (% % %))): Not documented!!!!
--->-->BiQuaternion((/ (% (Complex (Expression R)) %))): Not documented!!!!
--->-->BiQuaternion((/ (% % (Complex (Expression R))))): Not documented!!!!
--->-->BiQuaternion((coerce (% (Complex R)))): Not documented!!!!
--->-->BiQuaternion(): Missing Description
; compiling file "/var/aw/var/LatexWiki/BIQ.NRLIB/BIQ.lsp" (written 19 SEP 2023 08:17:38 AM):




; wrote /var/aw/var/LatexWiki/BIQ.NRLIB/BIQ.fasl
; compilation finished in 0:00:00.052
------------------------------------------------------------------------
   BiQuaternion is now explicitly exposed in frame initial 
   BiQuaternion will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/BIQ.NRLIB/BIQ


(1) -> )show BiQuaternion


 BiQuaternion(R: Join(OrderedSet,CommutativeRing)) is a domain constructor
 Abbreviation for BiQuaternion is BIQ 
 This constructor is exposed in this frame.
------------------------------- Operations --------------------------------


 ?? : (Integer, %) -> %               ?? : (%, %) -> %
 ?? : (PositiveInteger, %) -> %       ?+? : (%, %) -> %
 -? : % -> %                           ?-? : (%, %) -> %
 ?=? : (%, %) -> Boolean               1 : () -> %
 0 : () -> %                           ?^? : (%, PositiveInteger) -> %
 annihilate? : (%, %) -> Boolean       antiCommutator : (%, %) -> %
 associator : (%, %, %) -> %           coerce : Complex(R) -> %
 coerce : Integer -> %                 coerce : % -> OutputForm
 commutator : (%, %) -> %              conjugate : % -> %
 latex : % -> String                   one? : % -> Boolean
 opposite? : (%, %) -> Boolean         recip : % -> Union(%,"failed")
 sample : () -> %                      sig0 : () -> %
 sig1 : () -> %                        sig2 : () -> %
 sig3 : () -> %                        zero? : % -> Boolean
 ?~=? : (%, %) -> Boolean             
 ?? : (%, Fraction(Integer)) -> % if Complex(Expression(R)) has FIELD
 ?? : (Fraction(Integer), %) -> % if Complex(Expression(R)) has FIELD
 ?? : (%, Integer) -> % if Complex(Expression(R)) has LINEXP(INT)
 ?? : (%, Complex(Expression(R))) -> %
 ?? : (Complex(Expression(R)), %) -> %
 ?*? : (NonNegativeInteger, %) -> %
 ?/? : (%, Complex(Expression(R))) -> % if Complex(Expression(R)) has FIELD and Complex(Expression(R)) has RADCAT and Complex(Expression(R)) has TRANFUN
 ?/? : (Complex(Expression(R)), %) -> % if Complex(Expression(R)) has FIELD and Complex(Expression(R)) has RADCAT and Complex(Expression(R)) has TRANFUN
 ?/? : (%, %) -> % if Complex(Expression(R)) has FIELD and Complex(Expression(R)) has RADCAT and Complex(Expression(R)) has TRANFUN
 ?<? : (%, %) -> Boolean if Complex(Expression(R)) has ORDSET
 ?<=? : (%, %) -> Boolean if Complex(Expression(R)) has ORDSET
 ?>? : (%, %) -> Boolean if Complex(Expression(R)) has ORDSET
 ?>=? : (%, %) -> Boolean if Complex(Expression(R)) has ORDSET
 D : (%, Symbol, Symbol, Symbol) -> % if Complex(Expression(R)) has PDRING(SYMBOL)
 D : (%, (Complex(Expression(R)) -> Complex(Expression(R)))) -> %
 D : (%, (Complex(Expression(R)) -> Complex(Expression(R))), NonNegativeInteger) -> %
 D : (%, List(Symbol), List(NonNegativeInteger)) -> % if Complex(Expression(R)) has PDRING(SYMBOL)
 D : (%, Symbol, NonNegativeInteger) -> % if Complex(Expression(R)) has PDRING(SYMBOL)
 D : (%, List(Symbol)) -> % if Complex(Expression(R)) has PDRING(SYMBOL)
 D : (%, Symbol) -> % if Complex(Expression(R)) has PDRING(SYMBOL)
 D : (%, NonNegativeInteger) -> % if Complex(Expression(R)) has DIFRING
 D : % -> % if Complex(Expression(R)) has DIFRING
 ?^? : (%, Integer) -> % if Complex(Expression(R)) has FIELD
 ?^? : (%, NonNegativeInteger) -> %
 abs : % -> Complex(Expression(R)) if Complex(Expression(R)) has FIELD and Complex(Expression(R)) has RADCAT and Complex(Expression(R)) has TRANFUN or Complex(Expression(R)) has RNS
 associates? : (%, %) -> Boolean if Complex(Expression(R)) has ENTIRER
 characteristic : () -> NonNegativeInteger
 charthRoot : % -> Union(%,"failed") if Complex(Expression(R)) has CHARNZ
 coerce : Fraction(Integer) -> % if Complex(Expression(R)) has FIELD or Complex(Expression(R)) has RETRACT(FRAC(INT))
 coerce : Complex(Expression(R)) -> %
 convert : % -> InputForm if Complex(Expression(R)) has KONVERT(INFORM)
 differentiate : (%, (Complex(Expression(R)) -> Complex(Expression(R)))) -> %
 differentiate : (%, (Complex(Expression(R)) -> Complex(Expression(R))), NonNegativeInteger) -> %
 differentiate : (%, List(Symbol), List(NonNegativeInteger)) -> % if Complex(Expression(R)) has PDRING(SYMBOL)
 differentiate : (%, Symbol, NonNegativeInteger) -> % if Complex(Expression(R)) has PDRING(SYMBOL)
 differentiate : (%, List(Symbol)) -> % if Complex(Expression(R)) has PDRING(SYMBOL)
 differentiate : (%, Symbol) -> % if Complex(Expression(R)) has PDRING(SYMBOL)
 differentiate : (%, NonNegativeInteger) -> % if Complex(Expression(R)) has DIFRING
 differentiate : % -> % if Complex(Expression(R)) has DIFRING
 elt : (%, Complex(Expression(R))) -> % if Complex(Expression(R)) has ELTAB(COMPLEX(EXPR(R)),COMPLEX(EXPR(R)))
 eval : (%, Symbol, Complex(Expression(R))) -> % if Complex(Expression(R)) has IEVALAB(SYMBOL,COMPLEX(EXPR(R)))
 eval : (%, List(Symbol), List(Complex(Expression(R)))) -> % if Complex(Expression(R)) has IEVALAB(SYMBOL,COMPLEX(EXPR(R)))
 eval : (%, List(Equation(Complex(Expression(R))))) -> % if Complex(Expression(R)) has EVALAB(COMPLEX(EXPR(R)))
 eval : (%, Equation(Complex(Expression(R)))) -> % if Complex(Expression(R)) has EVALAB(COMPLEX(EXPR(R)))
 eval : (%, Complex(Expression(R)), Complex(Expression(R))) -> % if Complex(Expression(R)) has EVALAB(COMPLEX(EXPR(R)))
 eval : (%, List(Complex(Expression(R))), List(Complex(Expression(R)))) -> % if Complex(Expression(R)) has EVALAB(COMPLEX(EXPR(R)))
 exp : % -> % if Complex(Expression(R)) has FIELD and Complex(Expression(R)) has RADCAT and Complex(Expression(R)) has TRANFUN
 exquo : (%, %) -> Union(%,"failed") if Complex(Expression(R)) has ENTIRER
 imagI : % -> Complex(Expression(R))
 imagJ : % -> Complex(Expression(R))
 imagK : % -> Complex(Expression(R))
 inv : % -> % if Complex(Expression(R)) has FIELD
 leftPower : (%, NonNegativeInteger) -> %
 leftPower : (%, PositiveInteger) -> %
 leftRecip : % -> Union(%,"failed")
 listq : % -> List(Complex(Expression(R)))
 map : ((Complex(Expression(R)) -> Complex(Expression(R))), %) -> %
 matrixq : % -> SquareMatrix(2,Complex(Expression(R)))
 max : (%, %) -> % if Complex(Expression(R)) has ORDSET
 min : (%, %) -> % if Complex(Expression(R)) has ORDSET
 norm : % -> Complex(Expression(R))
 plenaryPower : (%, PositiveInteger) -> %
 qlist : List(Complex(Expression(R))) -> %
 quatern : (Complex(Expression(R)), Complex(Expression(R)), Complex(Expression(R)), Complex(Expression(R))) -> %
 rational : % -> Fraction(Integer) if Complex(Expression(R)) has INS
 rational? : % -> Boolean if Complex(Expression(R)) has INS
 rationalIfCan : % -> Union(Fraction(Integer),"failed") if Complex(Expression(R)) has INS
 real : % -> Complex(Expression(R))
 reducedSystem : Matrix(%) -> Matrix(Complex(Expression(R)))
 reducedSystem : (Matrix(%), Vector(%)) -> Record(mat: Matrix(Complex(Expression(R))),vec: Vector(Complex(Expression(R))))
 reducedSystem : (Matrix(%), Vector(%)) -> Record(mat: Matrix(Integer),vec: Vector(Integer)) if Complex(Expression(R)) has LINEXP(INT)
 reducedSystem : Matrix(%) -> Matrix(Integer) if Complex(Expression(R)) has LINEXP(INT)
 retract : % -> Complex(Expression(R))
 retract : % -> Fraction(Integer) if Complex(Expression(R)) has RETRACT(FRAC(INT))
 retract : % -> Integer if Complex(Expression(R)) has RETRACT(INT)
 retractIfCan : % -> Union(Complex(Expression(R)),"failed")
 retractIfCan : % -> Union(Fraction(Integer),"failed") if Complex(Expression(R)) has RETRACT(FRAC(INT))
 retractIfCan : % -> Union(Integer,"failed") if Complex(Expression(R)) has RETRACT(INT)
 rightPower : (%, NonNegativeInteger) -> %
 rightPower : (%, PositiveInteger) -> %
 rightRecip : % -> Union(%,"failed")
 rot : (Complex(Expression(R)), %) -> % if Complex(Expression(R)) has FIELD and Complex(Expression(R)) has RADCAT and Complex(Expression(R)) has TRANFUN
 siglist : % -> List(Complex(Expression(R)))
 smaller? : (%, %) -> Boolean if Complex(Expression(R)) has ORDSET
 subtractIfCan : (%, %) -> Union(%,"failed")
 unit? : % -> Boolean if Complex(Expression(R)) has ENTIRER
 unitCanonical : % -> % if Complex(Expression(R)) has ENTIRER
 unitNormal : % -> Record(unit: %,canonical: %,associate: %) if Complex(Expression(R)) has ENTIRER


(1) -> Q := BiQuaternion Integer


$$
BiQuaternion(Integer) 
\leqno(1)
$$
                                                                   Type: Type
q:Q := quatern(q0,q1,q2,q3)




$$
q0+{q1 \  i}+{q2 \  j}+{q3 \  k} 
\leqno(2)
$$
                                                  Type: BiQuaternion(Integer)
(3) -> Ft:=operator 'Ft; Fx:=operator 'Fx; Fy:=operator 'Fy; Fz:=operator 'Fz;
                                                          Type: BasicOperator
(4) -> F:Q:=Ft(x,y,z)sig0()+Fx(x,y,z)sig1()+Fy(x,y,z)sig2()+Fz(x,y,z)sig3()






$$
{Ft 
\left(
{x, \: y, \: z} 
\right)}
-{{Fz 
\left(
{x, \: y, \: z} 
\right)}
\  i \  i}+{{Fy 
\left(
{x, \: y, \: z} 
\right)}
\  i \  j}+{{Fx 
\left(
{x, \: y, \: z} 
\right)}
\  i \  k} 
\leqno(4)
$$
                                                  Type: BiQuaternion(Integer)
(5) -> siglist(D(F,x,y,z))




$$
\left[
{{{Fz \sb {{,3}}} 
\left(
{x, \: y, \: z} 
\right)}+{{Fy
\sb {{,2}}} 
\left(
{x, \: y, \: z} 
\right)}+{{Fx
\sb {{,1}}} 
\left(
{x, \: y, \: z} 
\right)}},
\: {{{Ft \sb {{,1}}} 
\left(
{x, \: y, \: z} 
\right)}+{{\left(
{{Fz \sb {{,2}}} 
\left(
{x, \: y, \: z} 
\right)}
-{{Fy \sb {{,3}}} 
\left(
{x, \: y, \: z} 
\right)}
\right)}
\  i}}, \: {{{Ft \sb {{,2}}} 
\left(
{x, \: y, \: z} 
\right)}+{{\left(
-{{Fz \sb {{,1}}} 
\left(
{x, \: y, \: z} 
\right)}+{{Fx
\sb {{,3}}} 
\left(
{x, \: y, \: z} 
\right)}
\right)}
\  i}}, \: {{{Ft \sb {{,3}}} 
\left(
{x, \: y, \: z} 
\right)}+{{\left(
{{Fy \sb {{,1}}} 
\left(
{x, \: y, \: z} 
\right)}
-{{Fx \sb {{,2}}} 
\left(
{x, \: y, \: z} 
\right)}
\right)}
\  i}} 
\right]
\leqno(5)
$$
                                     Type: List(Complex(Expression(Integer)))
(6) -> %i::Q
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edSet,CommutativeRing)): Ex-ports == Im-ple-men-ta-tion where C==>Complex
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