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Edit detail for SandBoxTensorProductPolynomial revision 7 of 12

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Editor: Bill Page
Time: 2008/08/26 13:45:48 GMT-7
Note: new spad version

added:
Here's another way to write this - maybe better this way as first
step to express associativity of the tensor product.
\begin{spad}
)abbrev package TPROD2 TensorProduct2
macro IE1 == IndexedExponents(VAR1)
macro IE2 == IndexedExponents(VAR2)
macro S == Sum(VAR1,VAR2)
macro IEP == IndexedExponents(S)
macro SMP == SparseMultivariatePolynomial(R,S)

TensorProduct2(R:Ring, VAR1: OrderedSet, VAR2: OrderedSet, P:PolynomialCategory(R,IE1,VAR1), Q:PolynomialCategory(R,IE2,VAR2)): with
    _\_/: (P,Q) -> SMP
  == add
    scanIndex1(x:IE1):IEP ==
      zero? x => 0
      monomial(leadingCoefficient(x), in1(leadingSupport(x))$S) + scanIndex1(reductum(x))
    scanIndex2(x:IE2):IEP ==
      zero? x => 0
      monomial(leadingCoefficient(x), in2(leadingSupport(x))$S) + scanIndex2(reductum(x))
    mapMonomial1(p:P):SMP ==
      monomial(coefficient(p,degree p),scanIndex1(degree(p)))$SMP
    mapMonomial2(q:Q):SMP ==
      monomial(coefficient(q,degree q),scanIndex2(degree(q)))$SMP
    scanPoly1(p:P):SMP ==
      p=0 => 0
      mapMonomial1(leadingMonomial(p))+scanPoly1(reductum p)
    scanPoly2(q:Q):SMP ==
      q=0 => 0
      mapMonomial2(leadingMonomial(q))+scanPoly2(reductum q)

    (p:P \/ q:Q):SMP == scanPoly1(p)*scanPoly2(q)
\end{spad}

\begin{axiom}
test( p\/q = r )
test( (p+q) \/ w = (p\/w) + (q\/w) )
test( p \/ (q+w) = (p\/q) + (p\/w) )
test( p \/ (23*w) = 23*(p\/w) )
test( (23*p) \/ w = 23*(p\/w) )
p\/q\/w
\end{axiom}


http://en.wikipedia.org/wiki/Tensor_product

A tensor product is "the most general bilinear operation" available in a specified domain of computation, satisfying:

LatexWiki Image(1)
LatexWiki Image(2)
LatexWiki Image(3)

We can use the domain constructor Sum [SandBoxSum]?

axiom
)lib SUM
Sum is now explicitly exposed in frame initial Sum will be automatically loaded when needed from /var/zope2/var/LatexWiki/SUM.NRLIB/code.o

First we can define some recursive operations on the polynomials

axiom
scanPoly(p,n) == _
  (p=0 => 0; mapMonomial(leadingMonomial(p),n)+scanPoly(reductum p,n))
Type: Void
axiom
mapMonomial(p,n) == _
  monomial(coefficient(p,degree p),scanIndex(degree(p),n))$SMP(Integer,Sum(Symbol,Symbol))
Type: Void
axiom
scanIndex(p,n) == _
  (zero? p => 0$IndexedExponents(Sum(Symbol,Symbol)); _
    monomial(leadingCoefficient(p), _
      if n=1 then in1(leadingSupport(p))$Sum(Symbol,Symbol) _
             else in2(leadingSupport(p))$Sum(Symbol,Symbol) _
    )$IndexedExponents(Sum(Symbol,Symbol))+ _
      scanIndex(reductum(p),n))
Type: Void

For example:

axiom
-- functions are first compiled here
--
scanPoly(x,1)
axiom
Compiling function scanIndex with type (IndexedExponents Symbol,
      Integer) -> IndexedExponents Sum(Symbol,Symbol) 
; (DEFUN |*2;scanIndex;1;initial| ...) is being compiled.
;; The variable |*2;scanIndex;1;initial;MV| is undefined.
;; The compiler will assume this variable is a global.
axiom
Compiling function mapMonomial with type (Polynomial Integer,Integer
      ) -> SparseMultivariatePolynomial(Integer,Sum(Symbol,Symbol)) 
; (DEFUN |*2;mapMonomial;1;initial| ...) is being compiled.
;; The variable |*2;mapMonomial;1;initial;MV| is undefined.
;; The compiler will assume this variable is a global.
axiom
Compiling function scanPoly with type (Polynomial Integer,Integer)
       -> SparseMultivariatePolynomial(Integer,Sum(Symbol,Symbol)) 
; (DEFUN |*2;scanPoly;3;initial| ...) is being compiled.
;; The variable |*2;scanPoly;3;initial;MV| is undefined.
;; The compiler will assume this variable is a global.
axiom
Compiling function scanPoly with type (Polynomial Integer,Integer)
       -> SparseMultivariatePolynomial(Integer,Sum(Symbol,Symbol)) 
;;; *** |*2;scanPoly;3;initial| REDEFINED ; (DEFUN |*2;scanPoly;3;initial| ...) is being compiled. ;; The variable |*2;scanPoly;3;initial;MV| is undefined. ;; The compiler will assume this variable is a global.
axiom
Compiling function scanPoly with type (Variable x,Integer) -> 
      SparseMultivariatePolynomial(Integer,Sum(Symbol,Symbol)) 
; (DEFUN |*2;scanPoly;1;initial| ...) is being compiled.
;; The variable |*2;scanPoly;1;initial;MV| is undefined.
;; The compiler will assume this variable is a global.
LatexWiki Image(4)
Type: SparseMultivariatePolynomial?(Integer,Sum(Symbol,Symbol))

injects the polynomial x in to the tensor product. So now the full tensor product is just:

axiom
tensorPoly(p,q) == _
  scanPoly(p,1)*scanPoly(q,2)
Type: Void

For example:

axiom
p:=2*x^2+3
LatexWiki Image(5)
Type: Polynomial Integer
axiom
q:=5*x*y+7*y+11
LatexWiki Image(6)
Type: Polynomial Integer
axiom
r:=tensorPoly(p,q)
axiom
Compiling function tensorPoly with type (Polynomial Integer,
      Polynomial Integer) -> SparseMultivariatePolynomial(Integer,Sum(
      Symbol,Symbol)) 
; (DEFUN |*2;tensorPoly;1;initial| ...) is being compiled.
;; The variable |*2;tensorPoly;1;initial;MV| is undefined.
;; The compiler will assume this variable is a global.
LatexWiki Image(7)
Type: SparseMultivariatePolynomial?(Integer,Sum(Symbol,Symbol))
axiom
monomials(r)
LatexWiki Image(8)
Type: List SparseMultivariatePolynomial?(Integer,Sum(Symbol,Symbol))

Demonstrating the axioms (1) (2) and (3) of the tensor product:

axiom
w:= 13*y^2+17*y+19
LatexWiki Image(9)
Type: Polynomial Integer
axiom
test( tensorPoly(p+q,w) = (tensorPoly(p,w) + tensorPoly(q,w)) )
LatexWiki Image(10)
Type: Boolean
axiom
test( tensorPoly(p,q+w) = (tensorPoly(p,q) + tensorPoly(p,w)) )
LatexWiki Image(11)
Type: Boolean
axiom
test( tensorPoly(p,23*w) = 23*tensorPoly(p,w) )
LatexWiki Image(12)
Type: Boolean
axiom
test( tensorPoly(23*p,w) = 23*tensorPoly(p,w) )
LatexWiki Image(13)
Type: Boolean

I suppose that we could give an inductive proof that this implementation of the tensor product of polynomials is correct ... but for now lets take this demonstration as reassurance.

Re-coding the interpreter functions as library package.

spad
)abbrev package TPROD TensorProduct
macro IE == IndexedExponents(VAR)
macro IEP == IndexedExponents(Sum(VAR,VAR))
macro SMP == SparseMultivariatePolynomial(R,Sum(VAR,VAR))
TensorProduct(R:Ring, VAR: OrderedSet, P:PolynomialCategory(R,IE,VAR)): with _\_/: (P,P) -> SMP == add scanIndex(x:IE,n:Integer):IEP == zero? x => 0 monomial(leadingCoefficient(x), _ if n=1 then in1(leadingSupport(x))$Sum(VAR,VAR) _ else in2(leadingSupport(x))$Sum(VAR,VAR) _ ) + scanIndex(reductum(x),n) mapMonomial(p:P,n:Integer):SMP == monomial(coefficient(p,degree p),scanIndex(degree(p),n))$SMP scanPoly(p:P,n:Integer):SMP == p=0 => 0 mapMonomial(leadingMonomial(p),n)+scanPoly(reductum p,n)
(p:P \/ q:P):SMP == scanPoly(p,1)*scanPoly(q,2)
spad
   Compiling OpenAxiom source code from file 
      /var/zope2/var/LatexWiki/7467299293205133182-25px007.spad using 
      Spad compiler.
   TPROD abbreviates package TensorProduct 
   processing macro definition IE ==> IndexedExponents VAR 
processing macro definition IEP ==> IndexedExponents Sum(VAR,VAR)
processing macro definition SMP ==> SparseMultivariatePolynomial(R,Sum(VAR,VAR))
------------------------------------------------------------------------ initializing NRLIB TPROD for TensorProduct compiling into NRLIB TPROD Adding $ modemaps Adding R modemaps Adding VAR modemaps Adding P modemaps Adding IndexedExponents Sum(VAR,VAR) modemaps Adding Integer modemaps Adding IndexedExponents VAR modemaps compiling local scanIndex : (IndexedExponents VAR,Integer) -> IndexedExponents Sum(VAR,VAR) Adding Boolean modemaps Adding NonNegativeInteger modemaps Adding Sum(VAR,VAR) modemaps Time: 0.11 SEC.
Adding SparseMultivariatePolynomial(R,Sum(VAR,VAR)) modemaps Adding Integer modemaps compiling local mapMonomial : (P,Integer) -> SparseMultivariatePolynomial(R,Sum(VAR,VAR)) Adding IndexedExponents VAR modemaps Time: 0.01 SEC.
Adding Integer modemaps compiling local scanPoly : (P,Integer) -> SparseMultivariatePolynomial(R,Sum(VAR,VAR)) Adding Boolean modemaps Time: 0.01 SEC.
compiling exported \/ : (P,P) -> SparseMultivariatePolynomial(R,Sum(VAR,VAR)) Adding Integer modemaps Adding NonNegativeInteger modemaps Adding PositiveInteger modemaps Adding Fraction Integer modemaps Time: 0.04 SEC.
(time taken in buildFunctor: 0)
;;; *** |TensorProduct| REDEFINED
;;; *** |TensorProduct| REDEFINED Time: 0 SEC.
Warnings: [1] scanIndex: not known that OrderedSet is of mode CATEGORY(domain,IF(has(VAR,Finite),IF(has(VAR,Finite),Finite,%noBranch),%noBranch),IF(has(VAR,Monoid),IF(has(VAR,Monoid),Monoid,%noBranch),%noBranch),IF(has(VAR,AbelianMonoid),IF(has(VAR,AbelianMonoid),AbelianMonoid,%noBranch),%noBranch),IF(has(VAR,CancellationAbelianMonoid),IF(has(VAR,CancellationAbelianMonoid),CancellationAbelianMonoid,%noBranch),%noBranch),IF(has(VAR,Group),IF(has(VAR,Group),Group,%noBranch),%noBranch),IF(has(VAR,AbelianGroup),IF(has(VAR,AbelianGroup),AbelianGroup,%noBranch),%noBranch),IF(has(VAR,OrderedAbelianMonoidSup),IF(has(VAR,OrderedAbelianMonoidSup),OrderedAbelianMonoidSup,%noBranch),%noBranch),IF(has(VAR,OrderedSet),IF(has(VAR,OrderedSet),OrderedSet,%noBranch),%noBranch),selectsum: % -> Union(acomp: VAR,bcomp: VAR),in1: VAR -> %,in2: VAR -> %) [2] mapMonomial: not known that OrderedSet is of mode CATEGORY(domain,IF(has(VAR,Finite),IF(has(VAR,Finite),Finite,%noBranch),%noBranch),IF(has(VAR,Monoid),IF(has(VAR,Monoid),Monoid,%noBranch),%noBranch),IF(has(VAR,AbelianMonoid),IF(has(VAR,AbelianMonoid),AbelianMonoid,%noBranch),%noBranch),IF(has(VAR,CancellationAbelianMonoid),IF(has(VAR,CancellationAbelianMonoid),CancellationAbelianMonoid,%noBranch),%noBranch),IF(has(VAR,Group),IF(has(VAR,Group),Group,%noBranch),%noBranch),IF(has(VAR,AbelianGroup),IF(has(VAR,AbelianGroup),AbelianGroup,%noBranch),%noBranch),IF(has(VAR,OrderedAbelianMonoidSup),IF(has(VAR,OrderedAbelianMonoidSup),OrderedAbelianMonoidSup,%noBranch),%noBranch),IF(has(VAR,OrderedSet),IF(has(VAR,OrderedSet),OrderedSet,%noBranch),%noBranch),selectsum: % -> Union(acomp: VAR,bcomp: VAR),in1: VAR -> %,in2: VAR -> %)
Cumulative Statistics for Constructor TensorProduct Time: 0.17 seconds
finalizing NRLIB TPROD Processing TensorProduct for Browser database: --->-->TensorProduct((\/ (SMP P P))): Not documented!!!! --->-->TensorProduct(constructor): Not documented!!!! --->-->TensorProduct(): Missing Description ------------------------------------------------------------------------ TensorProduct is now explicitly exposed in frame initial TensorProduct will be automatically loaded when needed from /var/zope2/var/LatexWiki/TPROD.NRLIB/code.o

axiom
test( p\/q = r )
LatexWiki Image(14)
Type: Boolean
axiom
test( (p+q) \/ w = (p\/w) + (q\/w) )
LatexWiki Image(15)
Type: Boolean
axiom
test( p \/ (q+w) = (p\/q) + (p\/w) )
LatexWiki Image(16)
Type: Boolean
axiom
test( p \/ (23*w) = 23*(p\/w) )
LatexWiki Image(17)
Type: Boolean
axiom
test( (23*p) \/ w = 23*(p\/w) )
LatexWiki Image(18)
Type: Boolean

Here's another way to write this - maybe better this way as first step to express associativity of the tensor product.

spad
)abbrev package TPROD2 TensorProduct2
macro IE1 == IndexedExponents(VAR1)
macro IE2 == IndexedExponents(VAR2)
macro S == Sum(VAR1,VAR2)
macro IEP == IndexedExponents(S)
macro SMP == SparseMultivariatePolynomial(R,S)
TensorProduct2(R:Ring, VAR1: OrderedSet, VAR2: OrderedSet, P:PolynomialCategory(R,IE1,VAR1), Q:PolynomialCategory(R,IE2,VAR2)): with _\_/: (P,Q) -> SMP == add scanIndex1(x:IE1):IEP == zero? x => 0 monomial(leadingCoefficient(x), in1(leadingSupport(x))$S) + scanIndex1(reductum(x)) scanIndex2(x:IE2):IEP == zero? x => 0 monomial(leadingCoefficient(x), in2(leadingSupport(x))$S) + scanIndex2(reductum(x)) mapMonomial1(p:P):SMP == monomial(coefficient(p,degree p),scanIndex1(degree(p)))$SMP mapMonomial2(q:Q):SMP == monomial(coefficient(q,degree q),scanIndex2(degree(q)))$SMP scanPoly1(p:P):SMP == p=0 => 0 mapMonomial1(leadingMonomial(p))+scanPoly1(reductum p) scanPoly2(q:Q):SMP == q=0 => 0 mapMonomial2(leadingMonomial(q))+scanPoly2(reductum q)
(p:P \/ q:Q):SMP == scanPoly1(p)*scanPoly2(q)
spad
   Compiling OpenAxiom source code from file 
      /var/zope2/var/LatexWiki/6972063761479466203-25px009.spad using 
      Spad compiler.
   TPROD2 abbreviates package TensorProduct2 
   processing macro definition IE1 ==> IndexedExponents VAR1 
processing macro definition IE2 ==> IndexedExponents VAR2
processing macro definition S ==> Sum(VAR1,VAR2)
processing macro definition IEP ==> IndexedExponents S
processing macro definition SMP ==> SparseMultivariatePolynomial(R,S)
------------------------------------------------------------------------ initializing NRLIB TPROD2 for TensorProduct2 compiling into NRLIB TPROD2 Adding $ modemaps Adding R modemaps Adding VAR1 modemaps Adding VAR2 modemaps Adding P modemaps Adding Q modemaps Adding IndexedExponents Sum(VAR1,VAR2) modemaps Adding IndexedExponents VAR1 modemaps compiling local scanIndex1 : IndexedExponents VAR1 -> IndexedExponents Sum(VAR1,VAR2) Adding Boolean modemaps Adding Integer modemaps Adding NonNegativeInteger modemaps Adding Sum(VAR1,VAR2) modemaps Time: 0.11 SEC.
Adding IndexedExponents VAR2 modemaps compiling local scanIndex2 : IndexedExponents VAR2 -> IndexedExponents Sum(VAR1,VAR2) Adding Boolean modemaps Adding Integer modemaps Adding NonNegativeInteger modemaps Adding Sum(VAR1,VAR2) modemaps Time: 0.02 SEC.
Adding SparseMultivariatePolynomial(R,Sum(VAR1,VAR2)) modemaps compiling local mapMonomial1 : P -> SparseMultivariatePolynomial(R,Sum(VAR1,VAR2)) Adding IndexedExponents VAR1 modemaps Time: 0.02 SEC.
compiling local mapMonomial2 : Q -> SparseMultivariatePolynomial(R,Sum(VAR1,VAR2)) Adding IndexedExponents VAR2 modemaps Time: 0.01 SEC.
compiling local scanPoly1 : P -> SparseMultivariatePolynomial(R,Sum(VAR1,VAR2)) Adding Boolean modemaps Time: 0 SEC.
compiling local scanPoly2 : Q -> SparseMultivariatePolynomial(R,Sum(VAR1,VAR2)) Adding Boolean modemaps Time: 0.01 SEC.
compiling exported \/ : (P,Q) -> SparseMultivariatePolynomial(R,Sum(VAR1,VAR2)) Adding Fraction Integer modemaps Time: 0.03 SEC.
(time taken in buildFunctor: 0)
;;; *** |TensorProduct2| REDEFINED
;;; *** |TensorProduct2| REDEFINED Time: 0.07 SEC.
Warnings: [1] scanIndex1: not known that OrderedSet is of mode CATEGORY(domain,IF(has(VAR1,Finite),IF(has(VAR2,Finite),Finite,%noBranch),%noBranch),IF(has(VAR1,Monoid),IF(has(VAR2,Monoid),Monoid,%noBranch),%noBranch),IF(has(VAR1,AbelianMonoid),IF(has(VAR2,AbelianMonoid),AbelianMonoid,%noBranch),%noBranch),IF(has(VAR1,CancellationAbelianMonoid),IF(has(VAR2,CancellationAbelianMonoid),CancellationAbelianMonoid,%noBranch),%noBranch),IF(has(VAR1,Group),IF(has(VAR2,Group),Group,%noBranch),%noBranch),IF(has(VAR1,AbelianGroup),IF(has(VAR2,AbelianGroup),AbelianGroup,%noBranch),%noBranch),IF(has(VAR1,OrderedAbelianMonoidSup),IF(has(VAR2,OrderedAbelianMonoidSup),OrderedAbelianMonoidSup,%noBranch),%noBranch),IF(has(VAR1,OrderedSet),IF(has(VAR2,OrderedSet),OrderedSet,%noBranch),%noBranch),selectsum: % -> Union(acomp: VAR1,bcomp: VAR2),in1: VAR1 -> %,in2: VAR2 -> %) [2] mapMonomial1: not known that OrderedSet is of mode CATEGORY(domain,IF(has(VAR1,Finite),IF(has(VAR2,Finite),Finite,%noBranch),%noBranch),IF(has(VAR1,Monoid),IF(has(VAR2,Monoid),Monoid,%noBranch),%noBranch),IF(has(VAR1,AbelianMonoid),IF(has(VAR2,AbelianMonoid),AbelianMonoid,%noBranch),%noBranch),IF(has(VAR1,CancellationAbelianMonoid),IF(has(VAR2,CancellationAbelianMonoid),CancellationAbelianMonoid,%noBranch),%noBranch),IF(has(VAR1,Group),IF(has(VAR2,Group),Group,%noBranch),%noBranch),IF(has(VAR1,AbelianGroup),IF(has(VAR2,AbelianGroup),AbelianGroup,%noBranch),%noBranch),IF(has(VAR1,OrderedAbelianMonoidSup),IF(has(VAR2,OrderedAbelianMonoidSup),OrderedAbelianMonoidSup,%noBranch),%noBranch),IF(has(VAR1,OrderedSet),IF(has(VAR2,OrderedSet),OrderedSet,%noBranch),%noBranch),selectsum: % -> Union(acomp: VAR1,bcomp: VAR2),in1: VAR1 -> %,in2: VAR2 -> %)
Cumulative Statistics for Constructor TensorProduct2 Time: 0.27 seconds
finalizing NRLIB TPROD2 Processing TensorProduct2 for Browser database: --->-->TensorProduct2((\/ (SMP P Q))): Not documented!!!! --->-->TensorProduct2(constructor): Not documented!!!! --->-->TensorProduct2(): Missing Description ------------------------------------------------------------------------ TensorProduct2 is now explicitly exposed in frame initial TensorProduct2 will be automatically loaded when needed from /var/zope2/var/LatexWiki/TPROD2.NRLIB/code.o

axiom
test( p\/q = r )
LatexWiki Image(19)
Type: Boolean
axiom
test( (p+q) \/ w = (p\/w) + (q\/w) )
LatexWiki Image(20)
Type: Boolean
axiom
test( p \/ (q+w) = (p\/q) + (p\/w) )
LatexWiki Image(21)
Type: Boolean
axiom
test( p \/ (23*w) = 23*(p\/w) )
LatexWiki Image(22)
Type: Boolean
axiom
test( (23*p) \/ w = 23*(p\/w) )
LatexWiki Image(23)
Type: Boolean
axiom
p\/q\/w
LatexWiki Image(24)
Type: SparseMultivariatePolynomial?(Integer,Sum(Sum(Symbol,Symbol),Symbol))