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SandBoxTensorProductPolynomial
  
last edited 11 years ago by test1

Edit detail for SandBoxTensorProductPolynomial revision 8 of 12
1 2 3 4 5 6 7 8 9 10 11 12
Editor: Bill Page
Time: 2008/08/26 13:49:28 GMT-7
Note: associativity 
removed:
-p\/q\/w

added:
Associativity of the tensor product means these two expressions
should be identical:
\begin{axiom}
(p\/q)\/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:


\begin{equation}
\label{eq1}
(v_1+v_2)\otimes w - (v_1\otimes w+v_2\otimes w) = 0
\end{equation}
\begin{equation}
\label{eq2}
v\otimes (w_1+w_2) - (v\otimes w_1+v\otimes w_2) = 0
\end{equation}
\begin{equation}
\label{eq3}
cv\otimes w=v\otimes cw=c(v\otimes w)
\end{equation}


We can use the domain constructor Sum [SandBoxSum]
\begin{axiom}
)lib SUM
\end{axiom}


First we can define some recursive operations on the polynomials
\begin{axiom}
scanPoly(p,n) == _
  (p=0 => 0; mapMonomial(leadingMonomial(p),n)+scanPoly(reductum p,n))
mapMonomial(p,n) == 
  monomial(coefficient(p,degree p),scanIndex(degree(p),n))$SMP(Integer,Sum(Symbol,Symbol))
scanIndex(p,n) == 
  (zero? p => 0$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))
\end{axiom}
For example:
\begin{axiom}
-- functions are first compiled here
--
scanPoly(x,1)
\end{axiom}
injects the polynomial x in to the tensor product. So
now the full tensor product is just:
\begin{axiom}
tensorPoly(p,q) == _
  scanPoly(p,1)*scanPoly(q,2)
\end{axiom}


For example:
\begin{axiom}
p:=2x^2+3
q:=5xy+7y+11
r:=tensorPoly(p,q)
monomials(r)
\end{axiom}


Demonstrating the axioms (1) (2) and (3) of the tensor product:
\begin{axiom}
w:= 13y^2+17y+19
test( tensorPoly(p+q,w) = (tensorPoly(p,w) + tensorPoly(q,w)) )
test( tensorPoly(p,q+w) = (tensorPoly(p,q) + tensorPoly(p,w)) )
test( tensorPoly(p,23w) = 23tensorPoly(p,w) )
test( tensorPoly(23p,w) = 23tensorPoly(p,w) )
\end{axiom}
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.
\begin{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)
\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 \/ (23w) = 23(p\/w) )
test( (23p) \/ w = 23(p\/w) )
\end{axiom}


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 \/ (23w) = 23(p\/w) )
test( (23p) \/ w = 23(p\/w) )
\end{axiom}


Associativity of the tensor product means these two expressions
should be identical:
\begin{axiom}
(p\/q)\/w
p\/(q\/w)
\end{axiom}




    Some or all expressions may not have rendered properly,
    because Axiom returned the following error:
Error: export AXIOM=/usr/local/lib/open-axiom/x86_64-unknown-linux/1.2.0-2008-05-25; export ALDORROOT=/usr/local/aldor/linux/1.1.0; export PATH=$ALDORROOT/bin:$PATH; export HOME=/var/zope2/var/LatexWiki; ulimit -t 240; $AXIOM/bin/AXIOMsys < /var/zope2/var/LatexWiki/6578669110478838235-25px.axm
Some or all expressions may not have rendered properly, because Latex returned the following error:
! Missing $ inserted.
<inserted text> 
                $
l.128     
           \/: (P,P) -> SMP
 Missing $ inserted.
<inserted text> 
                $
l.133 ...hen in1(leadingSupport(x))$Sum(VAR,VAR) _
 Missing $ inserted.
<inserted text> 
                $
l.134 ...lse in2(leadingSupport(x))$Sum(VAR,VAR) _




Overfull \hbox (30.44293pt too wide) in paragraph at lines 127--141
[]\OT1/cmr/m/n/12 TensorProduct(R:Ring, VAR: Or-dered-Set, P:PolynomialCategory
(R,IE,VAR)):


Overfull \hbox (127.18265pt too wide) in paragraph at lines 127--141
\OML/cmm/m/it/12 zero\OT1/cmr/m/n/12 ?\OML/cmm/m/it/12 x \OT1/cmr/m/n/12 =\OML/
cmm/m/it/12 > \OT1/cmr/m/n/12 0\OML/cmm/m/it/12 monomial\OT1/cmr/m/n/12 (\OML/c
mm/m/it/12 leadingCoefficient\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 x\OT1/cmr/m/n/12
 )\OML/cmm/m/it/12 ;[] fn \OT1/cmr/m/n/12 = 1\OML/cmm/m/it/12 thenin\OT1/cmr/m/
n/12 1(\OML/cmm/m/it/12 leadingSupport\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 x\OT1/c
mr/m/n/12 ))$Sum(VAR,VAR)


Overfull \hbox (89.29578pt too wide) in paragraph at lines 127--141
[]\OML/cmm/m/it/12 lsein\OT1/cmr/m/n/12 2(\OML/cmm/m/it/12 leadingSupport\OT1/c
mr/m/n/12 (\OML/cmm/m/it/12 x\OT1/cmr/m/n/12 ))$Sum(VAR,VAR) $[] + \OML/cmm/m/i
t/12 scanIndex\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 reductum\OT1/cmr/m/n/12 (\OML/c
mm/m/it/12 x\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 ; n\OT1/cmr/m/n/12 )\OML/cmm/m/it
/12 mapMonomial\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 p \OT1/cmr/m/n/12 :


Overfull \hbox (87.85938pt too wide) in paragraph at lines 127--141
\OML/cmm/m/it/12 P; n \OT1/cmr/m/n/12 : \OML/cmm/m/it/12 Integer\OT1/cmr/m/n/12
 ) : \OML/cmm/m/it/12 SMP \OT1/cmr/m/n/12 == \OML/cmm/m/it/12 monomial\OT1/cmr/
m/n/12 (\OML/cmm/m/it/12 coefficient\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 p; degree
p\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 ; scanIndex\OT1/cmr/m/n/12 (\OML/cmm/m/it/12
 degree\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 p\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 ; n
\OT1/cmr/m/n/12 ))$SMP


Overfull \hbox (135.91133pt too wide) in paragraph at lines 127--141
\OT1/cmr/m/n/12 scan-Poly(p:P,n:Integer):SMP == p=0 => 0 map-Mono-mial(leadingM
onomial(p),n)+scanPoly(reductum
[10] [11]
Overfull \hbox (53.30666pt too wide) in paragraph at lines 154--160
\OT1/cmr/m/n/12 nents(VAR1) macro IE2 == In-dexed-Ex-po-nents(VAR2) macro S == 
Sum(VAR1,VAR2)


Overfull \hbox (6.8212pt too wide) in paragraph at lines 154--160
\OT1/cmr/m/n/12 macro IEP == In-dexed-Ex-po-nents(S) macro SMP == Sparse-Mul-ti
-vari-atePoly-
 Missing $ inserted.
<inserted text> 
                $
l.162     
           \/: (P,Q) -> SMP
 Missing $ inserted.
<inserted text> 
                $
l.180 


Overfull \hbox (154.23055pt too wide) in paragraph at lines 161--180
[]\OT1/cmr/m/n/12 TensorProduct2(R:Ring, VAR1: Or-dered-Set, VAR2: Or-dered-Set
, P:PolynomialCategory(R,IE1,VAR1),


Overfull \hbox (65.84291pt too wide) in paragraph at lines 161--180
\OT1/cmr/m/n/12 Q:PolynomialCategory(R,IE2,VAR2)): with $[]\OML/cmm/m/it/12 = \
OT1/cmr/m/n/12 : (\OML/cmm/m/it/12 P; Q\OT1/cmr/m/n/12 )\OMS/cmsy/m/n/12 ^^@ \O
ML/cmm/m/it/12 > SMP \OT1/cmr/m/n/12 == \OML/cmm/m/it/12 addscanIndex\OT1/cmr/m
/n/12 1(\OML/cmm/m/it/12 x \OT1/cmr/m/n/12 :


Overfull \hbox (76.06493pt too wide) in paragraph at lines 161--180
\OML/cmm/m/it/12 IE\OT1/cmr/m/n/12 1) : \OML/cmm/m/it/12 IEP \OT1/cmr/m/n/12 ==
 \OML/cmm/m/it/12 zero\OT1/cmr/m/n/12 ?