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SandBoxTensorProductPolynomial
  
last edited 11 years ago by test1

Edit detail for SandBoxTensorProductPolynomial revision 6 of 12
1 2 3 4 5 6 7 8 9 10 11 12
Editor: Bill Page
Time: 2008/08/26 13:06:12 GMT-7
Note: spad version tests 
changed:
-    _/_\: (P,P) -> SMP
    _\_/: (P,P) -> SMP

changed:
-    _/_\(p:P,q:P):SMP == scanPoly(p,1)*scanPoly(q,2)
    (p:P \/ q:P):SMP == scanPoly(p,1)*scanPoly(q,2)

added:

\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) )
\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: 0.13 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: 0.01 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: 0.20 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