MathAction SandBoxTensorProductPolynomial

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

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

$\label{eq1} (v_1+v_2)\otimes w - (v_1\otimes w+v_2\otimes w) = 0$

(1)

$\label{eq2} v\otimes (w_1+w_2) - (v\otimes w_1+v\otimes w_2) = 0$

(2)

$\label{eq3} cv\otimes w=v\otimes cw=c(v\otimes w)$

(3)

We can use the domain constructor Sum SandBoxSum

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(1) -> )lib SUM

   >> System error:
   The value
  17359
is not of type
  LIST

First we can define some recursive operations on the polynomials

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scanPoly(p,n) == _
  (p=0 => 0; mapMonomial(leadingMonomial(p),n)+scanPoly(reductum p,n))

Type: Void

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mapMonomial(p,n) == _
  monomial(coefficient(p,degree p),scanIndex(degree(p),n))$SMP(Integer,Sum(Symbol,Symbol))

Type: Void

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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:

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-- functions are first compiled here
--
scanPoly(x,1)

   There are 1 exposed and 3 unexposed library operations named 
      leadingMonomial having 1 argument(s) but none was determined to 
      be applicable. Use HyperDoc Browse, or issue
                         )display op leadingMonomial
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.
   Cannot find a definition or applicable library operation named 
      leadingMonomial with argument type(s) 
                                 Variable(x)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.
   There are 1 exposed and 3 unexposed library operations named 
      leadingMonomial having 1 argument(s) but none was determined to 
      be applicable. Use HyperDoc Browse, or issue
                         )display op leadingMonomial
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named 
      leadingMonomial with argument type(s) 
                                 Variable(x)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

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

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tensorPoly(p,q) == _
  scanPoly(p,1)*scanPoly(q,2)

Type: Void

For example:

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p:=2*x^2+3

$\label{eq4}{2 \ {{x}^{2}}}+ 3$

(4)

Type: Polynomial(Integer)

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q:=5*x*y+7*y+11

$\label{eq5}{{\left({5 \ x}+ 7 \right)}\ y}+{11}$

(5)

Type: Polynomial(Integer)

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r:=tensorPoly(p,q)

   There are no library operations named Sum 
      Use HyperDoc Browse or issue
                                )what op Sum
      to learn if there is any operation containing " Sum " in its 
      name.
   Cannot find a definition or applicable library operation named Sum 
      with argument type(s) 
                                    Type
                                    Type

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.
   Cannot compile map: mapMonomial 
   We will attempt to interpret the code.
   Cannot compile map: scanPoly 
   We will attempt to interpret the code.
   There are no library operations named Sum 
      Use HyperDoc Browse or issue
                                )what op Sum
      to learn if there is any operation containing " Sum " in its 
      name.

   Cannot find a definition or applicable library operation named Sum 
      with argument type(s) 
                                    Type
                                    Type

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

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

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w:= 13*y^2+17*y+19

$\label{eq6}{{13}\ {{y}^{2}}}+{{17}\ y}+{19}$

(6)

Type: Polynomial(Integer)

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test( tensorPoly(p+q,w) = (tensorPoly(p,w) + tensorPoly(q,w)) )

   There are no library operations named Sum 
      Use HyperDoc Browse or issue
                                )what op Sum
      to learn if there is any operation containing " Sum " in its 
      name.
   Cannot find a definition or applicable library operation named Sum 
      with argument type(s) 
                                    Type
                                    Type

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.
   There are no library operations named Sum 
      Use HyperDoc Browse or issue
                                )what op Sum
      to learn if there is any operation containing " Sum " in its 
      name.

   Cannot find a definition or applicable library operation named Sum 
      with argument type(s) 
                                    Type
                                    Type

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

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
IE ==> IndexedExponents(VAR)
IEP ==> IndexedExponents(Sum(VAR,VAR))
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 FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/6318234874982058351-25px007.spad
      using old system compiler.
   TPROD abbreviates package TensorProduct 
------------------------------------------------------------------------
   initializing NRLIB TPROD for TensorProduct 
   compiling into NRLIB TPROD 
   compiling local scanIndex : (IndexedExponents VAR,Integer) -> IndexedExponents Sum(VAR,VAR)
   Internal Error
   Error while instantiating type SumVARVAR

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test( p\/q = r )

   There are 2 exposed and 1 unexposed library operations named \/ 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                               )display op \/
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named \/ 
      with argument type(s) 
                             Polynomial(Integer)
                             Polynomial(Integer)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

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
IE1 ==> IndexedExponents(VAR1)
IE2 ==> IndexedExponents(VAR2)
S ==> Sum(VAR1,VAR2)
IEP ==> IndexedExponents(S)
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 FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/4565958775469301848-25px009.spad
      using old system compiler.
   TPROD2 abbreviates package TensorProduct2 
------------------------------------------------------------------------
   initializing NRLIB TPROD2 for TensorProduct2 
   compiling into NRLIB TPROD2 
   compiling local scanIndex1 : IndexedExponents VAR1 -> IndexedExponents Sum(VAR1,VAR2)
   Internal Error
   Error while instantiating type SumVAR1VAR2

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test( p\/q = r )

   There are 2 exposed and 1 unexposed library operations named \/ 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                               )display op \/
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named \/ 
      with argument type(s) 
                             Polynomial(Integer)
                             Polynomial(Integer)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Associativity of the tensor product means these two expressions should be identical:

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(p\/q)\/w

   There are 2 exposed and 1 unexposed library operations named \/ 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                               )display op \/
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named \/ 
      with argument type(s) 
                             Polynomial(Integer)
                             Polynomial(Integer)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

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