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SandBoxTensorProductPolynomial

Edit detail for SandBoxTensorProductPolynomial revision 4 of 12

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Editor: Bill Page Time: 2008/08/25 14:58:25 GMT-7
Note: check axioms

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

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

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

(1)

(2)

(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.

(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

(5)

Type: Polynomial Integer

axiom

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

(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.

(7)

Type: SparseMultivariatePolynomial?(Integer,Sum(Symbol,Symbol))

axiom

monomials(r)

(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

(9)

Type: Polynomial Integer

axiom

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

(10)

Type: Boolean

axiom

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

(11)

Type: Boolean

axiom

test( tensorPoly(p,23*w) = 23*tensorPoly(p,w) )

(12)

Type: Boolean

axiom

test( tensorPoly(23*p,w) = 23*tensorPoly(p,w) )

(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.