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SandBoxTensorProductPolynomial

Edit detail for SandBoxTensorProductPolynomial revision 12 of 12

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Editor: test1 Time: 2013/04/25 19:55:41 GMT+0
Note:

changed:
-macro IE == IndexedExponents(VAR)
-macro IEP == IndexedExponents(Sum(VAR,VAR))
-macro SMP == SparseMultivariatePolynomial(R,Sum(VAR,VAR))
IE ==> IndexedExponents(VAR)
IEP ==> IndexedExponents(Sum(VAR,VAR))
SMP ==> SparseMultivariatePolynomial(R,Sum(VAR,VAR))

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

changed:
-macro IE1 == IndexedExponents(VAR1)
-macro IE2 == IndexedExponents(VAR2)
-macro S == Sum(VAR1,VAR2)
-macro IEP == IndexedExponents(S)
-macro SMP == SparseMultivariatePolynomial(R,S)
IE1 ==> IndexedExponents(VAR1)
IE2 ==> IndexedExponents(VAR2)
S ==> Sum(VAR1,VAR2)
IEP ==> IndexedExponents(S)
SMP ==> SparseMultivariatePolynomial(R,S)

changed:
-    (p:P \/ q:Q):SMP == scanPoly1(p)*scanPoly2(q)
    _\_/(p:P, q:Q) : SMP == scanPoly1(p)*scanPoly2(q)

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

fricas

(1) -> )lib SUM

   Sum is now explicitly exposed in frame initial 
   Sum will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/SUM.NRLIB/SUM

First we can define some recursive operations on the polynomials

fricas

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

Type: Void

fricas

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

Type: Void

fricas

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:

fricas

-- 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:

fricas

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)

fricas

r:=tensorPoly(p,q)

   >> System error:
   #<SB-SYS:FD-STREAM for "file /var/aw/var/LatexWiki/SUM.NRLIB/SUM.fasl" {10021C7463}>
is a fasl file compiled with SBCL 1.1.1, and can't be loaded into SBCL
2.2.9.debian.

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)) )

fricas

Compiling function tensorPoly with type (Polynomial(Integer), 
      Polynomial(Integer)) -> NonNegativeInteger 

   >> System error:
   The function BOOT::|*2;scanPoly;1;initial| is undefined.

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)
****** comp fails at level 7 with expression: ******
error in function scanIndex 

(SEQ (|:=| (|:| #1=#:G15 (|Boolean|)) (|zero?| |x|))
     (|exit| 1
      (IF #1#
          0
          (+
           (|monomial| (|leadingCoefficient| |x|)
            (IF (= |n| 1)
                | << |
                ((|Sel| (|Sum| VAR VAR) |in1|) (|leadingSupport| |x|))
                | >> |
                ((|Sel| (|Sum| VAR VAR) |in2|) (|leadingSupport| |x|))))
           (|scanIndex| (|reductum| |x|) |n|)))))
****** level 7  ******
$x:= ((Sel (Sum VAR VAR) in1) (leadingSupport x))
$m:= VAR
$f:=
((((#:G15 # #) (|n| # #) (|x| # #) (|scanIndex| #) ...)))

   >> Apparent user error:
   Cannot coerce x 
      of mode (IndexedExponents VAR) 
      to mode ##3

fricas

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)
****** comp fails at level 6 with expression: ******
error in function scanIndex1 

(SEQ (|:=| (|:| #1=#:G16 (|Boolean|)) (|zero?| |x|))
     (|exit| 1
      (IF #1#
          0
          (+
           (|monomial| (|leadingCoefficient| |x|) | << |
            ((|Sel| (|Sum| VAR1 VAR2) |in1|) (|leadingSupport| |x|)) | >> |)
           (|scanIndex1| (|reductum| |x|))))))
****** level 6  ******
$x:= ((Sel (Sum VAR1 VAR2) in1) (leadingSupport x))
$m:= VAR1
$f:=
((((#:G16 # #) (|x| # . #1=#) (|scanIndex1| #) (|x| . #1#) ...)))

   >> Apparent user error:
   Cannot coerce x 
      of mode (IndexedExponents VAR1) 
      to mode ##5

fricas

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:

fricas

(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.