Edit detail for LinearOperator revision 9 of 63

Linear transformations (operators) over n-dimensional cartesian vector spaces oveer a commutative ring . Members of this domain are morphisms . Products, co-products and composition (grafting) of morphisms is implemented. Operators are represented internally as tensors.

Operator composition and products can be visualized by directed graphs (read from top to bottom) such as:

      n = 3     inputs

      \  |  /         \  /   /         \              |
       \ | /           \/   /           \            / \
        \|/             \  /           / \          /   \
        / \              \/           /   \         \   /
       /   \              \          /   / \         \ /
      /     \              \        /   /   \         |

      m = 2     outputs

Lines (edges) in the graph represent vectors, nodes represent operators. Horizontal juxtaposition represents product. Vertical juxtaposition represents composition.

(1) -> <spad>

)abbrev domain LIN LinearOperator
LinearOperator(dim:NonNegativeInteger,K:CommutativeRing): Join(Ring,BiModule(K,K)) with
    arity: % -> DirectProduct(2,NonNegativeInteger)
    elt: (%,%) -> %
    id: ()->%
    inp: List K -> %
      ++ incoming vector
    inp: List % -> %
    out: List K -> %
      ++ output vector
    out: List % -> %
    coerce: SquareMatrix(dim,K) -> %
    _*: (%,NonNegativeInteger) -> %
  == add
    import List NonNegativeInteger
    T == CartesianTensor(1,dim,K)
    Rep == Record(n:NonNegativeInteger, m:NonNegativeInteger, t:T)
    rep(x:%):Rep == x pretend Rep
    per(x:Rep):% == x pretend %

    arity(x:%):DirectProduct(2,NonNegativeInteger) == directProduct [rep(x).n,rep(x).m]

    0 == per [0,0,0]

    (x:% + y:%):% ==
      rep(x).t=0 => per [rep(y).n,rep(y).m,rep(y).t]
      rep(y).t=0 => per [rep(x).n,rep(x).m,rep(x).t]
      rep(x).n ~= rep(y).n or rep(x).m ~= rep(y).m => error "arity"
      per [rep(x).n,rep(x).m,rep(x).t+rep(y).t]

    (x:% - y:%):% ==
      rep(x).t=0 => per [rep(y).n,rep(y).m,-rep(y).t]
      rep(y).t=0 => per [rep(x).n,rep(x).m,rep(x).t]
      rep(x).n ~= rep(y).n or rep(x).m ~= rep(y).m => error "arity"
      per [rep(x).n,rep(x).m,rep(x).t-rep(y).t]

    1 == per [0,0,1]

    --
    -- f*g : A^n -> A^{m+p} = f:A^n -> A^m * g:A^n -> A^p
    --
    (x:% * y:%):% ==
      output("* rep(x).n",rep(x).n::OutputForm)$OutputPackage
      output("* rep(y).n",rep(y).n::OutputForm)$OutputPackage
      rep(x).n ~= rep(y).n => error "arity"
      r := product(rep(x).t, rep(y).t)
      u := 1$DirectProduct(dim,K)::T
      ud:= product(u,kroneckerDelta()$T)
      for i in 1..rep(x).n repeat
        --output("rank",rank(r)::OutputForm)$OutputPackage
        --output("n",rep(x).n::OutputForm)$OutputPackage
        r := contract(contract(ud,1,r,rep(x).n+1),1,rep(x).n+1)
        --output("rank",rank(r)::OutputForm)$OutputPackage
      per [rep(x).n,rep(x).m+rep(y).m,r]

    -- repeated product
    ((x:%)^(p:NonNegativeInteger)):% ==
       q:=subtractIfCan(p,1)
       q case NonNegativeInteger => x^q * x
       per [rep(x).n,0,1]

    --
    -- f+g : A^{n+m} -> A^p = f:A^n -> A^p + g:A^m -> A^p
    --
    (x:% + y:%):% ==
      output("+ rep(x).m",rep(x).m::OutputForm)$OutputPackage
      output("+ rep(y).m",rep(y).m::OutputForm)$OutputPackage
      rep(x).m ~= rep(y).m  => error "arity"
      output("+ rep(x).n",rep(x).n::OutputForm)$OutputPackage
      output("+ rep(y).n",rep(y).n::OutputForm)$OutputPackage
      r := product(rep(x).t, rep(y).t)
      u := 1$DirectProduct(dim,K)::T
      du:= product(kroneckerDelta()$T,u)
      for i in 1..rep(y).m repeat
        output("+ rank",rank(r)::OutputForm)$OutputPackage
        output("+ rep(y).m",rep(y).m::OutputForm)$OutputPackage
        r := contract(contract(du,1,r,rep(y).m+1),1,rep(y).m+2)
        --output("rank",rank(r)::OutputForm)$OutputPackage
      per [rep(x).n+rep(y).n,rep(y).m,r]

    -- repeated sum
    (x:% * p:NonNegativeInteger):% ==
      q:=subtractIfCan(p,1)
      q case NonNegativeInteger => x*q + x
      per [0,rep(x).m,1]  -- need identity for + !!!

    (x:% = y:%):Boolean ==
      rep(x).n ~= rep(y).n or rep(x).m ~= rep(y).m => error "arity"
      rep(x).t = rep(y).t

    (x:K * y:%):% == per [rep(y).n,rep(y).m,x*rep(y).t]

    (x:% * y:K):% == per [rep(x).n,rep(x).m,rep(x).t*y]

    elt(x:%,y:%):% ==
      r:=product(rep(x).t,rep(y).t)
      yn:=rep(y).n  -- outputs of y
      xm:=rep(x).m  -- inputs of x
      while yn>0 and xm>0 repeat
        output("yn",yn::OutputForm)$OutputPackage
        output("xm",xm::OutputForm)$OutputPackage
        output("rank",rank(r)::OutputForm)$OutputPackage
        r:=contract(r,rep(y).n+xm,yn+rep(y).m)
        yn:=subtractIfCan(yn,1)::NonNegativeInteger
        xm:=subtractIfCan(xm,1)::NonNegativeInteger
      per [rep(x).n+xm,yn+rep(y).m,r]

    id():% == per [1,1,kroneckerDelta()$T]

    inp(x:List K):% == per [1,0,entries(x)::T]

    inp(x:List %):% ==
      #removeDuplicates([rep(y).n for y in x])~=1 or
        #removeDuplicates([rep(y).m for y in x])~=1 => error "arity"
      per [rep(first x).n+1,rep(first x).m,[rep(y).t for y in x]::T]$Rep

    out(x:List K):% == per [0,1,entries(x)::T]

    out(x:List %):% ==
      #removeDuplicates([rep(y).n for y in x])~=1 or
        #removeDuplicates([rep(y).m for y in x])~=1 => error "arity"
      per [rep(first x).n,rep(first x).m+1,[rep(y).t for y in x]::T]$Rep

    coerce(x:%):OutputForm == (rep(x).t)::OutputForm</spad>

Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/6154772542123042263-25px001.spad
      using old system compiler.
   LIN abbreviates domain LinearOperator 
------------------------------------------------------------------------
   initializing NRLIB LIN for LinearOperator 
   compiling into NRLIB LIN 
   importing List NonNegativeInteger
************* USER ERROR **********
available signatures for T$: 
    NONE
NEED T$: () -> ?
****** comp fails at level 1 with expression: ******
((DEF (T$) (NIL) (|CartesianTensor| 1 |dim| K)))
****** level 1  ******
x:= (DEF (T$) (NIL) (CartesianTensor (One) dim K))
m:= $EmptyMode
f:=
((((|#| #) (< #) (<= #) (= #) ...)))

   >> Apparent user error:
   unspecified error

Construction operators: input and output

L:=LIN(2,FRAC POLY INT)

   LinearOperator is an unknown constructor and so is unavailable. Did 
      you mean to use -> but type something different instead?

Products

A12p := A1 * A2; A12p::OutputForm = A1::OutputForm * A2::OutputForm

(1)

arity(A12p)::OutputForm = arity(A1)::OutputForm * arity(A2)::OutputForm

   There are 1 exposed and 0 unexposed library operations named arity 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op arity
      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 arity
      with argument type(s) 
                             Polynomial(Integer)

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

Powers

A3p:=(A1*A1)*A1

(2)

arity(A3p)

   There are 1 exposed and 0 unexposed library operations named arity 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op arity
      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 arity
      with argument type(s) 
                             Polynomial(Integer)

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

B3p:=(B1*B1)*B1

(3)

arity(B3p)

   There are 1 exposed and 0 unexposed library operations named arity 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op arity
      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 arity
      with argument type(s) 
                             Polynomial(Integer)

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

Sums

A12s := A1 + A2; A12s::OutputForm = A1::OutputForm + A2::OutputForm

(4)

arity(A12s)::OutputForm = arity(A1)::OutputForm + arity(A2)::OutputForm

   There are 1 exposed and 0 unexposed library operations named arity 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op arity
      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 arity
      with argument type(s) 
                             Polynomial(Integer)

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

Multiplication

A3s:=(A1+A1)+A1

(5)

arity(A3s)

   There are 1 exposed and 0 unexposed library operations named arity 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op arity
      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 arity
      with argument type(s) 
                             Polynomial(Integer)

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

B3s:=(B1+B1)+B1

(6)

arity(B3s)

   There are 1 exposed and 0 unexposed library operations named arity 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op arity
      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 arity
      with argument type(s) 
                             Polynomial(Integer)

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

Expected error

B1A1:=B1*A1

(7)

arity(B1A1)

   There are 1 exposed and 0 unexposed library operations named arity 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op arity
      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 arity
      with argument type(s) 
                             Polynomial(Integer)

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

Composition

AB11:=A1 B1

   There are no library operations named A1 
      Use HyperDoc Browse or issue
                                 )what op A1
      to learn if there is any operation containing " A1 " in its name.

   Cannot find a definition or applicable library operation named A1 
      with argument type(s) 
                                Variable(B1)

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

Multiple inputs and outputs

W:L:=out[inp[inp([script(w,[[i,j],[k]]) for j in 1..2])$L for i in 1..2] for k in 1..2]

   L is not a valid type.

  ___ _       _        _  _
  ___ _)  =   _\_ _    _\/_
  _/          ___ _)   _/\_

  Rep == Record(Dom:VectorSpace, Cod:VectorSpace, t:T)

But VectorSpace? is a category which would make Dom and Cod domains. The domains that currently satisfy VectorSpace? in Axiom are rather limited and seem oddly focused on number theory (finite fields). It is a good thing however that DirectProduct? is a conditional member of this cateogry.

DirectProduct(2,FRAC INT) has VectorSpace(FRAC INT)

   There are no library operations named VectorSpace 
      Use HyperDoc Browse or issue
                            )what op VectorSpace
      to learn if there is any operation containing " VectorSpace " in 
      its name.

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

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

Unfortunately:

DirectProduct(2,DirectProduct(2,FRAC INT)) has VectorSpace(FRAC INT)

   There are no library operations named VectorSpace 
      Use HyperDoc Browse or issue
                            )what op VectorSpace
      to learn if there is any operation containing " VectorSpace " in 
      its name.

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

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

The problem with VectorSpace? is that the domain [Vector]? is not a member of this category instead it satisfies VectoryCategory?.

Is it possible to treat tensors from CartesianTensor? as maps from VectorSpace? to VectorSpace?? We would like for example:

  T:DirectProduct(dim,DirectProduct(dim,FRAC INT)) -> DirectProduct(dim,FRAC INT))

To be a linear operator with two inputs and one output.