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	FriCAS Algebra
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LinearOperator

last edited 4 years ago by Bill Page

Edit detail for LinearOperator revision 2 of 63

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Editor: Bill Page Time: 2011/03/08 01:07:28 GMT-8
Note: examples

added:
    arity: % -> DirectProduct(2,NonNegativeInteger)

changed:
-    coerce: DirectProduct(dim,K) -> %
    coerce: Vector K -> %
    coerce: Vector % -> %

added:
    coerce: SquareMatrix(dim,K) -> %

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

changed:
-      yn:=rep(y).n
-      xm:=rep(x).m
      yn:=rep(y).n  -- outputs of y
      xm:=rep(x).m  -- inputs of x

changed:
-        r:=contract(r,rep(x).n+xm,yn+rep(y).m)
        print(yn::OutputForm)$PrintPackage
        print(xm::OutputForm)$PrintPackage
        r:=contract(r,rep(x).n+xm,yn+rep(x).m)

changed:
-    coerce(x:DirectProduct(dim,K)):% == per [0,1,x::T]
-    coerce(x:List K):% == per [1,0,x::T]
    coerce(x:Vector K):% == per [0,1,entries(x)::T]
    coerce(x:List K):% == per [1,0,entries(x)::T]

added:
    coerce(x:SquareMatrix(dim,K)):% == per [2,0,x::T]
    coerce(x:%):OutputForm == (rep(x).t)::OutputForm

added:

\begin{axiom}
Y:LIN(2,FRAC POLY INT):=[y1,y2]::List FRAC POLY INT
arity Y
A:LIN(2,FRAC POLY INT):=[a1,a2]::Vector FRAC POLY INT
arity A
X:LIN(2, FRAC POLY INT):=[[x11,x12],[x21,x22]]::SquareMatrix(2,FRAC POLY INT)
arity(X)
T:=A*Y
arity(T)
C:=A Y
arity(C)
D:= Y A
arity(D)
XA:=X A
arity(XA)
Z:LIN(2, FRAC POLY INT):=[A,A]
arity(Z)
\end{axiom}

fricas

(1) -> <spad>

fricas

)abbrev domain LIN LinearOperator
LinearOperator(dim:NonNegativeInteger,K:CommutativeRing): Join(Ring,BiModule(K,K)) with
    arity: % -> DirectProduct(2,NonNegativeInteger)
    elt: (%,%) -> %
    id: %
    coerce: Vector K -> %
    coerce: Vector % -> %
    coerce: List K -> %
    coerce: List % -> %
    coerce: SquareMatrix(dim,K) -> %
  == 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]
    (x:% * y:%):% == per [rep(x).n+rep(y).n,rep(x).m+rep(y).m,product(rep(x).t, rep(y).t)]
    (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
        print(yn::OutputForm)$PrintPackage
        print(xm::OutputForm)$PrintPackage
        r:=contract(r,rep(x).n+xm,yn+rep(x).m)
        yn:=subtractIfCan(yn,1)::NonNegativeInteger
        xm:=subtractIfCan(xm,1)::NonNegativeInteger
      per [rep(x).n+xm,yn+rep(y).m,r]
    id:% == per [1,1,1]
    coerce(x:Vector K):% == per [0,1,entries(x)::T]
    coerce(x:List K):% == per [1,0,entries(x)::T]
    coerce(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
    coerce(x:SquareMatrix(dim,K)):% == per [2,0,x::T]
    coerce(x:%):OutputForm == (rep(x).t)::OutputForm</spad>

fricas

Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/2311065628842128602-25px001.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

fricas

Y:LIN(2,FRAC POLY INT):=[y1,y2]::List FRAC POLY INT

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