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	FriCAS Algebra
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LinearOperator

last edited 2 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}

spad

)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

   Compiling FriCAS source code from file 
      /var/zope2/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
   compiling local rep : $ -> Record(n: NonNegativeInteger,m: NonNegativeInteger,t: CartesianTensor(One,dim,K))
      LIN;rep is replaced by x 
Time: 0.14 SEC.

   compiling local per : Record(n: NonNegativeInteger,m: NonNegativeInteger,t: CartesianTensor(One,dim,K)) -> $
      LIN;per is replaced by x 
Time: 0 SEC.

   compiling exported arity : $ -> DirectProduct(2,NonNegativeInteger)
Time: 0.02 SEC.

   compiling exported Zero : () -> $
Time: 0.01 SEC.

   compiling exported + : ($,$) -> $
Time: 0.26 SEC.

   compiling exported - : ($,$) -> $
Time: 0.02 SEC.

   compiling exported One : () -> $
Time: 0 SEC.

   compiling exported * : ($,$) -> $
Time: 0 SEC.

   compiling exported = : ($,$) -> Boolean
Time: 0.01 SEC.

   compiling exported * : (K,$) -> $
Time: 0.01 SEC.

   compiling exported * : ($,K) -> $
Time: 0.01 SEC.

   compiling exported elt : ($,$) -> $
Time: 0.01 SEC.

   compiling local id : () -> $
Time: 0 SEC.

   compiling exported coerce : Vector K -> $
Time: 0.06 SEC.

   compiling exported coerce : List K -> $
Time: 0.05 SEC.

   compiling exported coerce : List $ -> $
Time: 0.09 SEC.

   compiling exported coerce : SquareMatrix(dim,K) -> $
Time: 0 SEC.

   compiling exported coerce : $ -> OutputForm
Time: 0 SEC.

(time taken in buildFunctor:  10)

;;;     ***       |LinearOperator| REDEFINED

;;;     ***       |LinearOperator| REDEFINED
Time: 0.02 SEC.

   Cumulative Statistics for Constructor LinearOperator
      Time: 0.71 seconds

   finalizing NRLIB LIN 
   Processing LinearOperator for Browser database:
--->-->LinearOperator((arity ((DirectProduct 2 (NonNegativeInteger)) %))): Not documented!!!!
--->-->LinearOperator((elt (% % %))): Not documented!!!!
--->-->LinearOperator(): Not documented!!!!
--->-->LinearOperator((coerce (% (Vector K)))): Not documented!!!!
--->-->LinearOperator((coerce (% (Vector %)))): Not documented!!!!
--->-->LinearOperator((coerce (% (List K)))): Not documented!!!!
--->-->LinearOperator((coerce (% (List %)))): Not documented!!!!
--->-->LinearOperator((coerce (% (SquareMatrix dim K)))): Not documented!!!!
--->-->LinearOperator(constructor): Not documented!!!!
--->-->LinearOperator(): Missing Description
; compiling file "/var/zope2/var/LatexWiki/LIN.NRLIB/LIN.lsp" (written 08 MAR 2011 01:07:00 AM):
; compiling (/VERSIONCHECK 2)
; compiling (PUT (QUOTE |LIN;rep|) ...)
; compiling (DEFUN |LIN;rep| ...)
; compiling (PUT (QUOTE |LIN;per|) ...)
; compiling (DEFUN |LIN;per| ...)
; compiling (DEFUN |LIN;arity;$Dp;3| ...)
; compiling (DEFUN |LIN;Zero;$;4| ...)
; compiling (DEFUN |LIN;+;3$;5| ...)
; compiling (DEFUN |LIN;-;3$;6| ...)
; compiling (DEFUN |LIN;One;$;7| ...)
; compiling (DEFUN |LIN;*;3$;8| ...)
; compiling (DEFUN |LIN;=;2$B;9| ...)
; compiling (DEFUN |LIN;*;K2$;10| ...)
; compiling (DEFUN |LIN;*;$K$;11| ...)
; compiling (DEFUN |LIN;elt;3$;12| ...)
; compiling (DEFUN |LIN;id| ...)
; compiling (DEFUN |LIN;coerce;V$;14| ...)
; compiling (DEFUN |LIN;coerce;L$;15| ...)
; compiling (DEFUN |LIN;coerce;L$;16| ...)
; compiling (DEFUN |LIN;coerce;Sm$;17| ...)
; compiling (DEFUN |LIN;coerce;$Of;18| ...)
; compiling (DEFUN |LinearOperator| ...)
; compiling (DEFUN |LinearOperator;| ...)
; compiling (MAKEPROP (QUOTE |LinearOperator|) ...)

; /var/zope2/var/LatexWiki/LIN.NRLIB/LIN.fasl written
; compilation finished in 0:00:00.637
------------------------------------------------------------------------
   LinearOperator is now explicitly exposed in frame initial 
   LinearOperator will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/LIN.NRLIB/LIN

axiom

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

$\label{eq1}\left[ y 1, \: y 2 \right]$

(1)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity Y

$\label{eq2}\left[ 1, \: 0 \right]$

(2)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

A:LIN(2,FRAC POLY INT):=[a1,a2]::Vector FRAC POLY INT

$\label{eq3}\left[ a 1, \: a 2 \right]$

(3)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity A

$\label{eq4}\left[ 0, \: 1 \right]$

(4)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

X:LIN(2, FRAC POLY INT):=[[x11,x12],[x21,x22]]::SquareMatrix(2,FRAC POLY INT)

$\label{eq5}\left[ \begin{array}{cc} x 11 & x 12 \ x 21 & x 22$

(5)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity(X)

$\label{eq6}\left[ 2, \: 0 \right]$

(6)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

T:=A*Y

$\label{eq7}\left[ \begin{array}{cc} {a 1 \ y 1}&{a 1 \ y 2} \ {a 2 \ y 1}&{a 2 \ y 2}$

(7)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity(T)

$\label{eq8}\left[ 1, \: 1 \right]$

(8)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

C:=A Y

   1
   1

$\label{eq9}{a 2 \ y 2}+{a 1 \ y 1}$

(9)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity(C)

$\label{eq10}\left[ 0, \: 0 \right]$

(10)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

D:= Y A

$\label{eq11}\left[ \begin{array}{cc} {a 1 \ y 1}&{a 2 \ y 1} \ {a 1 \ y 2}&{a 2 \ y 2}$

(11)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity(D)

$\label{eq12}\left[ 1, \: 1 \right]$

(12)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

XA:=X A

$\label{eq13}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {a 1 \ x 11}&{a 2 \ x 11} \ {a 1 \ x 12}&{a 2 \ x 12}$

(13)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity(XA)

$\label{eq14}\left[ 2, \: 1 \right]$

(14)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

Z:LIN(2, FRAC POLY INT):=[A,A]

$\label{eq15}\left[ \begin{array}{cc} a 1 & a 2 \ a 1 & a 2$

(15)

Type: LinearOperator?(2,Fraction(Polynomial(Integer)))

axiom

arity(Z)

$\label{eq16}\left[ 1, \: 1 \right]$

(16)

Type: DirectProduct?(2,NonNegativeInteger?)