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	FriCAS Algebra
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LinearOperator

last edited 1 year ago by Bill Page

Edit detail for LinearOperator revision 4 of 63

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Editor: Bill Page Time: 2011/03/08 12:12:56 GMT-8
Note: product and co-product

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

changed:
-        --print(yn::OutputForm)$PrintPackage
-        --print(xm::OutputForm)$PrintPackage
        output("yn",yn::OutputForm)$OutputPackage
        output("xm",xm::OutputForm)$OutputPackage
        output("rank",rank(r)::OutputForm)$OutputPackage

added:
\end{axiom}

\begin{axiom}
XX1:=X1*X1
arity(XX1)
\end{axiom}
\begin{axiom}
YY1:=Y1*Y1
arity(YY1)
\end{axiom}
\begin{axiom}

added:
\end{axiom}

\begin{axiom}

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 -> %
    inp: List % -> %
    out: List K -> %
    out: 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:%):% ==
      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)$OutputPackage
      per [rep(x).n,rep(x).m+rep(y).m,r]
    (x:% + y:%):% ==
      rep(x).m ~= rep(y).m  => error "arity"
      r:=product(rep(x).t, rep(y).t)
      u:=1$DirectProduct(dim,K)::T
      du:=product(kroneckerDelta()$T,u)
      for i in 1..rep(y).m repeat
        --output("rank",rank(r)::OutputForm)$OutputPackage
        --output("m",rep(y).m::OutputForm)$OutputPackage
        r:=contract(contract(du,1,r,rep(y).m+1),1,rep(y).m+1)
        --output("rank",rank(r)::OutputForm)$OutputPackage
      per [rep(x).n+rep(y).m,rep(y).m,r]
    (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)$OutputPackage
        r:=contract(r,rep(y).n+xm,yn+rep(y).m)
        yn:=subtractIfCan(yn,1)::NonNegativeInteger
        xm:=subtractIfCan(xm,1)::NonNegativeInteger
      per [rep(x).n+xm,yn+rep(y).m,r]
    id():% == per [1,1,1]
    inp(x:List K):% == per [1,0,entries(x)::T]
    inp(x:List %):% ==
      #removeDuplicates([rep(y).n for y in x])~=1 or
        #removeDuplicates([rep(y).m for y in x])~=1 => error "arity"
      per [rep(first x).n+1,rep(first x).m,[rep(y).t for y in x]::T]$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/zope2/var/LatexWiki/3626382297542132014-25px001.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: 0.12 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: 0.24 SEC.

   compiling exported + : ($,$) -> $
Time: 0.03 SEC.

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

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

   compiling exported * : ($,$) -> $
Time: 0.03 SEC.

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

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

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

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

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

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

   compiling exported inp : List K -> $
Time: 0.08 SEC.

   compiling exported inp : List $ -> $
Time: 0.13 SEC.

   compiling exported out : List K -> $
Time: 0 SEC.

   compiling exported out : List $ -> $
Time: 0.08 SEC.

   compiling exported coerce : $ -> OutputForm
Time: 0.02 SEC.

(time taken in buildFunctor:  10)

;;;     ***       |LinearOperator| REDEFINED

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

   Cumulative Statistics for Constructor LinearOperator
      Time: 0.88 seconds

   finalizing NRLIB LIN 
   Processing LinearOperator for Browser database:
--->-->LinearOperator((arity ((DirectProduct 2 (NonNegativeInteger)) %))): Not documented!!!!
--->-->LinearOperator((elt (% % %))): Not documented!!!!
--->-->LinearOperator((id (%))): Not documented!!!!
--->-->LinearOperator((inp (% (List K)))): Not documented!!!!
--->-->LinearOperator((inp (% (List %)))): Not documented!!!!
--->-->LinearOperator((out (% (List K)))): Not documented!!!!
--->-->LinearOperator((out (% (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 12:12:38 PM):
; 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;+;3$;9| ...)
; compiling (DEFUN |LIN;=;2$B;10| ...)
; compiling (DEFUN |LIN;*;K2$;11| ...)
; compiling (DEFUN |LIN;*;$K$;12| ...)
; compiling (DEFUN |LIN;elt;3$;13| ...)
; compiling (DEFUN |LIN;id;$;14| ...)
; compiling (DEFUN |LIN;inp;L$;15| ...)
; compiling (DEFUN |LIN;inp;L$;16| ...)
; compiling (DEFUN |LIN;out;L$;17| ...)
; compiling (DEFUN |LIN;out;L$;18| ...)
; compiling (DEFUN |LIN;coerce;$Of;19| ...)
; compiling (DEFUN |LinearOperator| ...)
; compiling (DEFUN |LinearOperator;| ...)
; compiling (MAKEPROP (QUOTE |LinearOperator|) ...)

; /var/zope2/var/LatexWiki/LIN.NRLIB/LIN.fasl written
; compilation finished in 0:00:00.744
------------------------------------------------------------------------
   LinearOperator is now explicitly exposed in frame initial 
   LinearOperator will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/LIN.NRLIB/LIN

axiom

L:=LIN(2,FRAC POLY INT)

$\label{eq1}\hbox{\axiomType{LinearOperator}\ } (2, \hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Polynomial}\ } (\hbox{\axiomType{Integer}\ })))$

(1)

Type: Domain

axiom

X1:L:=inp[script(x,[[i]]) for i in 1..2]

$\label{eq2}\left[{x_{1}}, \:{x_{2}}\right]$

(2)

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

axiom

arity X1

$\label{eq3}\left[ 1, \: 0 \right]$

(3)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

Y1:L:=out[script(y,[[],[i]]) for i in 1..2]

$\label{eq4}\left[{y^{1}}, \:{y^{2}}\right]$

(4)

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

axiom

arity Y1

$\label{eq5}\left[ 0, \: 1 \right]$

(5)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

Y11:L:=inp[Y1,Y1]

$\label{eq6}\left[ \begin{array}{cc} {y^{1}}&{y^{2}} \ {y^{1}}&{y^{2}}$

(6)

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

axiom

arity(Y11)

$\label{eq7}\left[ 1, \: 1 \right]$

(7)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

XX1:=X1*X1

$\label{eq8}\left[{{2 \ {x_{1}}\ {x_{2}}}+{2 \ {{x_{1}}^2}}}, \:{{2 \ {{x_{2}}^2}}+{2 \ {x_{1}}\ {x_{2}}}}\right]$

(8)

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

axiom

arity(XX1)

$\label{eq9}\left[ 1, \: 0 \right]$

(9)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

YY1:=Y1*Y1

$\label{eq10}\left[ \begin{array}{cc} {{y^{1}}^2}&{{y^{1}}\ {y^{2}}} \ {{y^{1}}\ {y^{2}}}&{{y^{2}}^2}$

(10)

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

axiom

arity(YY1)

$\label{eq11}\left[ 0, \: 2 \right]$

(11)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

Y1X1:=Y1*X1

   >> Error detected within library code:
   arity

axiom

XY11:=X1 Y1

$\label{eq12}\left[ \begin{array}{cc} {{y^{1}}\ {x_{1}}}&{{y^{2}}\ {x_{1}}} \ {{y^{1}}\ {x_{2}}}&{{y^{2}}\ {x_{2}}}$

(12)

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

axiom

arity(XY11)

$\label{eq13}\left[ 1, \: 1 \right]$

(13)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

YX11:= Y1 X1

   yn 1
   xm 1
   rank 2

$\label{eq14}{{y^{2}}\ {x_{2}}}+{{y^{1}}\ {x_{1}}}$

(14)

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

axiom

arity(YX11)

$\label{eq15}\left[ 0, \: 0 \right]$

(15)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

X2:L:=inp[inp([script(x,[[i,j]]) for j in 1..2])$L for i in 1..2]

$\label{eq16}\left[ \begin{array}{cc} {x_{1, \: 1}}&{x_{1, \: 2}} \ {x_{2, \: 1}}&{x_{2, \: 2}}$

(16)

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

axiom

arity(X2)

$\label{eq17}\left[ 2, \: 0 \right]$

(17)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

XX21:=X2 X1

$\label{eq18}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {{x_{1}}\ {x_{1, \: 1}}}&{{x_{2}}\ {x_{1, \: 1}}} \ {{x_{1}}\ {x_{1, \: 2}}}&{{x_{2}}\ {x_{1, \: 2}}}$

(18)

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

axiom

arity(XX21)

$\label{eq19}\left[ 2, \: 1 \right]$

(19)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

Y2:L:=out[out([script(y,[[],[i,j]]) for j in 1..2])$L for i in 1..2]

$\label{eq20}\left[ \begin{array}{cc} {y^{1, \: 1}}&{y^{1, \: 2}} \ {y^{2, \: 1}}&{y^{2, \: 2}}$

(20)

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

axiom

arity(Y2)

$\label{eq21}\left[ 0, \: 2 \right]$

(21)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

XY22:=X2 Y2

$\label{eq22}\left[ \begin{array}{cc} {\left[ \begin{array}{cc} {{y^{1, \: 1}}\ {x_{1, \: 1}}}&{{y^{1, \: 2}}\ {x_{1, \: 1}}} \ {{y^{2, \: 1}}\ {x_{1, \: 1}}}&{{y^{2, \: 2}}\ {x_{1, \: 1}}}$

(22)

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

axiom

arity(XY22)

$\label{eq23}\left[ 2, \: 2 \right]$

(23)

Type: DirectProduct?(2,NonNegativeInteger?)

axiom

YX22:=Y2 X2

   yn 2
   xm 2
   rank 4
   yn 1
   xm 1
   rank 2

   >> Error detected within library code:
   Improper index for contraction