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)abbrev domain MODOP ModuleOperator
++ Author: Manuel Bronstein
++ Date Created: 15 May 1990
++ Date Last Updated: 17 June 1993
++ Description:
++ Algebra of ADDITIVE operators on a module.
ModuleOperator(R: Ring, M:LeftModule(R)): Exports == Implementation where
O ==> OutputForm
OP ==> BasicOperator
FG ==> FreeGroup OP
RM ==> Record(coef:R, monom:FG)
TERM ==> List RM
FAB ==> FreeAbelianGroup TERM
OPADJ ==> "%opAdjoint"
OPEVAL ==> "%opEval"
INVEVAL ==> "%invEval"
Exports ==> Join(Ring, RetractableTo R, RetractableTo OP,
Eltable(M, M)) with
if R has CharacteristicZero then CharacteristicZero
if R has CharacteristicNonZero then CharacteristicNonZero
if R has CommutativeRing then
Algebra(R)
adjoint: $ -> $
++ adjoint(op) returns the adjoint of the operator \spad{op}.
adjoint: ($, $) -> $
++ adjoint(op1, op2) sets the adjoint of op1 to be op2.
++ op1 must be a basic operator
conjug : R -> R
++ conjug(x)should be local but conditional
evaluate: ($, M -> M) -> $
++ evaluate(f, u +-> g u) attaches the map g to f.
++ f must be a basic operator
++ g MUST be additive, i.e. \spad{g(a + b) = g(a) + g(b)} for
++ any \spad{a}, \spad{b} in M.
++ This implies that \spad{g(n a) = n g(a)} for
++ any \spad{a} in M and integer \spad{n > 0}.
evaluateInverse: ($, M -> M) -> $
++ evaluateInverse(x,f) \undocumented
**: (OP, Integer) -> $
++ op**n \undocumented
**: ($, Integer) -> $
++ op**n \undocumented
opeval : (OP, M) -> M
++ opeval should be local but conditional
makeop : (R, FG) -> $
++ makeop should be local but conditional
Implementation ==> FAB add
import NoneFunctions1($)
import BasicOperatorFunctions1(M)
Rep := FAB
inv : TERM -> $
termeval : (TERM, M) -> M
rmeval : (RM, M) -> M
monomeval: (FG, M) -> M
opInvEval: (OP, M) -> M
mkop : (R, FG) -> $
termprod0: (Integer, TERM, TERM) -> $
termprod : (Integer, TERM, TERM) -> TERM
termcopy : TERM -> TERM
trm2O : (Integer, TERM) -> O
term2O : TERM -> O
rm2O : (R, FG) -> O
nocopy : OP -> $
1 == makeop(1, 1)
coerce(n:Integer):$ == n::R::$
coerce(r:R):$ == (zero? r => 0; makeop(r, 1))
coerce(op:OP):$ == nocopy copy op
nocopy(op:OP):$ == makeop(1, op::FG)
elt(x:$, r:M) == +/[t.exp * termeval(t.gen, r) for t in terms x]
rmeval(t, r) == t.coef * monomeval(t.monom, r)
termcopy t == [[rm.coef, rm.monom] for rm in t]
characteristic == characteristic$R
mkop(r, fg) == [[r, fg]$RM]$TERM :: $
evaluate(f, g) == nocopy setProperty(retract(f)@OP,OPEVAL,g pretend None)
if R has OrderedSet then
makeop(r, fg) == (r >= 0 => mkop(r, fg); - mkop(-r, fg))
else makeop(r, fg) == mkop(r, fg)
inv(t:TERM):$ ==
empty? t => 1
c := first(t).coef
m := first(t).monom
inv(rest t) * makeop(1, inv m) * (recip(c)::R::$)
x:$ ** i:Integer ==
i = 0 => 1
positive? i => expt(x,i pretend PositiveInteger)$RepeatedSquaring($)
(inv(retract(x)@TERM)) ** (-i)
evaluateInverse(f, g) ==
nocopy setProperty(retract(f)@OP, INVEVAL, g pretend None)
coerce(x:$):O ==
zero? x => (0$R)::O
reduce(_+, [trm2O(t.exp, t.gen) for t in terms x])$List(O)
trm2O(c, t) ==
one? c => term2O t
c = -1 => - term2O t
c::O * term2O t
term2O t ==
reduce(_*, [rm2O(rm.coef, rm.monom) for rm in t])$List(O)
rm2O(c, m) ==
one? c => m::O
one? m => c::O
c::O * m::O
x:$ * y:$ ==
+/[ +/[termprod0(t.exp * s.exp, t.gen, s.gen) for s in terms y]
for t in terms x]
termprod0(n, x, y) ==
n >= 0 => termprod(n, x, y)::$
- (termprod(-n, x, y)::$)
termprod(n, x, y) ==
lc := first(xx := termcopy x)
lc.coef := n * lc.coef
rm := last xx
one?(first(y).coef) =>
rm.monom := rm.monom * first(y).monom
concat!(xx, termcopy rest y)
one?(rm.monom) =>
rm.coef := rm.coef * first(y).coef
rm.monom := first(y).monom
concat!(xx, termcopy rest y)
concat!(xx, termcopy y)
if M has ExpressionSpace then
opeval(op, r) ==
(func := property(op, OPEVAL)) case "failed" => kernel(op, r)
((func::None) pretend (M -> M)) r
else
opeval(op, r) ==
(func := property(op, OPEVAL)) case "failed" =>
error "eval: operator has no evaluation function"
((func::None) pretend (M -> M)) r
opInvEval(op, r) ==
(func := property(op, INVEVAL)) case "failed" =>
error "eval: operator has no inverse evaluation function"
((func::None) pretend (M -> M)) r
termeval(t, r) ==
for rm in reverse t repeat r := rmeval(rm, r)
r
monomeval(m, r) ==
for rec in reverse! factors m repeat
e := rec.exp
g := rec.gen
positive? e =>
for i in 1..e repeat r := opeval(g, r)
negative? e =>
for i in 1..(-e) repeat r := opInvEval(g, r)
r
recip x ==
(r := retractIfCan(x)@Union(R, "failed")) case "failed" => "failed"
(r1 := recip(r::R)) case "failed" => "failed"
r1::R::$
retractIfCan(x:$):Union(R, "failed") ==
(r:= retractIfCan(x)@Union(TERM,"failed")) case "failed" => "failed"
empty?(t := r::TERM) => 0$R
empty? rest t =>
rm := first t
one?(rm.monom) => rm.coef
"failed"
"failed"
retractIfCan(x:$):Union(OP, "failed") ==
(r:= retractIfCan(x)@Union(TERM,"failed")) case "failed" => "failed"
empty?(t := r::TERM) => "failed"
empty? rest t =>
rm := first t
one?(rm.coef) => retractIfCan(rm.monom)
"failed"
"failed"
if R has CommutativeRing then
termadj : TERM -> $
rmadj : RM -> $
monomadj : FG -> $
opadj : OP -> $
r:R * x:$ == r::$ * x
x:$ * r:R == x * (r::$)
adjoint x == +/[t.exp * termadj(t.gen) for t in terms x]
rmadj t == conjug(t.coef) * monomadj(t.monom)
adjoint(op, adj) == nocopy setProperty(retract(op)@OP, OPADJ, adj::None)
termadj t ==
ans:$ := 1
for rm in t repeat ans := rmadj(rm) * ans
ans
monomadj m ==
ans:$ := 1
for rec in factors m repeat ans := (opadj(rec.gen) ** rec.exp) * ans
ans
opadj op ==
(adj := property(op, OPADJ)) case "failed" =>
error "adjoint: operator does not have a defined adjoint"
(adj::None) pretend $
if R has conjugate:R -> R then conjug r == conjugate r else conjug r == r
)abbrev domain OP Operator
++ Author: Manuel Bronstein
++ Date Created: 15 May 1990
++ Date Last Updated: 12 February 1993
++ Description:
++ Algebra of ADDITIVE operators over a ring.
Operator(R: Ring) == ModuleOperator(R,R)
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