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--All rights reserved.
--Copyright (C) 2007-2010, Gabriel Dos Reis.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
-- - Redistributions of source code must retain the above copyright
-- notice, this list of conditions and the following disclaimer.
--
-- - Redistributions in binary form must reproduce the above copyright
-- notice, this list of conditions and the following disclaimer in
-- the documentation and/or other materials provided with the
-- distribution.
--
-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
-- names of its contributors may be used to endorse or promote products
-- derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
)abbrev category IDPC IndexedDirectProductCategory
++ Author: James Davenport, Gabriel Dos Reis
++ Date Created:
++ Date Last Updated: June 28, 2010
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This category represents the direct product of some set with
++ respect to an ordered indexing set.
IndexedDirectProductCategory(A:BasicType,S:OrderedType): Category ==
BasicType with
if A has SetCategory and S has SetCategory then SetCategory
map: (A -> A, %) -> %
++ map(f,z) returns the new element created by applying the
++ function f to each component of the direct product element z.
monomial: (A, S) -> %
++ monomial(a,s) constructs a direct product element with the s
++ component set to \spad{a}
leadingCoefficient: % -> A
++ leadingCoefficient(z) returns the coefficient of the leading
++ (with respect to the ordering on the indexing set)
++ monomial of z.
++ Error: if z has no support.
leadingSupport: % -> S
++ leadingSupport(z) returns the index of leading
++ (with respect to the ordering on the indexing set) monomial of z.
++ Error: if z has no support.
reductum: % -> %
++ reductum(z) returns a new element created by removing the
++ leading coefficient/support pair from the element z.
++ Error: if z has no support.
terms: % -> List Pair(S,A)
++ \spad{terms x} returns the list of terms in \spad{x}.
++ Each term is a pair of a support (the first component)
++ and the corresponding value (the second component).
)abbrev domain IDPO IndexedDirectProductObject
++ Author: James Davenport, Gabriel Dos Reis
++ Date Created:
++ Date Last Updated: June 28, 2010
++ Description:
++ Indexed direct products of objects over a set \spad{A}
++ of generators indexed by an ordered set S. All items have finite support.
IndexedDirectProductObject(A,S): Public == Private where
A: BasicType
S: OrderedType
Public == IndexedDirectProductCategory(A,S)
Private == add
Term == Pair(S,A)
Rep == List Term
-- Return the index of a term
termIndex(t: Term): S == first t
-- Return the value of a term
termValue(t: Term): A == second t
x = y ==
x' := rep x
y' := rep y
while not null x' and not null y' repeat
termIndex first x' ~= termIndex first y' => return false
termValue first x' ~= termValue first y' => return false
x' := rest x'
y' := rest y'
null x' and null y'
if A has CoercibleTo OutputForm and S has CoercibleTo OutputForm then
coerce(x:%):OutputForm ==
bracket [rarrow(termIndex(t)::OutputForm, termValue(t)::OutputForm)
for t in rep x]
-- sample():% == [[sample()$S,sample()$A]$Term]$Rep
monomial(r,s) == per [[s,r]]
map(f,x) == per [[termIndex tm,f termValue tm] for tm in rep x]
reductum x ==
per rest rep x
leadingCoefficient x ==
null rep x =>
error "Can't take leadingCoefficient of empty product element"
termValue first rep x
leadingSupport x ==
null rep x =>
error "Can't take leadingCoefficient of empty product element"
termIndex first rep x
terms x == rep x
)abbrev domain IDPAM IndexedDirectProductAbelianMonoid
++ Indexed direct products of abelian monoids over an abelian monoid \spad{A} of
++ generators indexed by the ordered set S. All items have finite support.
++ Only non-zero terms are stored.
IndexedDirectProductAbelianMonoid(A:AbelianMonoid,S:OrderedType):
Join(AbelianMonoid,IndexedDirectProductCategory(A,S))
== IndexedDirectProductObject(A,S) add
--representations
Term == Pair(S,A)
import Term
import List Term
termIndex(t: Term): S == first t
termValue(t: Term): A == second t
r: A
n: NonNegativeInteger
f: A -> A
s: S
0 == nil$List(Term) pretend %
zero? x == null terms x
import %tail: List Term -> List Term from Foreign Builtin
qsetrest!: (List Term, List Term) -> List Term
qsetrest!(l, e) ==
%store(%tail l,e)$Foreign(Builtin)
-- PERFORMANCE CRITICAL; Should build list up
-- by merging 2 sorted lists. Doing this will
-- avoid the recursive calls (very useful if there is a
-- large number of vars in a polynomial.
x + y ==
x' := terms x
y' := terms y
null x' => y
null y' => x
endcell: List Term := nil
res: List Term := nil
while not empty? x' and not empty? y' repeat
newcell: List Term := nil
if termIndex x'.first = termIndex y'.first then
r := termValue x'.first + termValue y'.first
if not zero? r then
newcell := cons([termIndex x'.first, r], empty())
x' := rest x'
y' := rest y'
else if termIndex x'.first > termIndex y'.first then
newcell := cons(x'.first, empty())
x' := rest x'
else
newcell := cons(y'.first, empty())
y' := rest y'
if not empty? newcell then
if not empty? endcell then
qsetrest!(endcell, newcell)
endcell := newcell
else
res := newcell;
endcell := res
end :=
empty? x' => y'
x'
if empty? res then res := end
else qsetrest!(endcell, end)
res pretend %
n * x ==
n = 0 => 0
n = 1 => x
[[termIndex u,a] for u in terms x
| not zero?(a:=n * termValue u)] pretend %
monomial(r,s) ==
zero? r => 0
[[s,r]] pretend %
map(f,x) ==
[[termIndex tm,a] for tm in terms x
| not zero?(a:=f termValue tm)] pretend %
reductum x ==
null terms x => 0
rest(terms x) pretend %
leadingCoefficient x ==
null terms x => 0
termValue terms(x).first
pair2Term(t: Pair(A,S)): Term ==
[second t, first t]
)abbrev domain IDPOAM IndexedDirectProductOrderedAbelianMonoid
++ Indexed direct products of ordered abelian monoids \spad{A} of
++ generators indexed by the ordered set S.
++ The inherited order is lexicographical.
++ All items have finite support: only non-zero terms are stored.
IndexedDirectProductOrderedAbelianMonoid(A:OrderedAbelianMonoid,S:OrderedType):
Join(OrderedAbelianMonoid,IndexedDirectProductCategory(A,S))
== IndexedDirectProductAbelianMonoid(A,S) add
--representations
Term:= Record(k:S,c:A)
Rep:= List Term
x,y: %
x<y ==
empty? y => false
empty? x => true -- note careful order of these two lines
y.first.k > x.first.k => true
y.first.k < x.first.k => false
y.first.c > x.first.c => true
y.first.c < x.first.c => false
x.rest < y.rest
)abbrev domain IDPOAMS IndexedDirectProductOrderedAbelianMonoidSup
++ Indexed direct products of ordered abelian monoid sups \spad{A},
++ generators indexed by the ordered set S.
++ All items have finite support: only non-zero terms are stored.
IndexedDirectProductOrderedAbelianMonoidSup(A:OrderedAbelianMonoidSup,S:OrderedSet):
Join(OrderedAbelianMonoidSup,IndexedDirectProductCategory(A,S))
== IndexedDirectProductOrderedAbelianMonoid(A,S) add
--representations
Term:= Record(k:S,c:A)
Rep:= List Term
x,y: %
r: A
s: S
subtractIfCan(x,y) ==
empty? y => x
empty? x => "failed"
x.first.k < y.first.k => "failed"
x.first.k > y.first.k =>
t:= subtractIfCan(x.rest, y)
t case "failed" => "failed"
cons( x.first, t)
u:=subtractIfCan(x.first.c, y.first.c)
u case "failed" => "failed"
zero? u => subtractIfCan(x.rest, y.rest)
t:= subtractIfCan(x.rest, y.rest)
t case "failed" => "failed"
cons([x.first.k,u],t)
sup(x,y) ==
empty? y => x
empty? x => y
x.first.k < y.first.k => cons(y.first,sup(x,y.rest))
x.first.k > y.first.k => cons(x.first,sup(x.rest,y))
u:=sup(x.first.c, y.first.c)
cons([x.first.k,u],sup(x.rest,y.rest))
)abbrev domain IDPAG IndexedDirectProductAbelianGroup
++ Indexed direct products of abelian groups over an abelian group \spad{A} of
++ generators indexed by the ordered set S.
++ All items have finite support: only non-zero terms are stored.
IndexedDirectProductAbelianGroup(A:AbelianGroup,S:OrderedType):
Join(AbelianGroup,IndexedDirectProductCategory(A,S))
== IndexedDirectProductAbelianMonoid(A,S) add
--representations
Term == Pair(S,A)
termIndex(t: Term): S == first t
termValue(t: Term): A == second t
-x == [[termIndex u,-termValue u] for u in terms x] pretend %
n:Integer * x:% ==
n = 0 => 0
n = 1 => x
[[termIndex u,a] for u in terms x
| not zero?(a := n * termValue u)] pretend %
import %tail: List Term -> List Term from Foreign Builtin
qsetrest!: (List Term, List Term) -> List Term
qsetrest!(l, e) ==
%store(%tail l,e)$Foreign(Builtin)
x - y ==
x' := terms x
y' := terms y
null x' => -y
null y' => x
endcell: List Term := nil
res: List Term := nil
while not empty? x' and not empty? y' repeat
newcell: List Term := nil
if termIndex x'.first = termIndex y'.first then
r := termValue x'.first - termValue y'.first
if not zero? r then
newcell := cons([termIndex x'.first, r], empty())
x' := rest x'
y' := rest y'
else if termIndex x'.first > termIndex y'.first then
newcell := cons(x'.first, empty())
x' := rest x'
else
newcell := cons([termIndex y'.first,-termValue y'.first], empty())
y' := rest y'
if not empty? newcell then
if not empty? endcell then
qsetrest!(endcell, newcell)
endcell := newcell
else
res := newcell;
endcell := res
end :=
empty? x' => terms(-(y' pretend %))
x'
if empty? res then res := end
else qsetrest!(endcell, end)
res pretend %
|