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--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 ULSCCAT UnivariateLaurentSeriesConstructorCategory
++ Author: Clifton J. Williamson
++ Date Created: 6 February 1990
++ Date Last Updated: June 18, 2010
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords: series, Laurent, Taylor
++ Examples:
++ References:
++ Description:
++ This is a category of univariate Laurent series constructed from
++ univariate Taylor series. A Laurent series is represented by a pair
++ \spad{[n,f(x)]}, where n is an arbitrary integer and \spad{f(x)}
++ is a Taylor series. This pair represents the Laurent series
++ \spad{x**n * f(x)}.
UnivariateLaurentSeriesConstructorCategory(Coef,UTS):_
Category == Definition where
Coef: Ring
UTS : UnivariateTaylorSeriesCategory Coef
I ==> Integer
Definition ==> Join(UnivariateLaurentSeriesCategory(Coef),_
RetractableTo UTS, CoercibleFrom UTS) with
laurent: (I,UTS) -> %
++ \spad{laurent(n,f(x))} returns \spad{x**n * f(x)}.
degree: % -> I
++ \spad{degree(f(x))} returns the degree of the lowest order term of
++ \spad{f(x)}, which may have zero as a coefficient.
taylorRep: % -> UTS
++ \spad{taylorRep(f(x))} returns \spad{g(x)}, where
++ \spad{f = x**n * g(x)} is represented by \spad{[n,g(x)]}.
removeZeroes: % -> %
++ \spad{removeZeroes(f(x))} removes leading zeroes from the
++ representation of the Laurent series \spad{f(x)}.
++ A Laurent series is represented by (1) an exponent and
++ (2) a Taylor series which may have leading zero coefficients.
++ When the Taylor series has a leading zero coefficient, the
++ 'leading zero' is removed from the Laurent series as follows:
++ the series is rewritten by increasing the exponent by 1 and
++ dividing the Taylor series by its variable.
++ Note: \spad{removeZeroes(f)} removes all leading zeroes from f
removeZeroes: (I,%) -> %
++ \spad{removeZeroes(n,f(x))} removes up to n leading zeroes from
++ the Laurent series \spad{f(x)}.
++ A Laurent series is represented by (1) an exponent and
++ (2) a Taylor series which may have leading zero coefficients.
++ When the Taylor series has a leading zero coefficient, the
++ 'leading zero' is removed from the Laurent series as follows:
++ the series is rewritten by increasing the exponent by 1 and
++ dividing the Taylor series by its variable.
taylor: % -> UTS
++ taylor(f(x)) converts the Laurent series f(x) to a Taylor series,
++ if possible. Error: if this is not possible.
taylorIfCan: % -> Union(UTS,"failed")
++ \spad{taylorIfCan(f(x))} converts the Laurent series \spad{f(x)}
++ to a Taylor series, if possible. If this is not possible,
++ "failed" is returned.
if Coef has Field then QuotientFieldCategory(UTS)
--++ the quotient field of univariate Taylor series over a field is
--++ the field of Laurent series
add
zero? x == zero? taylorRep x
retract(x:%):UTS == taylor x
retractIfCan(x:%):Union(UTS,"failed") == taylorIfCan x
)abbrev domain ULSCONS UnivariateLaurentSeriesConstructor
++ Authors: Bill Burge, Clifton J. Williamson
++ Date Created: August 1988
++ Date Last Updated: 17 June 1996
++ Fix History:
++ 14 June 1996: provided missing exquo: (%,%) -> % (Frederic Lehobey)
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords: series, Laurent, Taylor
++ Examples:
++ References:
++ Description:
++ This package enables one to construct a univariate Laurent series
++ domain from a univariate Taylor series domain. Univariate
++ Laurent series are represented by a pair \spad{[n,f(x)]}, where n is
++ an arbitrary integer and \spad{f(x)} is a Taylor series. This pair
++ represents the Laurent series \spad{x**n * f(x)}.
UnivariateLaurentSeriesConstructor(Coef,UTS):_
Exports == Implementation where
Coef : Ring
UTS : UnivariateTaylorSeriesCategory Coef
I ==> Integer
L ==> List
NNI ==> NonNegativeInteger
OUT ==> OutputForm
P ==> Polynomial Coef
RF ==> Fraction Polynomial Coef
RN ==> Fraction Integer
ST ==> Stream Coef
TERM ==> Record(k:I,c:Coef)
monom ==> monomial$UTS
EFULS ==> ElementaryFunctionsUnivariateLaurentSeries(Coef,UTS,%)
STTAYLOR ==> StreamTaylorSeriesOperations Coef
Exports ==> UnivariateLaurentSeriesConstructorCategory(Coef,UTS)
Implementation ==> add
--% representation
Rep := Record(expon:I,ps:UTS)
getExpon : % -> I
getUTS : % -> UTS
getExpon x == x.expon
getUTS x == x.ps
--% creation and destruction
laurent(n,psr) == [n,psr]
taylorRep x == getUTS x
degree x == getExpon x
0 == laurent(0,0)
1 == laurent(0,1)
monomial(s,e) == laurent(e,s::UTS)
coerce(uts:UTS):% == laurent(0,uts)
coerce(r:Coef):% == r :: UTS :: %
coerce(i:I):% == i :: Coef :: %
taylorIfCan uls ==
n := getExpon uls
negative? n =>
uls := removeZeroes(-n,uls)
negative? getExpon(uls) => "failed"
getUTS uls
n = 0 => getUTS uls
getUTS(uls) * monom(1,n :: NNI)
taylor uls ==
(uts := taylorIfCan uls) case "failed" =>
error "taylor: Laurent series has a pole"
uts :: UTS
termExpon: TERM -> I
termExpon term == term.k
termCoef: TERM -> Coef
termCoef term == term.c
rec: (I,Coef) -> TERM
rec(exponent,coef) == [exponent,coef]
recs: (ST,I) -> Stream TERM
recs(st,n) == delay
empty? st => empty()
zero? (coef := frst st) => recs(rst st,n + 1)
concat(rec(n,coef),recs(rst st,n + 1))
terms x == recs(coefficients getUTS x,getExpon x)
recsToCoefs: (Stream TERM,I) -> ST
recsToCoefs(st,n) == delay
empty? st => empty()
term := frst st; ex := termExpon term
n = ex => concat(termCoef term,recsToCoefs(rst st,n + 1))
concat(0,recsToCoefs(rst st,n + 1))
series st ==
empty? st => 0
ex := termExpon frst st
laurent(ex,series recsToCoefs(st,ex))
--% normalizations
removeZeroes x ==
empty? coefficients(xUTS := getUTS x) => 0
coefficient(xUTS,0) = 0 =>
removeZeroes laurent(getExpon(x) + 1,quoByVar xUTS)
x
removeZeroes(n,x) ==
n <= 0 => x
empty? coefficients(xUTS := getUTS x) => 0
coefficient(xUTS,0) = 0 =>
removeZeroes(n - 1,laurent(getExpon(x) + 1,quoByVar xUTS))
x
--% predicates
x = y ==
%peq(x,y)$Foreign(Builtin) => true
(expDiff := getExpon(x) - getExpon(y)) = 0 =>
getUTS(x) = getUTS(y)
abs(expDiff) > _$streamCount$Lisp => false
positive? expDiff =>
getUTS(x) * monom(1,expDiff :: NNI) = getUTS(y)
getUTS(y) * monom(1,(- expDiff) :: NNI) = getUTS(x)
pole? x ==
(n := degree x) >= 0 => false
x := removeZeroes(-n,x)
negative? degree x
--% arithmetic
x + y ==
n := getExpon(x) - getExpon(y)
n >= 0 =>
laurent(getExpon y,getUTS(y) + getUTS(x) * monom(1,n::NNI))
laurent(getExpon x,getUTS(x) + getUTS(y) * monom(1,(-n)::NNI))
x - y ==
n := getExpon(x) - getExpon(y)
n >= 0 =>
laurent(getExpon y,getUTS(x) * monom(1,n::NNI) - getUTS(y))
laurent(getExpon x,getUTS(x) - getUTS(y) * monom(1,(-n)::NNI))
x:% * y:% == laurent(getExpon x + getExpon y,getUTS x * getUTS y)
x:% ** n:NNI ==
zero? n =>
zero? x => error "0 ** 0 is undefined"
1
laurent(n * getExpon(x),getUTS(x) ** n)
recip x ==
x := removeZeroes(1000,x)
zero? coefficient(x,d := degree x) => "failed"
(uts := recip getUTS x) case "failed" => "failed"
laurent(-d,uts :: UTS)
elt(uls1:%,uls2:%) ==
(uts := taylorIfCan uls2) case "failed" =>
error "elt: second argument must have positive order"
uts2 := uts :: UTS
not zero? coefficient(uts2,0) =>
error "elt: second argument must have positive order"
if negative?(deg := getExpon uls1) then uls1 := removeZeroes(-deg,uls1)
negative?(deg := getExpon uls1) =>
(recipr := recip(uts2 :: %)) case "failed" =>
error "elt: second argument not invertible"
uts1 := taylor(uls1 * monomial(1,-deg))
(elt(uts1,uts2) :: %) * (recipr :: %) ** ((-deg) :: NNI)
elt(taylor uls1,uts2) :: %
eval(uls:%,r:Coef) ==
if negative?(n := getExpon uls) then uls := removeZeroes(-n,uls)
uts := getUTS uls
negative?(n := getExpon uls) =>
zero? r => error "eval: 0 raised to negative power"
(recipr := recip r) case "failed" =>
error "eval: non-unit raised to negative power"
(recipr :: Coef) ** ((-n) :: NNI) *$STTAYLOR eval(uts,r)
zero? n => eval(uts,r)
r ** (n :: NNI) *$STTAYLOR eval(uts,r)
--% values
variable x == variable getUTS x
center x == center getUTS x
coefficient(x,n) ==
a := n - getExpon(x)
a >= 0 => coefficient(getUTS x,a :: NNI)
0
elt(x:%,n:I) == coefficient(x,n)
--% other functions
order x == getExpon x + order getUTS x
order(x,n) ==
negative?(m := n - (e := getExpon x)) => n
e + order(getUTS x,m :: NNI)
truncate(x,n) ==
negative?(m := n - (e := getExpon x)) => 0
laurent(e,truncate(getUTS x,m :: NNI))
truncate(x,n1,n2) ==
if n2 < n1 then (n1,n2) := (n2,n1)
negative?(m1 := n1 - (e := getExpon x)) => truncate(x,n2)
laurent(e,truncate(getUTS x,m1 :: NNI,(n2 - e) :: NNI))
if Coef has IntegralDomain then
rationalFunction(x,n) ==
negative?(m := n - (e := getExpon x)) => 0
poly := polynomial(getUTS x,m :: NNI) :: RF
zero? e => poly
v := variable(x) :: RF; c := center(x) :: P :: RF
positive? e => poly * (v - c) ** (e :: NNI)
poly / (v - c) ** ((-e) :: NNI)
rationalFunction(x,n1,n2) ==
if n2 < n1 then (n1,n2) := (n2,n1)
negative?(m1 := n1 - (e := getExpon x)) => rationalFunction(x,n2)
poly := polynomial(getUTS x,m1 :: NNI,(n2 - e) :: NNI) :: RF
zero? e => poly
v := variable(x) :: RF; c := center(x) :: P :: RF
positive? e => poly * (v - c) ** (e :: NNI)
poly / (v - c) ** ((-e) :: NNI)
-- La fonction < exquo > manque dans laurent.spad,
--les lignes suivantes le mettent en evidence :
--
--ls := laurent(0,series [i for i in 1..])$ULS(INT,x,0)
---- missing function in laurent.spad of Axiom 2.0a version of
---- Friday March 10, 1995 at 04:15:22 on 615:
--exquo(ls,ls)
--
-- Je l'ai ajoutee a laurent.spad.
--
--Frederic Lehobey
x exquo y ==
x := removeZeroes(1000,x)
y := removeZeroes(1000,y)
zero? coefficient(y, d := degree y) => "failed"
(uts := (getUTS x) exquo (getUTS y)) case "failed" => "failed"
laurent(degree x-d,uts :: UTS)
if Coef has coerce: Symbol -> Coef then
if Coef has "**": (Coef,I) -> Coef then
approximate(x,n) ==
negative?(m := n - (e := getExpon x)) => 0
app := approximate(getUTS x,m :: NNI)
zero? e => app
app * ((variable(x) :: Coef) - center(x)) ** e
complete x == laurent(getExpon x,complete getUTS x)
extend(x,n) ==
e := getExpon x
negative?(m := n - e) => x
laurent(e,extend(getUTS x,m :: NNI))
map(f:Coef -> Coef,x:%) == laurent(getExpon x,map(f,getUTS x))
multiplyCoefficients(f,x) ==
e := getExpon x
laurent(e,multiplyCoefficients(f(e + #1),getUTS x))
multiplyExponents(x,n) ==
laurent(n * getExpon x,multiplyExponents(getUTS x,n))
differentiate x ==
e := getExpon x
laurent(e - 1,multiplyCoefficients((e + #1) :: Coef,getUTS x))
if Coef has PartialDifferentialRing(Symbol) then
differentiate(x:%,s:Symbol) ==
(s = variable(x)) => differentiate x
map(differentiate(#1,s),x) - differentiate(center x,s)*differentiate(x)
characteristic == characteristic$Coef
if Coef has Field then
retract(x:%):UTS == taylor x
retractIfCan(x:%):Union(UTS,"failed") == taylorIfCan x
(x:%) ** (n:I) ==
zero? n =>
zero? x => error "0 ** 0 is undefined"
1
positive? n => laurent(n * getExpon(x),getUTS(x) ** (n :: NNI))
xInv := inv x; minusN := (-n) :: NNI
laurent(minusN * getExpon(xInv),getUTS(xInv) ** minusN)
(x:UTS) * (y:%) == (x :: %) * y
(x:%) * (y:UTS) == x * (y :: %)
inv x ==
(xInv := recip x) case "failed" =>
error "multiplicative inverse does not exist"
xInv :: %
(x:%) / (y:%) ==
(yInv := recip y) case "failed" =>
error "inv: multiplicative inverse does not exist"
x * (yInv :: %)
(x:UTS) / (y:UTS) == (x :: %) / (y :: %)
numer x ==
(n := degree x) >= 0 => taylor x
x := removeZeroes(-n,x)
(n := degree x) = 0 => taylor x
getUTS x
denom x ==
(n := degree x) >= 0 => 1
x := removeZeroes(-n,x)
(n := degree x) = 0 => 1
monom(1,(-n) :: NNI)
--% algebraic and transcendental functions
if Coef has Algebra Fraction Integer then
coerce(r:RN) == r :: Coef :: %
if Coef has Field then
(x:%) ** (r:RN) == x **$EFULS r
exp x == exp(x)$EFULS
log x == log(x)$EFULS
sin x == sin(x)$EFULS
cos x == cos(x)$EFULS
tan x == tan(x)$EFULS
cot x == cot(x)$EFULS
sec x == sec(x)$EFULS
csc x == csc(x)$EFULS
asin x == asin(x)$EFULS
acos x == acos(x)$EFULS
atan x == atan(x)$EFULS
acot x == acot(x)$EFULS
asec x == asec(x)$EFULS
acsc x == acsc(x)$EFULS
sinh x == sinh(x)$EFULS
cosh x == cosh(x)$EFULS
tanh x == tanh(x)$EFULS
coth x == coth(x)$EFULS
sech x == sech(x)$EFULS
csch x == csch(x)$EFULS
asinh x == asinh(x)$EFULS
acosh x == acosh(x)$EFULS
atanh x == atanh(x)$EFULS
acoth x == acoth(x)$EFULS
asech x == asech(x)$EFULS
acsch x == acsch(x)$EFULS
ratInv: I -> Coef
ratInv n ==
zero? n => 1
inv(n :: RN) :: Coef
integrate x ==
not zero? coefficient(x,-1) =>
error "integrate: series has term of order -1"
e := getExpon x
laurent(e + 1,multiplyCoefficients(ratInv(e + 1 + #1),getUTS x))
if Coef has integrate: (Coef,Symbol) -> Coef and _
Coef has variables: Coef -> List Symbol then
integrate(x:%,s:Symbol) ==
(s = variable(x)) => integrate x
not entry?(s,variables center x) => map(integrate(#1,s),x)
error "integrate: center is a function of variable of integration"
if Coef has TranscendentalFunctionCategory and _
Coef has PrimitiveFunctionCategory and _
Coef has AlgebraicallyClosedFunctionSpace Integer then
integrateWithOneAnswer: (Coef,Symbol) -> Coef
integrateWithOneAnswer(f,s) ==
res := integrate(f,s)$FunctionSpaceIntegration(I,Coef)
res case Coef => res :: Coef
first(res :: List Coef)
integrate(x:%,s:Symbol) ==
(s = variable(x)) => integrate x
not entry?(s,variables center x) =>
map(integrateWithOneAnswer(#1,s),x)
error "integrate: center is a function of variable of integration"
termOutput:(I,Coef,OUT) -> OUT
termOutput(k,c,vv) ==
-- creates a term c * vv ** k
k = 0 => c :: OUT
mon :=
k = 1 => vv
vv ** (k :: OUT)
c = 1 => mon
c = -1 => -mon
(c :: OUT) * mon
showAll?:() -> Boolean
-- check a global Lisp variable
showAll?() == true
termsToOutputForm:(I,ST,OUT) -> OUT
termsToOutputForm(m,uu,xxx) ==
l : L OUT := empty()
empty? uu => (0$Coef) :: OUT
count : NNI := _$streamCount$Lisp
n : NNI := 0
while n <= count and not empty? uu repeat
if frst(uu) ~= 0 then
l := concat(termOutput((n :: I) + m,frst(uu),xxx),l)
uu := rst uu
n := n + 1
if showAll?() then
n := count + 1
while explicitEntries? uu and _
not eq?(uu,rst uu) repeat
if frst(uu) ~= 0 then
l := concat(termOutput((n::I) + m,frst(uu),xxx),l)
uu := rst uu
n := n + 1
l :=
explicitlyEmpty? uu => l
eq?(uu,rst uu) and frst uu = 0 => l
concat(prefix("O" :: OUT,[xxx ** ((n :: I) + m) :: OUT]),l)
empty? l => (0$Coef) :: OUT
reduce("+",reverse! l)
coerce(x:%):OUT ==
x := removeZeroes(_$streamCount$Lisp,x)
m := degree x
uts := getUTS x
p := coefficients uts
var := variable uts; cen := center uts
xxx :=
zero? cen => var :: OUT
paren(var :: OUT - cen :: OUT)
termsToOutputForm(m,p,xxx)
)abbrev domain ULS UnivariateLaurentSeries
++ Author: Clifton J. Williamson
++ Date Created: 18 January 1990
++ Date Last Updated: 21 September 1993
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords: series, Laurent
++ Examples:
++ References:
++ Description: Dense Laurent series in one variable
++ \spadtype{UnivariateLaurentSeries} is a domain representing Laurent
++ series in one variable with coefficients in an arbitrary ring. The
++ parameters of the type specify the coefficient ring, the power series
++ variable, and the center of the power series expansion. For example,
++ \spad{UnivariateLaurentSeries(Integer,x,3)} represents Laurent series in
++ \spad{(x - 3)} with integer coefficients.
UnivariateLaurentSeries(Coef,var,cen): Exports == Implementation where
Coef : Ring
var : Symbol
cen : Coef
I ==> Integer
UTS ==> UnivariateTaylorSeries(Coef,var,cen)
Exports ==> Join(UnivariateLaurentSeriesConstructorCategory(Coef,UTS),_
PartialDifferentialDomain(%,Variable var)) with
coerce: Variable(var) -> %
++ \spad{coerce(var)} converts the series variable \spad{var} into a
++ Laurent series.
if Coef has Algebra Fraction Integer then
integrate: (%,Variable(var)) -> %
++ \spad{integrate(f(x))} returns an anti-derivative of the power
++ series \spad{f(x)} with constant coefficient 0.
++ We may integrate a series when we can divide coefficients
++ by integers.
Implementation ==> UnivariateLaurentSeriesConstructor(Coef,UTS) add
variable x == var
center x == cen
coerce(v:Variable(var)) ==
zero? cen => monomial(1,1)
monomial(1,1) + monomial(cen,0)
differentiate(x:%,v:Variable(var)) == differentiate x
if Coef has Algebra Fraction Integer then
integrate(x:%,v:Variable(var)) == integrate x
)abbrev package ULS2 UnivariateLaurentSeriesFunctions2
++ Author: Clifton J. Williamson
++ Date Created: 5 March 1990
++ Date Last Updated: 5 March 1990
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords: Laurent series, map
++ Examples:
++ References:
++ Description: Mapping package for univariate Laurent series
++ This package allows one to apply a function to the coefficients of
++ a univariate Laurent series.
UnivariateLaurentSeriesFunctions2(Coef1,Coef2,var1,var2,cen1,cen2):_
Exports == Implementation where
Coef1 : Ring
Coef2 : Ring
var1: Symbol
var2: Symbol
cen1: Coef1
cen2: Coef2
ULS1 ==> UnivariateLaurentSeries(Coef1, var1, cen1)
ULS2 ==> UnivariateLaurentSeries(Coef2, var2, cen2)
UTS1 ==> UnivariateTaylorSeries(Coef1, var1, cen1)
UTS2 ==> UnivariateTaylorSeries(Coef2, var2, cen2)
UTSF2 ==> UnivariateTaylorSeriesFunctions2(Coef1, Coef2, UTS1, UTS2)
Exports ==> with
map: (Coef1 -> Coef2,ULS1) -> ULS2
++ \spad{map(f,g(x))} applies the map f to the coefficients of the Laurent
++ series \spad{g(x)}.
Implementation ==> add
map(f,ups) == laurent(degree ups, map(f, taylorRep ups)$UTSF2)
|