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)abbrev domain ITAYLOR InnerTaylorSeries
++ Author: Clifton J. Williamson
++ Date Created: 21 December 1989
++ Date Last Updated: October 26, 2009
++ Basic Operations:
++ Related Domains: UnivariateTaylorSeries(Coef,var,cen)
++ Also See:
++ AMS Classifications:
++ Keywords: stream, dense Taylor series
++ Examples:
++ References:
++ Description: Internal package for dense Taylor series.
++ This is an internal Taylor series type in which Taylor series
++ are represented by a \spadtype{Stream} of \spadtype{Ring} elements.
++ For univariate series, the \spad{Stream} elements are the Taylor
++ coefficients. For multivariate series, the \spad{n}th Stream element
++ is a form of degree n in the power series variables.
InnerTaylorSeries(Coef): Exports == Implementation where
Coef : Ring
I ==> Integer
NNI ==> NonNegativeInteger
ST ==> Stream Coef
STT ==> StreamTaylorSeriesOperations Coef
Exports == Join(Ring, BiModule(Coef,Coef)) with
coefficients: % -> Stream Coef
++\spad{coefficients(x)} returns a stream of ring elements.
++ When x is a univariate series, this is a stream of Taylor
++ coefficients. When x is a multivariate series, the
++ \spad{n}th element of the stream is a form of
++ degree n in the power series variables.
series: Stream Coef -> %
++\spad{series(s)} creates a power series from a stream of
++ ring elements.
++ For univariate series types, the stream s should be a stream
++ of Taylor coefficients. For multivariate series types, the
++ stream s should be a stream of forms the \spad{n}th element
++ of which is a
++ form of degree n in the power series variables.
pole?: % -> Boolean
++\spad{pole?(x)} tests if the series x has a pole.
++ Note: this is false when x is a Taylor series.
order: % -> NNI
++\spad{order(x)} returns the order of a power series x,
++ i.e. the degree of the first non-zero term of the series.
order: (%,NNI) -> NNI
++\spad{order(x,n)} returns the minimum of n and the order of x.
* : (%,Integer)->%
++\spad{x*i} returns the product of integer i and the series x.
if Coef has IntegralDomain then IntegralDomain
--++ An IntegralDomain provides 'exquo'
Implementation == add
Rep == ST
-- In what follows, we will be calling operations on Streams
-- which are NOT defined in the package Stream. Thus, it is
-- necessary to explicitly pass back and forth between Rep and %.
series st == per st
0 == per coerce(0)$STT
1 == per coerce(1)$STT
x = y ==
-- tests if two power series are equal
-- difference must be a finite stream of zeroes of length <= n + 1,
-- where n = $streamCount$Lisp
st : ST := rep(x - y)
n : I := _$streamCount$Lisp
for i in 0..n repeat
empty? st => return true
frst st ~= 0 => return false
st := rst st
empty? st
coefficients x == rep x
x + y == per(rep x +$STT rep y)
x - y == per(rep x -$STT rep y)
(x:%) * (y:%) == per(rep x *$STT rep y)
- x == per(-$STT rep x)
(i:I) * (x:%) == per((i::Coef) *$STT rep x)
(x:%) * (i:I) == per(rep x *$STT (i::Coef))
(c:Coef) * (x:%) == per(c *$STT rep x)
(x:%) * (c:Coef) == per(rep x *$STT c)
recip x ==
(rec := recip$STT rep x) case "failed" => "failed"
per(rec :: ST)
if Coef has IntegralDomain then
x exquo y ==
(quot := rep x exquo$STT rep y) case "failed" => "failed"
per(quot :: ST)
x:% ** n:NNI ==
n = 0 => 1
expt(x,n :: PositiveInteger)$RepeatedSquaring(%)
characteristic == characteristic$Coef
pole? x == false
iOrder: (ST,NNI,NNI) -> NNI
iOrder(st,n,n0) ==
(n = n0) or (empty? st) => n0
zero? frst st => iOrder(rst st,n + 1,n0)
n
order(x,n) == iOrder(rep x,0,n)
iOrder2: (ST,NNI) -> NNI
iOrder2(st,n) ==
empty? st => error "order: series has infinite order"
zero? frst st => iOrder2(rst st,n + 1)
n
order x == iOrder2(rep x,0)
zero? x == empty? rep x
)abbrev domain UTS UnivariateTaylorSeries
++ Author: Clifton J. Williamson
++ Date Created: 21 December 1989
++ Date Last Updated: June 18, 2010
++ Basic Operations:
++ Related Domains: UnivariateLaurentSeries(Coef,var,cen), UnivariatePuiseuxSeries(Coef,var,cen)
++ Also See:
++ AMS Classifications:
++ Keywords: dense, Taylor series
++ Examples:
++ References:
++ Description: Dense Taylor series in one variable
++ \spadtype{UnivariateTaylorSeries} is a domain representing Taylor
++ 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,
++ \spadtype{UnivariateTaylorSeries}(Integer,x,3) represents
++ Taylor series in
++ \spad{(x - 3)} with \spadtype{Integer} coefficients.
UnivariateTaylorSeries(Coef,var,cen): Exports == Implementation where
Coef : Ring
var : Symbol
cen : Coef
I ==> Integer
NNI ==> NonNegativeInteger
P ==> Polynomial Coef
RN ==> Fraction Integer
ST ==> Stream
STT ==> StreamTaylorSeriesOperations Coef
TERM ==> Record(k:NNI,c:Coef)
UP ==> UnivariatePolynomial(var,Coef)
Exports ==> Join(UnivariateTaylorSeriesCategory(Coef),_
PartialDifferentialDomain(%,Variable var)) with
coerce: UP -> %
++\spad{coerce(p)} converts a univariate polynomial p in the variable
++\spad{var} to a univariate Taylor series in \spad{var}.
univariatePolynomial: (%,NNI) -> UP
++\spad{univariatePolynomial(f,k)} returns a univariate polynomial
++ consisting of the sum of all terms of f of degree \spad{<= k}.
coerce: Variable(var) -> %
++\spad{coerce(var)} converts the series variable \spad{var} into a
++ Taylor series.
lagrange: % -> %
++\spad{lagrange(g(x))} produces the Taylor series for \spad{f(x)}
++ where \spad{f(x)} is implicitly defined as \spad{f(x) = x*g(f(x))}.
lambert: % -> %
++\spad{lambert(f(x))} returns \spad{f(x) + f(x^2) + f(x^3) + ...}.
++ This function is used for computing infinite products.
++ \spad{f(x)} should have zero constant coefficient.
++ If \spad{f(x)} is a Taylor series with constant term 1, then
++ \spad{product(n = 1..infinity,f(x^n)) = exp(log(lambert(f(x))))}.
oddlambert: % -> %
++\spad{oddlambert(f(x))} returns \spad{f(x) + f(x^3) + f(x^5) + ...}.
++ \spad{f(x)} should have a zero constant coefficient.
++ This function is used for computing infinite products.
++ If \spad{f(x)} is a Taylor series with constant term 1, then
++ \spad{product(n=1..infinity,f(x^(2*n-1)))=exp(log(oddlambert(f(x))))}.
evenlambert: % -> %
++\spad{evenlambert(f(x))} returns \spad{f(x^2) + f(x^4) + f(x^6) + ...}.
++ \spad{f(x)} should have a zero constant coefficient.
++ This function is used for computing infinite products.
++ If \spad{f(x)} is a Taylor series with constant term 1, then
++ \spad{product(n=1..infinity,f(x^(2*n))) = exp(log(evenlambert(f(x))))}.
generalLambert: (%,I,I) -> %
++\spad{generalLambert(f(x),a,d)} returns \spad{f(x^a) + f(x^(a + d)) +
++ f(x^(a + 2 d)) + ... }. \spad{f(x)} should have zero constant
++ coefficient and \spad{a} and d should be positive.
revert: % -> %
++ \spad{revert(f(x))} returns a Taylor series \spad{g(x)} such that
++ \spad{f(g(x)) = g(f(x)) = x}. Series \spad{f(x)} should have constant
++ coefficient 0 and invertible 1st order coefficient.
multisect: (I,I,%) -> %
++\spad{multisect(a,b,f(x))} selects the coefficients of
++ \spad{x^((a+b)*n+a)}, and changes this monomial to \spad{x^n}.
invmultisect: (I,I,%) -> %
++\spad{invmultisect(a,b,f(x))} substitutes \spad{x^((a+b)*n)}
++ for \spad{x^n} and multiples by \spad{x^b}.
if Coef has Algebra Fraction Integer then
integrate: (%,Variable(var)) -> %
++ \spad{integrate(f(x),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 ==> InnerTaylorSeries(Coef) add
Rep := Stream Coef
--% creation and destruction of series
stream: % -> Stream Coef
stream x == x pretend Stream(Coef)
coerce(v:Variable(var)) ==
zero? cen => monomial(1,1)
monomial(1,1) + monomial(cen,0)
coerce(n:I) == n :: Coef :: %
coerce(r:Coef) == coerce(r)$STT
monomial(c,n) == monom(c,n)$STT
getExpon: TERM -> NNI
getExpon term == term.k
getCoef: TERM -> Coef
getCoef term == term.c
rec: (NNI,Coef) -> TERM
rec(expon,coef) == [expon,coef]
recs: (ST Coef,NNI) -> ST TERM
recs(st,n) == delay$ST(TERM)
empty? st => empty()
zero? (coef := frst st) => recs(rst st,n + 1)
concat(rec(n,coef),recs(rst st,n + 1))
terms x == recs(stream x,0)
recsToCoefs: (ST TERM,NNI) -> ST Coef
recsToCoefs(st,n) == delay
empty? st => empty()
term := frst st; expon := getExpon term
n = expon => concat(getCoef term,recsToCoefs(rst st,n + 1))
concat(0,recsToCoefs(st,n + 1))
series(st: ST TERM) == recsToCoefs(st,0)
stToPoly: (ST Coef,P,NNI,NNI) -> P
stToPoly(st,term,n,n0) ==
(n > n0) or (empty? st) => 0
frst(st) * term ** n + stToPoly(rst st,term,n + 1,n0)
polynomial(x,n) == stToPoly(stream x,(var :: P) - (cen :: P),0,n)
polynomial(x,n1,n2) ==
if n1 > n2 then (n1,n2) := (n2,n1)
stToPoly(rest(stream x,n1),(var :: P) - (cen :: P),n1,n2)
stToUPoly: (ST Coef,UP,NNI,NNI) -> UP
stToUPoly(st,term,n,n0) ==
(n > n0) or (empty? st) => 0
frst(st) * term ** n + stToUPoly(rst st,term,n + 1,n0)
univariatePolynomial(x,n) ==
stToUPoly(stream x,monomial(1,1)$UP - monomial(cen,0)$UP,0,n)
coerce(p:UP) ==
zero? p => 0
if not zero? cen then
p := p(monomial(1,1)$UP + monomial(cen,0)$UP)
st : ST Coef := empty()
oldDeg : NNI := degree(p) + 1
while not zero? p repeat
deg := degree p
delta := (oldDeg - deg - 1) :: NNI
for i in 1..delta repeat st := concat(0$Coef,st)
st := concat(leadingCoefficient p,st)
oldDeg := deg; p := reductum p
for i in 1..oldDeg repeat st := concat(0$Coef,st)
st
if Coef has coerce: Symbol -> Coef then
if Coef has "**": (Coef,NNI) -> Coef then
stToCoef: (ST Coef,Coef,NNI,NNI) -> Coef
stToCoef(st,term,n,n0) ==
(n > n0) or (empty? st) => 0
frst(st) * term ** n + stToCoef(rst st,term,n + 1,n0)
approximate(x,n) ==
stToCoef(stream x,(var :: Coef) - cen,0,n)
--% values
variable x == var
center s == cen
coefficient(x,n) ==
-- Cannot use elt! Should return 0 if stream doesn't have it.
u := stream x
while not empty? u and positive? n repeat
u := rst u
n := (n - 1) :: NNI
empty? u or n ~= 0 => 0
frst u
elt(x:%,n:NNI) == coefficient(x,n)
--% functions
map(f,x) == map(f,x)$Rep
eval(x:%,r:Coef) == eval(stream x,r-cen)$STT
differentiate x == deriv(stream x)$STT
differentiate(x:%,v:Variable(var)) == differentiate 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)
multiplyCoefficients(f,x) == gderiv(f,stream x)$STT
lagrange x == lagrange(stream x)$STT
lambert x == lambert(stream x)$STT
oddlambert x == oddlambert(stream x)$STT
evenlambert x == evenlambert(stream x)$STT
generalLambert(x:%,a:I,d:I) == generalLambert(stream x,a,d)$STT
extend(x,n) == extend(x,n+1)$Rep
complete x == complete(x)$Rep
truncate(x,n) == first(stream x,n + 1)$Rep
truncate(x,n1,n2) ==
if n2 < n1 then (n1,n2) := (n2,n1)
m := (n2 - n1) :: NNI
st := first(rest(stream x,n1)$Rep,m + 1)$Rep
for i in 1..n1 repeat st := concat(0$Coef,st)
st
elt(x:%,y:%) == compose(stream x,stream y)$STT
revert x == revert(stream x)$STT
multisect(a,b,x) == multisect(a,b,stream x)$STT
invmultisect(a,b,x) == invmultisect(a,b,stream x)$STT
multiplyExponents(x,n) == invmultisect(n,0,x)
quoByVar x == (empty? x => 0; rst x)
if Coef has IntegralDomain then
unit? x == unit? coefficient(x,0)
if Coef has Field then
if Coef is RN then
(x:%) ** (s:Coef) == powern(s,stream x)$STT
else
(x:%) ** (s:Coef) == power(s,stream x)$STT
if Coef has Algebra Fraction Integer then
coerce(r:RN) == r :: Coef :: %
integrate x == integrate(0,stream x)$STT
integrate(x:%,v:Variable(var)) == integrate x
if Coef has integrate: (Coef,Symbol) -> Coef and _
Coef has variables: Coef -> List Symbol then
integrate(x:%,s:Symbol) ==
(s = variable(x)) => integrate x
not entry?(s,variables center x) => map(integrate(#1,s),x)
error "integrate: center is a function of variable of integration"
if Coef has TranscendentalFunctionCategory and _
Coef has PrimitiveFunctionCategory and _
Coef has AlgebraicallyClosedFunctionSpace Integer then
integrateWithOneAnswer: (Coef,Symbol) -> Coef
integrateWithOneAnswer(f,s) ==
res := integrate(f,s)$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"
--% OutputForms
-- We use the default coerce: % -> OutputForm in UTSCAT&
)abbrev package UTS2 UnivariateTaylorSeriesFunctions2
++ Author: Clifton J. Williamson
++ Date Created: 9 February 1990
++ Date Last Updated: 9 February 1990
++ Basic Operations:
++ Related Domains: UnivariateTaylorSeries(Coef1,var,cen)
++ Also See:
++ AMS Classifications:
++ Keywords: Taylor series, map
++ Examples:
++ References:
++ Description: Mapping package for univariate Taylor series.
++ This package allows one to apply a function to the coefficients of
++ a univariate Taylor series.
UnivariateTaylorSeriesFunctions2(Coef1,Coef2,UTS1,UTS2):_
Exports == Implementation where
Coef1 : Ring
Coef2 : Ring
UTS1 : UnivariateTaylorSeriesCategory Coef1
UTS2 : UnivariateTaylorSeriesCategory Coef2
ST2 ==> StreamFunctions2(Coef1,Coef2)
Exports ==> with
map: (Coef1 -> Coef2,UTS1) -> UTS2
++\spad{map(f,g(x))} applies the map f to the coefficients of
++ the Taylor series \spad{g(x)}.
Implementation ==> add
map(f,uts) == series map(f,coefficients uts)$ST2
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