)abbrev category UPXSCCA UnivariatePuiseuxSeriesConstructorCategory
++ Author: Clifton J. Williamson
++ Date Created: 6 February 1990
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords: series, Puiseux, Laurent
++ Examples:
++ References:
++ Description:
++   This is a category of univariate Puiseux series constructed
++   from univariate Laurent series.  A Puiseux series is represented
++   by a pair \spad{[r, f(x)]}, where r is a positive rational number and
++   \spad{f(x)} is a Laurent series.  This pair represents the Puiseux
++   series \spad{f(x^r)}.
UnivariatePuiseuxSeriesConstructorCategory(Coef, ULS) : _
 Category == Definition where
  Coef : Ring
  ULS  : UnivariateLaurentSeriesCategory Coef
  I  ==> Integer
  RN ==> Fraction Integer

  Definition ==> Join(UnivariatePuiseuxSeriesCategory(Coef), _
                      RetractableTo ULS) with
    puiseux : (RN, ULS) -> %
      ++ \spad{puiseux(r, f(x))} returns \spad{f(x^r)}.
    rationalPower : % -> RN
      ++ \spad{rationalPower(f(x))} returns r where the Puiseux series
      ++ \spad{f(x) = g(x^r)}.
    laurentRep  : % -> ULS
      ++ \spad{laurentRep(f(x))} returns \spad{g(x)} where the Puiseux series
      ++ \spad{f(x) = g(x^r)} is represented by \spad{[r, g(x)]}.
    degree : % -> RN
      ++ \spad{degree(f(x))} returns the degree of the leading term of the
      ++ Puiseux series \spad{f(x)}, which may have zero as a coefficient.
    coerce : ULS -> %
      ++ \spad{coerce(f(x))} converts the Laurent series \spad{f(x)} to a
      ++ Puiseux series.
    laurent : % -> ULS
      ++ \spad{laurent(f(x))} converts the Puiseux series \spad{f(x)} to a
      ++ Laurent series if possible. Error: if this is not possible.
    laurentIfCan : % -> Union(ULS,"failed")
      ++ \spad{laurentIfCan(f(x))} converts the Puiseux series \spad{f(x)}
      ++ to a Laurent series if possible.
      ++ If this is not possible, "failed" is returned.

   add

     zero? x == zero? laurentRep x
     retract(x : %) : ULS == laurent x
     retractIfCan(x:%):Union(ULS,"failed") == laurentIfCan x

)abbrev domain UPXSCONS UnivariatePuiseuxSeriesConstructor
++ Author: Clifton J. Williamson
++ Date Created: 9 May 1989
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords: series, Puiseux, Laurent
++ Examples:
++ References:
++ Description:
++   This package enables one to construct a univariate Puiseux series
++   domain from a univariate Laurent series domain. Univariate
++   Puiseux series are represented by a pair \spad{[r, f(x)]}, where r is
++   a positive rational number and \spad{f(x)} is a Laurent series.
++   This pair represents the Puiseux series \spad{f(x^r)}.

UnivariatePuiseuxSeriesConstructor(Coef, ULS) : _
 Exports == Implementation where
  Coef  : Ring
  ULS   : UnivariateLaurentSeriesCategory Coef
  I     ==> Integer
  L     ==> List
  NNI   ==> NonNegativeInteger
  OUT   ==> OutputForm
  PI    ==> PositiveInteger
  RN    ==> Fraction Integer
  ST    ==> Stream Coef
  LTerm ==> Record(k : I, c : Coef)
  PTerm ==> Record(k : RN, c : Coef)
  ST2LP ==> StreamFunctions2(LTerm, PTerm)
  ST2PL ==> StreamFunctions2(PTerm, LTerm)

  Exports ==> UnivariatePuiseuxSeriesConstructorCategory(Coef, ULS)

  Implementation ==> add

--% representation

    Rep := Record(expon : RN, lSeries : ULS)

    getExpon : % -> RN
    getULS  : % -> ULS

    getExpon pxs == pxs.expon
    getULS   pxs == pxs.lSeries

--% creation and destruction

    puiseux(n, ls)   == [n, ls]
    laurentRep x    == getULS x
    rationalPower x == getExpon x
    degree x        == getExpon(x) * degree(getULS(x))

    0 == puiseux(1, 0)
    1 == puiseux(1, 1)

    monomial(c, k) ==
      k = 0 => c :: %
      k < 0 => puiseux(-k, monomial(c, -1))
      puiseux(k, monomial(c, 1))

    coerce(ls : ULS) == puiseux(1, ls)
    coerce(r : Coef) == r :: ULS  :: %
    coerce(i : I)    == i :: Coef :: %

    laurentIfCan upxs ==
      r := getExpon upxs
      (denom r) = 1 =>
        multiplyExponents(getULS upxs, numer(r) :: PI)
      "failed"

    laurent upxs ==
      (uls := laurentIfCan upxs) case "failed" =>
        error "laurent: Puiseux series has fractional powers"
      uls :: ULS

    multExp : (RN, LTerm) -> PTerm
    multExp(r, lTerm) == [r * lTerm.k, lTerm.c]

    terms upxs == map((t1 : LTerm) : PTerm +-> multExp(getExpon upxs, t1), terms getULS upxs)$ST2LP

    clearDen : (I, PTerm) -> LTerm
    clearDen(n, lTerm) ==
      (int := retractIfCan(n * lTerm.k)@Union(I,"failed")) case "failed" =>
        error "series: inappropriate denominator"
      [int :: I, lTerm.c]

    series(n, stream) ==
      str := map((t1 : PTerm) : LTerm +-> clearDen(n, t1), stream)$ST2PL
      puiseux(1/n, series str)

--% normalizations

    rewrite : (%, PI) -> %
    rewrite(upxs, m) ==
      -- rewrites a series in x^r as a series in x^(r/m)
      puiseux((getExpon upxs)*(1/m), multiplyExponents(getULS upxs, m))

    ratGcd : (RN, RN) -> RN
    ratGcd(r1, r2) ==
      -- if r1 = prod(p prime, p ^ ep(r1)) and
      -- if r2 = prod(p prime, p ^ ep(r2)), then
      -- ratGcd(r1, r2) = prod(p prime, p ^ min(ep(r1), ep(r2)))
      gcd(numer r1, numer r2) / lcm(denom r1, denom r2)

    withNewExpon : (%, RN) -> %
    withNewExpon(upxs, r) ==
      rewrite(upxs, qcoerce(numer(getExpon(upxs)/r)))

--% predicates

    upxs1 = upxs2 ==
      r1 := getExpon upxs1; r2 := getExpon upxs2
      ls1 := getULS upxs1; ls2 := getULS upxs2
      (r1 = r2) => (ls1 = ls2)
      r := ratGcd(r1, r2)
      m1 := qcoerce(numer(getExpon(upxs1)/r))@PI
      m2 := qcoerce(numer(getExpon(upxs2)/r))@PI
      multiplyExponents(ls1, m1) = multiplyExponents(ls2, m2)

    pole? upxs == pole? getULS upxs

--% arithmetic

    applyFcn : ((ULS, ULS) -> ULS, %, %) -> %
    applyFcn(op, pxs1, pxs2) ==
      r1 := getExpon pxs1; r2 := getExpon pxs2
      ls1 := getULS pxs1; ls2 := getULS pxs2
      (r1 = r2) => puiseux(r1, op(ls1, ls2))
      r := ratGcd(r1, r2)
      m1 := qcoerce(numer(getExpon(pxs1)/r))@PI
      m2 := qcoerce(numer(getExpon(pxs2)/r))@PI
      puiseux(r, op(multiplyExponents(ls1, m1), multiplyExponents(ls2, m2)))

    pxs1 + pxs2     == applyFcn(_+$ULS, pxs1, pxs2)
    pxs1 - pxs2     == applyFcn(_-$ULS, pxs1, pxs2)
    pxs1 : % * pxs2 : % == applyFcn(_*$ULS, pxs1, pxs2)

    pxs : % ^ n : NNI == puiseux(getExpon pxs, getULS(pxs)^n)

    recip pxs ==
      rec := recip getULS pxs
      rec case "failed" => "failed"
      puiseux(getExpon pxs, rec :: ULS)

    RATALG : Boolean := Coef has Algebra(Fraction Integer)

    elt(upxs1 : %, upxs2 : %) ==
      uls1 := laurentRep upxs1; uls2 := laurentRep upxs2
      r1 := rationalPower upxs1; r2 := rationalPower upxs2
      (n := retractIfCan(r1)@Union(Integer,"failed")) case Integer =>
        puiseux(r2, uls1(uls2 ^ r1))
      RATALG =>
        if zero? (coef := coefficient(uls2, deg := degree uls2)) then
          deg := order(uls2, deg + 1000)
          zero? (coef := coefficient(uls2, deg)) =>
            error "elt: series with many leading zero coefficients"
        -- a fractional power of a Laurent series may not be defined:
        -- if f(x) = c * x^n + ..., then f(x) ^ (p/q) will be defined
        -- only if q divides n
        b := lcm(denom r1, deg); c := b quo deg
        mon : ULS := monomial(1, c)
        uls2 := elt(uls2, mon) ^ r1
        puiseux(r2*(1/c), elt(uls1, uls2))
      error "elt: rational powers not available for this coefficient domain"

    if Coef has "^": (Coef,Integer) -> Coef and
       Coef has "^": (Coef, RN) -> Coef then
         eval(upxs : %, a : Coef) == eval(getULS upxs, a ^ getExpon(upxs))

    if Coef has Field then

      pxs1 : % / pxs2 : % == applyFcn((z1 : ULS, z2 : ULS) : ULS +-> z1 /$ULS z2, pxs1, pxs2)

      inv upxs ==
        (invUpxs := recip upxs) case "failed" =>
          error "inv: multiplicative inverse does not exist"
        invUpxs :: %

--% values

    variable upxs == variable getULS upxs
    center   upxs == center   getULS upxs

    coefficient(upxs, rn) ==
      (denom(n := rn / getExpon upxs)) = 1 =>
        coefficient(getULS upxs, numer n)
      0

    elt(upxs : %, rn : RN) == coefficient(upxs, rn)

--% other functions

    roundDown : RN -> I
    roundDown rn ==
      -- returns the largest integer <= rn
      (den := denom rn) = 1 => numer rn
      n := (num := numer rn) quo den
      positive?(num) => n
      n - 1

    roundUp : RN -> I
    roundUp rn ==
      -- returns the smallest integer >= rn
      (den := denom rn) = 1 => numer rn
      n := (num := numer rn) quo den
      positive?(num) => n + 1
      n

    order upxs == getExpon upxs * order getULS upxs
    order(upxs, r) ==
      e := getExpon upxs
      ord := order(getULS upxs, n := roundDown(r / e))
      ord = n =>
          coefficient(getULS upxs, n) = 0 => r
          ord*e
      ord * e

    truncate(upxs, r) ==
      e := getExpon upxs
      puiseux(e, truncate(getULS upxs, roundDown(r / e)))

    truncate(upxs, r1, r2) ==
      e := getExpon upxs
      puiseux(e, truncate(getULS upxs, roundUp(r1 / e), roundDown(r2 / e)))

    complete upxs == puiseux(getExpon upxs, complete getULS upxs)
    extend(upxs, r) ==
      e := getExpon upxs
      puiseux(e, extend(getULS upxs, roundDown(r / e)))

    map(fcn, upxs) == puiseux(getExpon upxs, map(fcn, getULS upxs))

    characteristic() == characteristic()$Coef

    -- multiplyCoefficients(f, upxs) ==
      -- r := getExpon upxs
      -- puiseux(r, multiplyCoefficients(f(#1 * r), getULS upxs))

    multiplyExponents(upxs : %, n : RN) ==
      puiseux(n * getExpon(upxs), getULS upxs)
    multiplyExponents(upxs : %, n : PI) ==
      puiseux(n * getExpon(upxs), getULS upxs)

    if Coef has "*": (Fraction Integer, Coef) -> Coef then

      differentiate upxs ==
        r := getExpon upxs
        puiseux(r, differentiate getULS upxs) * monomial(r :: Coef, r-1)

      if Coef has PartialDifferentialRing(Symbol) then

        differentiate(upxs : %, s : Symbol) ==
          (s = variable(upxs)) => differentiate upxs
          dcds := differentiate(center upxs, s)
          map((z1 : Coef) : Coef +-> differentiate(z1, s), upxs) - dcds*differentiate(upxs)

    if Coef has Algebra Fraction Integer then

      coerce(r : RN) == r :: Coef :: %

      ratInv : RN -> Coef
      ratInv r ==
        zero? r => 1
        inv(r) :: Coef

      integrate upxs ==
        not zero? coefficient(upxs, -1) =>
          error "integrate: series has term of order -1"
        r := getExpon upxs
        uls := getULS upxs
        uls := multiplyCoefficients((z1 : Integer) : Coef +-> ratInv(z1 * r + 1), uls)
        monomial(1, 1) * puiseux(r, uls)

      if Coef has integrate : (Coef, Symbol) -> Coef and _
         Coef has variables : Coef -> List Symbol then

        integrate(upxs : %, s : Symbol) ==
          (s = variable(upxs)) => integrate upxs
          not entry?(s, variables center upxs) => map((z1 : Coef) : Coef +-> integrate(z1, s), upxs)
          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(upxs : %, s : Symbol) ==
          (s = variable(upxs)) => integrate upxs
          not entry?(s, variables center upxs) =>
            map((z1 : Coef) : Coef +-> integrateWithOneAnswer(z1, s), upxs)
          error "integrate: center is a function of variable of integration"

      if Coef has Field then
         (upxs : %) ^ (q : RN) ==
           num := numer q; den := denom q
           den = 1 => upxs ^ num
           r := rationalPower upxs; uls := laurentRep upxs
           deg := degree uls
           if zero?(coef := coefficient(uls, deg)) then
             deg := order(uls, deg + 1000)
             zero?(coef := coefficient(uls, deg)) =>
               error "power of series with many leading zero coefficients"
           ulsPow := (uls * monomial(1, -deg)$ULS) ^ q
           puiseux(r, ulsPow) * monomial(1, deg*q*r)

      applyUnary : (ULS -> ULS, %) -> %
      applyUnary(fcn, upxs) ==
        puiseux(rationalPower upxs, fcn laurentRep upxs)

      exp upxs   == applyUnary(exp, upxs)
      log upxs   == applyUnary(log, upxs)
      sin upxs   == applyUnary(sin, upxs)
      cos upxs   == applyUnary(cos, upxs)
      tan upxs   == applyUnary(tan, upxs)
      cot upxs   == applyUnary(cot, upxs)
      sec upxs   == applyUnary(sec, upxs)
      csc upxs   == applyUnary(csc, upxs)
      asin upxs  == applyUnary(asin, upxs)
      acos upxs  == applyUnary(acos, upxs)
      atan upxs  == applyUnary(atan, upxs)
      acot upxs  == applyUnary(acot, upxs)
      asec upxs  == applyUnary(asec, upxs)
      acsc upxs  == applyUnary(acsc, upxs)
      sinh upxs  == applyUnary(sinh, upxs)
      cosh upxs  == applyUnary(cosh, upxs)
      tanh upxs  == applyUnary(tanh, upxs)
      coth upxs  == applyUnary(coth, upxs)
      sech upxs  == applyUnary(sech, upxs)
      csch upxs  == applyUnary(csch, upxs)
      asinh upxs == applyUnary(asinh, upxs)
      acosh upxs == applyUnary(acosh, upxs)
      atanh upxs == applyUnary(atanh, upxs)
      acoth upxs == applyUnary(acoth, upxs)
      asech upxs == applyUnary(asech, upxs)
      acsch upxs == applyUnary(acsch, upxs)

)abbrev domain UPXS UnivariatePuiseuxSeries
++ Author: Clifton J. Williamson
++ Date Created: 28 January 1990
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords: series, Puiseux
++ Examples:
++ References:
++ Description: Dense Puiseux series in one variable
++   \spadtype{UnivariatePuiseuxSeries} is a domain representing Puiseux
++   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{UnivariatePuiseuxSeries(Integer, x, 3)} represents Puiseux series in
++   \spad{(x - 3)} with \spadtype{Integer} coefficients.
UnivariatePuiseuxSeries(Coef, var, cen) : Exports == Implementation where
  Coef : Ring
  var  : Symbol
  cen  : Coef
  I   ==> Integer
  L   ==> List
  NNI ==> NonNegativeInteger
  OUT ==> OutputForm
  RN  ==> Fraction Integer
  ST  ==> Stream Coef
  UTS ==> UnivariateTaylorSeries(Coef, var, cen)
  ULS ==> UnivariateLaurentSeries(Coef, var, cen)

  Exports ==> Join(UnivariatePuiseuxSeriesConstructorCategory(Coef, ULS), _
                   RetractableTo UTS) with
    coerce : Variable(var) -> %
      ++ coerce(var) converts the series variable \spad{var} into a
      ++ Puiseux series.
    differentiate : (%, Variable(var)) -> %
      ++ \spad{differentiate(f(x), x)} returns the derivative of
      ++ \spad{f(x)} with respect to \spad{x}.
    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 ==> UnivariatePuiseuxSeriesConstructor(Coef, ULS) add

    Rep := Record(expon : RN, lSeries : ULS)

    getExpon : % -> RN
    getExpon pxs == pxs.expon

    variable upxs == var
    center   upxs == cen

    coerce(uts : UTS) == uts :: ULS :: %

    retractIfCan(upxs:%):Union(UTS,"failed") ==
      (ulsIfCan := retractIfCan(upxs)@Union(ULS,"failed")) case "failed" =>
        "failed"
      retractIfCan(ulsIfCan :: ULS)

    --retract(upxs: %): UTS ==
      --(ulsIfCan := retractIfCan(upxs)@Union(ULS,"failed")) case "failed" =>
        --error "retractIfCan: series has fractional exponents"
      --utsIfCan := retractIfCan(ulsIfCan :: ULS)@Union(UTS,"failed")
      --utsIfCan case "failed" =>
        --error "retractIfCan: series has negative exponents"
      --utsIfCan :: UTS

    coerce(v : Variable(var)) ==
      zero? cen => monomial(1, 1)
      monomial(1, 1) + monomial(cen, 0)

    if Coef has "*": (Fraction Integer, Coef) -> Coef then
      differentiate(upxs : %, v : Variable(var)) == differentiate upxs

    if Coef has Algebra Fraction Integer then
      integrate(upxs : %, v : Variable(var)) == integrate upxs

    if Coef has coerce : Symbol -> Coef then
      if Coef has "^": (Coef,RN) -> Coef then

        roundDown : RN -> I
        roundDown rn ==
          -- returns the largest integer <= rn
          (den := denom rn) = 1 => numer rn
          n := (num := numer rn) quo den
          positive?(num) => n
          n - 1

        stToCoef : (ST, 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)

        approximateLaurent : (ULS, Coef, I) -> Coef
        approximateLaurent(x, term, n) ==
          (m := n - (e := degree x)) < 0 => 0
          app := stToCoef(coefficients taylorRep x, term, 0, m :: NNI)
          zero? e => app
          app * term ^ (e :: RN)

        approximate(x, r) ==
          e := rationalPower(x)
          term := ((variable(x) :: Coef) - center(x)) ^ e
          approximateLaurent(laurentRep x, term, roundDown(r / e))

    termOutput : (RN, 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?() == _$streamsShowAll$Lisp

    termsToOutputForm : (RN, RN, ST, OUT) -> OUT
    termsToOutputForm(m, rat, uu, xxx) ==
      l : L OUT := empty()
      empty? uu => 0 :: OUT
      n : NNI; count : NNI := _$streamCount$Lisp
      for n in 0..count while not empty? uu repeat
        if frst(uu) ~= 0 then
          l := concat(termOutput((n :: I) * rat + m, frst uu, xxx), l)
        uu := rst uu
      if showAll?() then
        uu1 := uu
        for n in (count + 1).. while explicitEntries? uu and _
               not eq?(uu1, rst uu) repeat
          if frst(uu) ~= 0 then
            l := concat(termOutput((n :: I) * rat + m, frst uu, xxx), l)
          if odd?(n) then uu1 := rst(uu1)
          uu := rst uu
      l :=
        explicitlyEmpty? uu => l
        eq?(uu, rst uu) and frst uu = 0 => l
        concat(prefix('O :: OUT, [xxx^(((n::I)*rat + m) :: OUT)]), l)
      empty? l => 0 :: OUT
      reduce("+",reverse! l)

    coerce(upxs : %) : OUT ==
      rat := getExpon upxs; uls := laurentRep upxs
      count : I := _$streamCount$Lisp
      uls := removeZeroes(_$streamCount$Lisp, uls)
      m : RN := (degree uls) * rat
      p := coefficients taylorRep uls
      xxx :=
        zero? cen => var :: OUT
        paren(var :: OUT - cen :: OUT)
      termsToOutputForm(m, rat, p, xxx)

)abbrev package UPXS2 UnivariatePuiseuxSeriesFunctions2
++ Mapping package for univariate Puiseux series
++ Author: Scott C. Morrison
++ Date Created: 5 April 1991
++ Keywords: Puiseux series, map
++ Examples:
++ References:
++ Description:
++   Mapping package for univariate Puiseux series.
++   This package allows one to apply a function to the coefficients of
++   a univariate Puiseux series.
UnivariatePuiseuxSeriesFunctions2(Coef1, Coef2, var1, var2, cen1, cen2) : _
 Exports == Implementation where
  Coef1 : Ring
  Coef2 : Ring
  var1 : Symbol
  var2 : Symbol
  cen1 : Coef1
  cen2 : Coef2
  UPS1  ==> UnivariatePuiseuxSeries(Coef1, var1, cen1)
  UPS2  ==> UnivariatePuiseuxSeries(Coef2, var2, cen2)
  ULSP2 ==> UnivariateLaurentSeriesFunctions2(Coef1, Coef2, var1, var2, cen1, cen2)

  Exports ==> with
    map : (Coef1 -> Coef2, UPS1) -> UPS2
      ++ \spad{map(f, g(x))} applies the map f to the coefficients of the
      ++ Puiseux series \spad{g(x)}.

  Implementation ==> add

    map(f, ups) == puiseux(rationalPower ups, map(f, laurentRep ups)$ULSP2)

--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
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--
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--      the documentation and/or other materials provided with the
--      distribution.
--
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--      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
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--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
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