path: root/src/algebra/intaux.spad.pamphlet

\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra intaux.spad}
\author{Barry Trager, Manuel Bronstein}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{domain IR IntegrationResult}
<<domain IR IntegrationResult>>=
)abbrev domain IR IntegrationResult
++ The result of a transcendental integration.
++ Author: Barry Trager, Manuel Bronstein
++ Date Created: 1987
++ Date Last Updated: 12 August 1992
++ Description:
++ If a function f has an elementary integral g, then g can be written
++ in the form \spad{g = h + c1 log(u1) + c2 log(u2) + ... + cn log(un)}
++ where h, which is in the same field than f, is called the rational
++ part of the integral, and \spad{c1 log(u1) + ... cn log(un)} is called the
++ logarithmic part of the integral. This domain manipulates integrals
++ represented in that form, by keeping both parts separately. The logs
++ are not explicitly computed.
++ Keywords: integration.
++ Examples: )r RATINT INPUT
IntegrationResult(F:Field): Exports == Implementation where
  O   ==> OutputForm
  B   ==> Boolean
  Z   ==> Integer
  Q   ==> Fraction Integer
  SE  ==> Symbol
  UP  ==> SparseUnivariatePolynomial F
  LOG ==> Record(scalar:Q, coeff:UP, logand:UP)
  NE  ==> Record(integrand:F, intvar:F)

  Exports ==> (Module Q, RetractableTo F) with
    mkAnswer: (F, List LOG, List NE) -> %
      ++ mkAnswer(r,l,ne) creates an integration result from
      ++ a rational part r, a logarithmic part l, and a non-elementary part ne.
    ratpart : % -> F
      ++ ratpart(ir) returns the rational part of an integration result
    logpart : % -> List LOG
      ++ logpart(ir) returns the logarithmic part of an integration result
    notelem : % -> List NE
      ++ notelem(ir) returns the non-elementary part of an integration result
    elem?   : % -> B
      ++ elem?(ir) tests if an integration result is elementary over F?
    integral: (F, F) -> %
      ++ integral(f,x) returns the formal integral of f with respect to x
    differentiate: (%, F -> F) -> F
      ++ differentiate(ir,D) differentiates ir with respect to the derivation D.
    if F has PartialDifferentialRing(SE) then
      differentiate: (%, Symbol) -> F
        ++ differentiate(ir,x) differentiates ir with respect to x
    if F has RetractableTo Symbol then
      integral: (F, Symbol) -> %
        ++ integral(f,x) returns the formal integral of f with respect to x

  Implementation ==> add
    Rep := Record(ratp: F, logp: List LOG, nelem: List NE)

    timelog : (Q, LOG) -> LOG
    timene  : (Q, NE)  -> NE
    LOG2O   : LOG      -> O
    NE2O    : NE       -> O
    Q2F     : Q        -> F
    nesimp  : List NE  -> List NE
    neselect: (List NE, F) -> F
    pLogDeriv: (LOG, F -> F) -> F
    pNeDeriv : (NE,  F -> F) -> F


    alpha:O := new()$Symbol :: O

    - u               == (-1$Z) * u
    0                 == mkAnswer(0, empty(), empty())
    coerce(x:F):%     == mkAnswer(x, empty(), empty())
    ratpart u         == u.ratp
    logpart u         == u.logp
    notelem u         == u.nelem
    elem? u           == empty? notelem u
    mkAnswer(x, l, n) == [x, l, nesimp n]
    timelog(r, lg)    == [r * lg.scalar, lg.coeff, lg.logand]
    integral(f:F,x:F) == (zero? f => 0; mkAnswer(0, empty(), [[f, x]]))
    timene(r, ne)     == [Q2F(r) * ne.integrand, ne.intvar]
    n:Z * u:%         == (n::Q) * u
    Q2F r             == numer(r)::F / denom(r)::F
    neselect(l, x)    == +/[ne.integrand for ne in l | ne.intvar = x]

    if F has RetractableTo Symbol then
      integral(f:F, x:Symbol):% == integral(f, x::F)

    LOG2O rec ==
      one? degree rec.coeff =>
        -- deg 1 minimal poly doesn't get sigma
        lastc := - coefficient(rec.coeff, 0) / coefficient(rec.coeff, 1)
        lg    := (rec.logand) lastc
        logandp := prefix('log::O, [lg::O])
        (cc := Q2F(rec.scalar) * lastc) = 1 => logandp
        cc = -1 => - logandp
        cc::O * logandp
      coeffp:O := (outputForm(rec.coeff, alpha) = 0::Z::O)@O
      logandp := alpha * prefix('log::O, [outputForm(rec.logand, alpha)])
      if not one?(cc := Q2F(rec.scalar)) then
        logandp := cc::O * logandp
      sum(logandp, coeffp)

    nesimp l ==
      [[u,x] for x in removeDuplicates!([ne.intvar for ne in l]$List(F))
                                           | (u := neselect(l, x)) ~= 0]

    if (F has LiouvillianFunctionCategory) and (F has RetractableTo Symbol) then
      retractIfCan u ==
        empty? logpart u =>
          ratpart u +
             +/[integral(ne.integrand, retract(ne.intvar)@Symbol)$F
                for ne in notelem u]
        "failed"

    else
      retractIfCan u ==
        elem? u and empty? logpart u => ratpart u
        "failed"

    r:Q * u:% ==
      r = 0 => 0
      mkAnswer(Q2F(r) * ratpart u, map(timelog(r, #1), logpart u),
                                          map(timene(r, #1), notelem u))

    -- Initial attempt, quick and dirty, no simplification
    u + v ==
      mkAnswer(ratpart u + ratpart v, concat(logpart u, logpart v),
                                    nesimp concat(notelem u, notelem v))

    if F has PartialDifferentialRing(Symbol) then
      differentiate(u:%, x:Symbol):F == differentiate(u, differentiate(#1, x))

    differentiate(u:%, derivation:F -> F):F ==
      derivation ratpart u +
          +/[pLogDeriv(log, derivation) for log in logpart u]
               + (+/[pNeDeriv(ne, derivation) for ne in notelem u])

    pNeDeriv(ne, derivation) ==
      one? derivation(ne.intvar) => ne.integrand
      zero? derivation(ne.integrand) => 0
      error "pNeDeriv: cannot differentiate not elementary part into F"

    pLogDeriv(log, derivation) ==
      map(derivation, log.coeff) ~= 0 =>
        error "pLogDeriv: can only handle logs with constant coefficients"
      one?(n := degree(log.coeff)) =>
        c   := - (leadingCoefficient reductum log.coeff)
                                        / (leadingCoefficient log.coeff)
        ans := (log.logand) c
        Q2F(log.scalar) * c * derivation(ans) / ans
      numlog := map(derivation, log.logand)
      diflog := extendedEuclidean(log.logand, log.coeff,
                                    numlog)::Record(coef1:UP, coef2:UP)
      algans := diflog.coef1
      ans:F := 0
      for i in 0..(n-1) repeat
        algans := algans * monomial(1, 1) rem log.coeff
        ans := ans + coefficient(algans, i)
      Q2F(log.scalar) * ans

    coerce(u:%):O ==
      (r := retractIfCan u) case F => r::F::O
      l := reverse! [LOG2O f for f in logpart u]$List(O)
      if ratpart u ~= 0 then l := concat(ratpart(u)::O, l)
      if not elem? u then l := concat([NE2O f for f in notelem u], l)
      null l => 0@F::O
      reduce("+", l)

    NE2O ne ==
      int((ne.integrand)::O * hconcat ['d::O, (ne.intvar)::O])

@
\section{package IR2 IntegrationResultFunctions2}
<<package IR2 IntegrationResultFunctions2>>=
)abbrev package IR2 IntegrationResultFunctions2
++ Internally used by the integration packages
++ Author: Manuel Bronstein
++ Date Created: 1987
++ Date Last Updated: 12 August 1992
++ Keywords: integration.
IntegrationResultFunctions2(E, F): Exports == Implementation where
  E : Field
  F : Field

  SE  ==> Symbol
  Q   ==> Fraction Integer
  IRE ==> IntegrationResult E
  IRF ==> IntegrationResult F
  UPE ==> SparseUnivariatePolynomial E
  UPF ==> SparseUnivariatePolynomial F
  NEE ==> Record(integrand:E, intvar:E)
  NEF ==> Record(integrand:F, intvar:F)
  LGE ==> Record(scalar:Q, coeff:UPE, logand:UPE)
  LGF ==> Record(scalar:Q, coeff:UPF, logand:UPF)
  NLE ==> Record(coeff:E, logand:E)
  NLF ==> Record(coeff:F, logand:F)
  UFE ==> Union(Record(mainpart:E, limitedlogs:List NLE), "failed")
  URE ==> Union(Record(ratpart:E, coeff:E), "failed")
  UE  ==> Union(E, "failed")

  Exports ==> with
    map: (E -> F, IRE) -> IRF
	++ map(f,ire) \undocumented
    map: (E -> F, URE) -> Union(Record(ratpart:F, coeff:F), "failed")
	++ map(f,ure) \undocumented
    map: (E -> F,  UE) -> Union(F, "failed")
	++ map(f,ue) \undocumented
    map: (E -> F, UFE) ->
               Union(Record(mainpart:F, limitedlogs:List NLF), "failed")
	++ map(f,ufe) \undocumented

  Implementation ==> add
    import SparseUnivariatePolynomialFunctions2(E, F)

    NEE2F: (E -> F, NEE) -> NEF
    LGE2F: (E -> F, LGE) -> LGF
    NLE2F: (E -> F, NLE) -> NLF

    NLE2F(func, r)         == [func(r.coeff), func(r.logand)]
    NEE2F(func, n)         == [func(n.integrand), func(n.intvar)]
    map(func:E -> F, u:UE) == (u case "failed" => "failed"; func(u::E))

    map(func:E -> F, ir:IRE) ==
      mkAnswer(func ratpart ir, [LGE2F(func, f) for f in logpart ir],
                                   [NEE2F(func, g) for g in notelem ir])

    map(func:E -> F, u:URE) ==
      u case "failed" => "failed"
      [func(u.ratpart), func(u.coeff)]

    map(func:E -> F, u:UFE) ==
      u case "failed" => "failed"
      [func(u.mainpart), [NLE2F(func, f) for f in u.limitedlogs]]

    LGE2F(func, lg) ==
      [lg.scalar, map(func, lg.coeff), map(func, lg.logand)]

@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--Copyright (C) 2007-2009, 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.
@
<<*>>=
<<license>>

-- SPAD files for the integration world should be compiled in the
-- following order:
--
--   INTAUX  rderf  intrf  rdeef  intef  irexpand  integrat

<<domain IR IntegrationResult>>
<<package IR2 IntegrationResultFunctions2>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}