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authordos-reis <gdr@axiomatics.org>2011-09-20 06:29:26 +0000
committerdos-reis <gdr@axiomatics.org>2011-09-20 06:29:26 +0000
commit255be06767355e3b41acd75990c6b90270b8f2bd (patch)
tree0a032d40d7cb2213e206d70beb23dc74ccf4a230 /src/algebra/d02Package.spad.pamphlet
parent12c856f9901ef3d6d82fb99855ecdf3e0b91484b (diff)
downloadopen-axiom-255be06767355e3b41acd75990c6b90270b8f2bd.tar.gz
* algebra/asp.spad.pamphlet: Remove.
* algebra/c02.spad.pamphlet: Likewise. * algebra/c05.spad.pamphlet: Likewise. * algebra/c06.spad.pamphlet: Likewise. * algebra/d01.spad.pamphlet: Likewise. * algebra/d02.spad.pamphlet: Likewise. * algebra/d03.spad.pamphlet: Likewise. * algebra/e01.spad.pamphlet: Likewise. * algebra/e02.spad.pamphlet: Likewise. * algebra/e04.spad.pamphlet: Likewise. * algebra/f01.spad.pamphlet: Likewise. * algebra/f02.spad.pamphlet: Likewise. * algebra/f04.spad.pamphlet: Likewise. * algebra/f07.spad.pamphlet: Likewise. * algebra/s.spad.pamphlet: Likewise. * algebra/d01Package.spad.pamphlet: Likewise. * algebra/d02Package.spad.pamphlet: Likewise. * algebra/d03Package.spad.pamphlet: Likewise. * algebra/e04Package.spad.pamphlet: Likewise. * algebra/d01agents.spad.pamphlet: Likewise. * algebra/d01routine.spad.pamphlet: Likewise. * algebra/d01transform.spad.pamphlet: Likewise. * algebra/d01weights.spad.pamphlet: Likewise. * algebra/d02agents.spad.pamphlet: Likewise. * algebra/d02routine.spad.pamphlet: Likewise. * algebra/d03agents.spad.pamphlet: Likewise. * algebra/d03routine.spad.pamphlet: Likewise. * algebra/e04agents.spad.pamphlet: Likewise. * algebra/e04routine.spad.pamphlet: Likewise.
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-\documentclass{article}
-\usepackage{open-axiom}
-\begin{document}
-\title{\$SPAD/src/algebra d02Package.spad}
-\author{Brian Dupee}
-\maketitle
-\begin{abstract}
-\end{abstract}
-\eject
-\tableofcontents
-\eject
-\section{package ODEPACK AnnaOrdinaryDifferentialEquationPackage}
-<<package ODEPACK AnnaOrdinaryDifferentialEquationPackage>>=
-)abbrev package ODEPACK AnnaOrdinaryDifferentialEquationPackage
-++ Author: Brian Dupee
-++ Date Created: February 1995
-++ Date Last Updated: December 1997
-++ Basic Operations: solve, measure
-++ Description:
-++ \axiomType{AnnaOrdinaryDifferentialEquationPackage} is a \axiom{package}
-++ of functions for the \axiom{category} \axiomType{OrdinaryDifferentialEquationsSolverCategory}
-++ with \axiom{measure}, and \axiom{solve}.
-++
-EDF ==> Expression DoubleFloat
-LDF ==> List DoubleFloat
-MDF ==> Matrix DoubleFloat
-DF ==> DoubleFloat
-FI ==> Fraction Integer
-EFI ==> Expression Fraction Integer
-SOCDF ==> Segment OrderedCompletion DoubleFloat
-VEDF ==> Vector Expression DoubleFloat
-VEF ==> Vector Expression Float
-EF ==> Expression Float
-LF ==> List Float
-F ==> Float
-VDF ==> Vector DoubleFloat
-VMF ==> Vector MachineFloat
-MF ==> MachineFloat
-LS ==> List Symbol
-ST ==> String
-LST ==> List String
-INT ==> Integer
-RT ==> RoutinesTable
-ODEA ==> Record(xinit:DF,xend:DF,fn:VEDF,yinit:LDF,intvals:LDF,_
- g:EDF,abserr:DF,relerr:DF)
-IFL ==> List(Record(ifail:Integer,instruction:String))
-Entry ==> Record(chapter:String, type:String, domainName: String,
- defaultMin:F, measure:F, failList:IFL, explList:LST)
-Measure ==> Record(measure:F,name:String, explanations:List String)
-
-AnnaOrdinaryDifferentialEquationPackage(): with
- solve:(NumericalODEProblem) -> Result
- ++ solve(odeProblem) is a top level ANNA function to solve numerically a
- ++ system of ordinary differential equations i.e. equations for the
- ++ derivatives Y[1]'..Y[n]' defined in terms of X,Y[1]..Y[n], together
- ++ with starting values for X and Y[1]..Y[n] (called the initial
- ++ conditions), a final value of X, an accuracy requirement and any
- ++ intermediate points at which the result is required.
- ++
- ++ It iterates over the \axiom{domains} of
- ++ \axiomType{OrdinaryDifferentialEquationsSolverCategory}
- ++ to get the name and other
- ++ relevant information of the the (domain of the) numerical
- ++ routine likely to be the most appropriate,
- ++ i.e. have the best \axiom{measure}.
- ++
- ++ The method used to perform the numerical
- ++ process will be one of the routines contained in the NAG numerical
- ++ Library. The function predicts the likely most effective routine
- ++ by checking various attributes of the system of ODE's and calculating
- ++ a measure of compatibility of each routine to these attributes.
- ++
- ++ It then calls the resulting `best' routine.
- solve:(NumericalODEProblem,RT) -> Result
- ++ solve(odeProblem,R) is a top level ANNA function to solve numerically a
- ++ system of ordinary differential equations i.e. equations for the
- ++ derivatives Y[1]'..Y[n]' defined in terms of X,Y[1]..Y[n], together
- ++ with starting values for X and Y[1]..Y[n] (called the initial
- ++ conditions), a final value of X, an accuracy requirement and any
- ++ intermediate points at which the result is required.
- ++
- ++ It iterates over the \axiom{domains} of
- ++ \axiomType{OrdinaryDifferentialEquationsSolverCategory} contained in
- ++ the table of routines \axiom{R} to get the name and other
- ++ relevant information of the the (domain of the) numerical
- ++ routine likely to be the most appropriate,
- ++ i.e. have the best \axiom{measure}.
- ++
- ++ The method used to perform the numerical
- ++ process will be one of the routines contained in the NAG numerical
- ++ Library. The function predicts the likely most effective routine
- ++ by checking various attributes of the system of ODE's and calculating
- ++ a measure of compatibility of each routine to these attributes.
- ++
- ++ It then calls the resulting `best' routine.
- solve:(VEF,F,F,LF) -> Result
- ++ solve(f,xStart,xEnd,yInitial) is a top level ANNA function to solve numerically a
- ++ system of ordinary differential equations i.e. equations for the
- ++ derivatives Y[1]'..Y[n]' defined in terms of X,Y[1]..Y[n], together
- ++ with a starting value for X and Y[1]..Y[n] (called the initial
- ++ conditions) and a final value of X. A default value
- ++ is used for the accuracy requirement.
- ++
- ++ It iterates over the \axiom{domains} of
- ++ \axiomType{OrdinaryDifferentialEquationsSolverCategory} contained in
- ++ the table of routines \axiom{R} to get the name and other
- ++ relevant information of the the (domain of the) numerical
- ++ routine likely to be the most appropriate,
- ++ i.e. have the best \axiom{measure}.
- ++
- ++ The method used to perform the numerical
- ++ process will be one of the routines contained in the NAG numerical
- ++ Library. The function predicts the likely most effective routine
- ++ by checking various attributes of the system of ODE's and calculating
- ++ a measure of compatibility of each routine to these attributes.
- ++
- ++ It then calls the resulting `best' routine.
- solve:(VEF,F,F,LF,F) -> Result
- ++ solve(f,xStart,xEnd,yInitial,tol) is a top level ANNA function to solve
- ++ numerically a system of ordinary differential equations, \axiom{f}, i.e.
- ++ equations for the derivatives Y[1]'..Y[n]' defined in terms
- ++ of X,Y[1]..Y[n] from \axiom{xStart} to \axiom{xEnd} with the initial
- ++ values for Y[1]..Y[n] (\axiom{yInitial}) to a tolerance \axiom{tol}.
- ++
- ++ It iterates over the \axiom{domains} of
- ++ \axiomType{OrdinaryDifferentialEquationsSolverCategory} contained in
- ++ the table of routines \axiom{R} to get the name and other
- ++ relevant information of the the (domain of the) numerical
- ++ routine likely to be the most appropriate,
- ++ i.e. have the best \axiom{measure}.
- ++
- ++ The method used to perform the numerical
- ++ process will be one of the routines contained in the NAG numerical
- ++ Library. The function predicts the likely most effective routine
- ++ by checking various attributes of the system of ODE's and calculating
- ++ a measure of compatibility of each routine to these attributes.
- ++
- ++ It then calls the resulting `best' routine.
- solve:(VEF,F,F,LF,EF,F) -> Result
- ++ solve(f,xStart,xEnd,yInitial,G,tol) is a top level ANNA function to solve
- ++ numerically a system of ordinary differential equations, \axiom{f}, i.e.
- ++ equations for the derivatives Y[1]'..Y[n]' defined in terms
- ++ of X,Y[1]..Y[n] from \axiom{xStart} to \axiom{xEnd} with the initial
- ++ values for Y[1]..Y[n] (\axiom{yInitial}) to a tolerance \axiom{tol}.
- ++ The calculation will stop if the function G(X,Y[1],..,Y[n]) evaluates to zero before
- ++ X = xEnd.
- ++
- ++ It iterates over the \axiom{domains} of
- ++ \axiomType{OrdinaryDifferentialEquationsSolverCategory} contained in
- ++ the table of routines \axiom{R} to get the name and other
- ++ relevant information of the the (domain of the) numerical
- ++ routine likely to be the most appropriate,
- ++ i.e. have the best \axiom{measure}.
- ++
- ++ The method used to perform the numerical
- ++ process will be one of the routines contained in the NAG numerical
- ++ Library. The function predicts the likely most effective routine
- ++ by checking various attributes of the system of ODE's and calculating
- ++ a measure of compatibility of each routine to these attributes.
- ++
- ++ It then calls the resulting `best' routine.
- solve:(VEF,F,F,LF,LF,F) -> Result
- ++ solve(f,xStart,xEnd,yInitial,intVals,tol) is a top level ANNA function to solve
- ++ numerically a system of ordinary differential equations, \axiom{f}, i.e.
- ++ equations for the derivatives Y[1]'..Y[n]' defined in terms
- ++ of X,Y[1]..Y[n] from \axiom{xStart} to \axiom{xEnd} with the initial
- ++ values for Y[1]..Y[n] (\axiom{yInitial}) to a tolerance \axiom{tol}.
- ++ The values of Y[1]..Y[n] will be output for the values of X in
- ++ \axiom{intVals}.
- ++
- ++ It iterates over the \axiom{domains} of
- ++ \axiomType{OrdinaryDifferentialEquationsSolverCategory} contained in
- ++ the table of routines \axiom{R} to get the name and other
- ++ relevant information of the the (domain of the) numerical
- ++ routine likely to be the most appropriate,
- ++ i.e. have the best \axiom{measure}.
- ++
- ++ The method used to perform the numerical
- ++ process will be one of the routines contained in the NAG numerical
- ++ Library. The function predicts the likely most effective routine
- ++ by checking various attributes of the system of ODE's and calculating
- ++ a measure of compatibility of each routine to these attributes.
- ++
- ++ It then calls the resulting `best' routine.
- solve:(VEF,F,F,LF,EF,LF,F) -> Result
- ++ solve(f,xStart,xEnd,yInitial,G,intVals,tol) is a top level ANNA function to solve
- ++ numerically a system of ordinary differential equations, \axiom{f}, i.e.
- ++ equations for the derivatives Y[1]'..Y[n]' defined in terms
- ++ of X,Y[1]..Y[n] from \axiom{xStart} to \axiom{xEnd} with the initial
- ++ values for Y[1]..Y[n] (\axiom{yInitial}) to a tolerance \axiom{tol}.
- ++ The values of Y[1]..Y[n] will be output for the values of X in
- ++ \axiom{intVals}. The calculation will stop if the function
- ++ G(X,Y[1],..,Y[n]) evaluates to zero before X = xEnd.
- ++
- ++ It iterates over the \axiom{domains} of
- ++ \axiomType{OrdinaryDifferentialEquationsSolverCategory} contained in
- ++ the table of routines \axiom{R} to get the name and other
- ++ relevant information of the the (domain of the) numerical
- ++ routine likely to be the most appropriate,
- ++ i.e. have the best \axiom{measure}.
- ++
- ++ The method used to perform the numerical
- ++ process will be one of the routines contained in the NAG numerical
- ++ Library. The function predicts the likely most effective routine
- ++ by checking various attributes of the system of ODE's and calculating
- ++ a measure of compatibility of each routine to these attributes.
- ++
- ++ It then calls the resulting `best' routine.
- solve:(VEF,F,F,LF,EF,LF,F,F) -> Result
- ++ solve(f,xStart,xEnd,yInitial,G,intVals,epsabs,epsrel) is a top level ANNA function to solve
- ++ numerically a system of ordinary differential equations, \axiom{f}, i.e.
- ++ equations for the derivatives Y[1]'..Y[n]' defined in terms
- ++ of X,Y[1]..Y[n] from \axiom{xStart} to \axiom{xEnd} with the initial
- ++ values for Y[1]..Y[n] (\axiom{yInitial}) to an absolute error
- ++ requirement \axiom{epsabs} and relative error \axiom{epsrel}.
- ++ The values of Y[1]..Y[n] will be output for the values of X in
- ++ \axiom{intVals}. The calculation will stop if the function
- ++ G(X,Y[1],..,Y[n]) evaluates to zero before X = xEnd.
- ++
- ++ It iterates over the \axiom{domains} of
- ++ \axiomType{OrdinaryDifferentialEquationsSolverCategory} contained in
- ++ the table of routines \axiom{R} to get the name and other
- ++ relevant information of the the (domain of the) numerical
- ++ routine likely to be the most appropriate,
- ++ i.e. have the best \axiom{measure}.
- ++
- ++ The method used to perform the numerical
- ++ process will be one of the routines contained in the NAG numerical
- ++ Library. The function predicts the likely most effective routine
- ++ by checking various attributes of the system of ODE's and calculating
- ++ a measure of compatibility of each routine to these attributes.
- ++
- ++ It then calls the resulting `best' routine.
- measure:(NumericalODEProblem) -> Measure
- ++ measure(prob) is a top level ANNA function for identifying the most
- ++ appropriate numerical routine from those in the routines table
- ++ provided for solving the numerical ODE
- ++ problem defined by \axiom{prob}.
- ++
- ++ It calls each \axiom{domain} of \axiom{category}
- ++ \axiomType{OrdinaryDifferentialEquationsSolverCategory} in turn to
- ++ calculate all measures and returns the best i.e. the name of
- ++ the most appropriate domain and any other relevant information.
- ++ It predicts the likely most effective NAG numerical
- ++ Library routine to solve the input set of ODEs
- ++ by checking various attributes of the system of ODEs and calculating
- ++ a measure of compatibility of each routine to these attributes.
- measure:(NumericalODEProblem,RT) -> Measure
- ++ measure(prob,R) is a top level ANNA function for identifying the most
- ++ appropriate numerical routine from those in the routines table
- ++ provided for solving the numerical ODE
- ++ problem defined by \axiom{prob}.
- ++
- ++ It calls each \axiom{domain} listed in \axiom{R} of \axiom{category}
- ++ \axiomType{OrdinaryDifferentialEquationsSolverCategory} in turn to
- ++ calculate all measures and returns the best i.e. the name of
- ++ the most appropriate domain and any other relevant information.
- ++ It predicts the likely most effective NAG numerical
- ++ Library routine to solve the input set of ODEs
- ++ by checking various attributes of the system of ODEs and calculating
- ++ a measure of compatibility of each routine to these attributes.
-
- == add
-
- import ODEA,NumericalODEProblem
-
- f2df:F -> DF
- ef2edf:EF -> EDF
- preAnalysis:(ODEA,RT) -> RT
- zeroMeasure:Measure -> Result
- measureSpecific:(ST,RT,ODEA) -> Record(measure:F,explanations:ST)
- solveSpecific:(ODEA,ST) -> Result
- changeName:(Result,ST) -> Result
- recoverAfterFail:(ODEA,RT,Measure,Integer,Result) -> Record(a:Result,b:Measure)
-
- f2df(f:F):DF == (convert(f)@DF)$F
-
- ef2edf(f:EF):EDF == map(f2df,f)$ExpressionFunctions2(F,DF)
-
- preAnalysis(args:ODEA,t:RT):RT ==
- rt := selectODEIVPRoutines(t)$RT
- if positive?(# variables(args.g)) then
- changeMeasure(rt,d02bbf@Symbol,getMeasure(rt,d02bbf@Symbol)*0.8)
- if positive?(# args.intvals) then
- changeMeasure(rt,d02bhf@Symbol,getMeasure(rt,d02bhf@Symbol)*0.8)
- rt
-
- zeroMeasure(m:Measure):Result ==
- a := coerce(0$F)$AnyFunctions1(F)
- text := coerce("Zero Measure")$AnyFunctions1(ST)
- r := construct([[result@Symbol,a],[method@Symbol,text]])$Result
- concat(measure2Result m,r)$ExpertSystemToolsPackage
-
- measureSpecific(name:ST,R:RT,ode:ODEA):Record(measure:F,explanations:ST) ==
- name = "d02bbfAnnaType" => measure(R,ode)$d02bbfAnnaType
- name = "d02bhfAnnaType" => measure(R,ode)$d02bhfAnnaType
- name = "d02cjfAnnaType" => measure(R,ode)$d02cjfAnnaType
- name = "d02ejfAnnaType" => measure(R,ode)$d02ejfAnnaType
- error("measureSpecific","invalid type name: " name)$ErrorFunctions
-
- measure(Ode:NumericalODEProblem,R:RT):Measure ==
- ode:ODEA := retract(Ode)$NumericalODEProblem
- sofar := 0$F
- best := "none" :: ST
- routs := copy R
- routs := preAnalysis(ode,routs)
- empty?(routs)$RT =>
- error("measure", "no routines found")$ErrorFunctions
- rout := inspect(routs)$RT
- e := retract(rout.entry)$AnyFunctions1(Entry)
- meth := empty()$LST
- for i in 1..# routs repeat
- rout := extract!(routs)$RT
- e := retract(rout.entry)$AnyFunctions1(Entry)
- n := e.domainName
- if e.defaultMin > sofar then
- m := measureSpecific(n,R,ode)
- if m.measure > sofar then
- sofar := m.measure
- best := n
- str:LST := [string(rout.key)$Symbol "measure: "
- outputMeasure(m.measure)$ExpertSystemToolsPackage " - "
- m.explanations]
- else
- str := [string(rout.key)$Symbol " is no better than other routines"]
- meth := append(meth,str)$LST
- [sofar,best,meth]
-
- measure(ode:NumericalODEProblem):Measure == measure(ode,routines()$RT)
-
- solveSpecific(ode:ODEA,n:ST):Result ==
- n = "d02bbfAnnaType" => ODESolve(ode)$d02bbfAnnaType
- n = "d02bhfAnnaType" => ODESolve(ode)$d02bhfAnnaType
- n = "d02cjfAnnaType" => ODESolve(ode)$d02cjfAnnaType
- n = "d02ejfAnnaType" => ODESolve(ode)$d02ejfAnnaType
- error("solveSpecific","invalid type name: " n)$ErrorFunctions
-
- changeName(ans:Result,name:ST):Result ==
- sy:Symbol := coerce(name "Answer")$Symbol
- anyAns:Any := coerce(ans)$AnyFunctions1(Result)
- construct([[sy,anyAns]])$Result
-
- recoverAfterFail(ode:ODEA,routs:RT,m:Measure,iint:Integer,r:Result):
- Record(a:Result,b:Measure) ==
- while positive?(iint) repeat
- routineName := m.name
- s := recoverAfterFail(routs,routineName(1..6),iint)$RT
- s case "failed" => iint := 0
- if s = "increase tolerance" then
- ode.relerr := ode.relerr*(10.0::DF)
- ode.abserr := ode.abserr*(10.0::DF)
- if s = "decrease tolerance" then
- ode.relerr := ode.relerr/(10.0::DF)
- ode.abserr := ode.abserr/(10.0::DF)
- (s = "no action")@Boolean => iint := 0
- fl := coerce(s)$AnyFunctions1(ST)
- flrec:Record(key:Symbol,entry:Any):=[failure@Symbol,fl]
- m2 := measure(ode::NumericalODEProblem,routs)
- zero?(m2.measure) => iint := 0
- r2:Result := solveSpecific(ode,m2.name)
- m := m2
- insert!(flrec,r2)$Result
- r := concat(r2,changeName(r,routineName))$ExpertSystemToolsPackage
- iany := search(ifail@Symbol,r2)$Result
- iany case "failed" => iint := 0
- iint := retract(iany)$AnyFunctions1(Integer)
- [r,m]
-
- solve(Ode:NumericalODEProblem,t:RT):Result ==
- ode:ODEA := retract(Ode)$NumericalODEProblem
- routs := copy(t)$RT
- m := measure(Ode,routs)
- zero?(m.measure) => zeroMeasure m
- r := solveSpecific(ode,n := m.name)
- iany := search(ifail@Symbol,r)$Result
- iint := 0$Integer
- if (iany case Any) then
- iint := retract(iany)$AnyFunctions1(Integer)
- if positive?(iint) then
- tu:Record(a:Result,b:Measure) := recoverAfterFail(ode,routs,m,iint,r)
- r := tu.a
- m := tu.b
- r := concat(measure2Result m,r)$ExpertSystemToolsPackage
- expl := getExplanations(routs,n(1..6))$RoutinesTable
- expla := coerce(expl)$AnyFunctions1(LST)
- explaa:Record(key:Symbol,entry:Any) := ["explanations"::Symbol,expla]
- r := concat(construct([explaa]),r)
- iflist := showIntensityFunctions(ode)$ODEIntensityFunctionsTable
- iflist case "failed" => r
- concat(iflist2Result iflist, r)$ExpertSystemToolsPackage
-
- solve(ode:NumericalODEProblem):Result == solve(ode,routines()$RT)
-
- solve(f:VEF,xStart:F,xEnd:F,yInitial:LF,G:EF,intVals:LF,epsabs:F,epsrel:F):Result ==
- d:ODEA := [f2df xStart,f2df xEnd,vector([ef2edf e for e in members f])$VEDF,
- [f2df i for i in yInitial], [f2df j for j in intVals],
- ef2edf G,f2df epsabs,f2df epsrel]
- solve(d::NumericalODEProblem,routines()$RT)
-
- solve(f:VEF,xStart:F,xEnd:F,yInitial:LF,G:EF,intVals:LF,tol:F):Result ==
- solve(f,xStart,xEnd,yInitial,G,intVals,tol,tol)
-
- solve(f:VEF,xStart:F,xEnd:F,yInitial:LF,intVals:LF,tol:F):Result ==
- solve(f,xStart,xEnd,yInitial,1$EF,intVals,tol)
-
- solve(f:VEF,xStart:F,xEnd:F,y:LF,G:EF,tol:F):Result ==
- solve(f,xStart,xEnd,y,G,empty()$LF,tol)
-
- solve(f:VEF,xStart:F,xEnd:F,yInitial:LF,tol:F):Result ==
- solve(f,xStart,xEnd,yInitial,1$EF,empty()$LF,tol)
-
- solve(f:VEF,xStart:F,xEnd:F,yInitial:LF):Result == solve(f,xStart,xEnd,yInitial,1.0e-4)
-
-@
-\section{License}
-<<license>>=
---Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
---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>>
-
-<<package ODEPACK AnnaOrdinaryDifferentialEquationPackage>>
-@
-\eject
-\begin{thebibliography}{99}
-\bibitem{1} nothing
-\end{thebibliography}
-\end{document}