\documentclass{article} \usepackage{open-axiom} \begin{document} \title{\$SPAD/src/algebra transsolve.spad} \author{Waldemar Wiwianka, Martin Rubey} \maketitle \begin{abstract} \end{abstract} \eject \tableofcontents \eject \section{package SOLVETRA TransSolvePackage} <>= )abbrev package SOLVETRA TransSolvePackage ++ Author: W. Wiwianka, Martin Rubey ++ Date Created: Summer 1991 ++ Change History: 9/91 ++ Basic Operations: solve ++ Related Constructors: RadicalSolvePackage, FloatingRealPackage ++ Keywords: ++ Description: ++ This package tries to find solutions of equations of type Expression(R). ++ This means expressions involving transcendental, exponential, logarithmic ++ and nthRoot functions. ++ After trying to transform different kernels to one kernel by applying ++ several rules, it calls zerosOf for the SparseUnivariatePolynomial in ++ the remaining kernel. ++ For example the expression \spad{sin(x)*cos(x)-2} will be transformed to ++ \spad{-2 tan(x/2)**4 -2 tan(x/2)**3 -4 tan(x/2)**2 +2 tan(x/2) -2} ++ by using the function normalize and then to ++ \spad{-2 tan(x)**2 + tan(x) -2} ++ with help of subsTan. This function tries to express the given function ++ in terms of \spad{tan(x/2)} to express in terms of \spad{tan(x)} . ++ Other examples are the expressions \spad{sqrt(x+1)+sqrt(x+7)+1} or ++ \spad{sqrt(sin(x))+1} . TransSolvePackage(R) : Exports == Implementation where R : Join(EuclideanDomain, RetractableTo Integer, LinearlyExplicitRingOver Integer, CharacteristicZero) I ==> Integer NNI ==> NonNegativeInteger RE ==> Expression R EQ ==> Equation S ==> Symbol V ==> Variable L ==> List K ==> Kernel RE SUP ==> SparseUnivariatePolynomial C ==> Complex F ==> Float INT ==> Interval SMP ==> SparseMultivariatePolynomial Exports == with solve : RE -> L EQ RE ++ solve(expr) finds the solutions of the equation expr = 0 ++ where expr is a function of type Expression(R) ++ with respect to the unique symbol x appearing in eq. solve : EQ RE -> L EQ RE ++ solve(eq) finds the solutions of the equation eq ++ where eq is an equation of functions of type Expression(R) ++ with respect to the unique symbol x appearing in eq. solve : ( EQ RE , S ) -> L EQ RE ++ solve(eq,x) finds the solutions of the equation eq ++ where eq is an equation of functions of type Expression(R) ++ with respect to the symbol x. solve : ( RE , S) -> L EQ RE ++ solve(expr,x) finds the solutions of the equation expr = 0 ++ with respect to the symbol x where expr is a function ++ of type Expression(R). solve : (L EQ RE, L S) -> L L EQ RE ++ solve(leqs, lvar) returns a list of solutions to the list of ++ equations leqs with respect to the list of symbols lvar. -- solve : (L EQ RE, L Kernel RE) -> L L EQ RE -- ++ solve(leqs, lker) returns a list of solutions to the list -- ++ of equations leqs with respect to the list of kernels lker. Implementation == add import ACF import HomogeneousAggregate(R) import AlgebraicManipulations(R, RE) import TranscendentalManipulations(R, RE) import TrigonometricManipulations(R, RE) import ElementaryFunctionStructurePackage(R, RE) import SparseUnivariatePolynomial(R) import LinearSystemMatrixPackage(RE,Vector RE,Vector RE,Matrix RE) import TransSolvePackageService(R) import MultivariateFactorize(K, IndexedExponents K, R, SMP(R, K)) ---- Local Function Declarations ---- solveInner : (RE, S) -> L EQ RE tryToTrans : ( RE , S) -> RE eliminateKernRoot: (RE , K) -> RE eliminateRoot: (RE , S) -> RE combineLog : ( RE , S ) -> RE testLog : ( RE , S ) -> Boolean splitExpr : ( RE ) -> L RE buildnexpr : ( RE , S ) -> L RE logsumtolog : RE -> RE logexpp : ( RE , RE ) -> RE testRootk : ( RE, S) -> Boolean testkernel : ( RE , S ) -> Boolean funcinv : ( RE , RE ) -> Union(RE,"failed") testTrig : ( RE , S ) -> Boolean testHTrig : ( RE , S ) -> Boolean tableXkernels : ( RE , S ) -> L RE subsTan : ( RE , S ) -> RE -- exported functions solve(oside: RE) : L EQ RE == zero? oside => error "equation is always satisfied" lv := variables oside empty? lv => error "inconsistent equation" #lv>1 => error "too many variables" solve(oside,lv.first) solve(equ:EQ RE) : L EQ RE == solve(lhs(equ)-rhs(equ)) solve(equ:EQ RE, x:S) : L EQ RE == oneside:=lhs(equ)-rhs(equ) solve(oneside,x) testZero?(lside:RE,sol:EQ RE):Boolean == if R has QuotientFieldCategory(Integer) then retractIfCan(rhs sol)@Union(Integer,"failed") case "failed" => true else retractIfCan(rhs sol)@Union(Fraction Integer,"failed") case "failed" => true zero? eval(lside,sol) => true false solve(lside: RE, x:S) : L EQ RE == [sol for sol in solveInner(lside,x) | testZero?(lside,sol)] solveInner(lside: RE, x:S) : L EQ RE == lside:=eliminateRoot(lside,x) ausgabe1:=tableXkernels(lside,x) X:=new()@Symbol Y:=new()@Symbol::RE (#ausgabe1) = 1 => bigX:= (first ausgabe1)::RE eq1:=eval(lside,bigX=(X::RE)) -- Type : Expression R f:=univariate(eq1,first kernels (X::RE)) -- Type : Fraction SparseUnivariatePolynomial Expression R lfatt:= factors factorPolynomial numer f lr:L RE := "append" /[zerosOf(fatt.factor,x) for fatt in lfatt] -- Type : List Expression R r1:=[]::L RE for i in 1..#lr repeat finv := funcinv(bigX,lr(i)) if finv case RE then r1:=cons(finv::RE,r1) bigX_back:=funcinv(bigX,bigX)::RE if not testkernel(bigX_back,x) then if bigX = bigX_back then return []::L EQ RE return "append"/[solve(bigX_back-ri, x) for ri in r1] newlist:=[]::L EQ RE for i in 1..#r1 repeat elR := eliminateRoot((numer(bigX_back - r1(i))::RE ),x) f:=univariate(elR, kernel(x)) -- Type : Fraction SparseUnivariatePolynomial Expression R lfatt:= factors factorPolynomial numer f secondsol:="append" /[zerosOf(ff.factor,x) for ff in lfatt] for j in 1..#secondsol repeat newlist:=cons((x::RE)=rootSimp( secondsol(j) ),newlist) newlist newlside:=tryToTrans(lside,x) ::RE listofkernels:=tableXkernels(newlside,x) (#listofkernels) = 1 => solve(newlside,x) lfacts := factors factor(numer lside) #lfacts > 1 => sols : L EQ RE := [] for frec in lfacts repeat sols := append(solve(frec.factor :: RE, x), sols) sols return []::L EQ RE -- local functions -- This function was suggested by Manuel Bronstein as a simpler -- alternative to normalize. simplifyingLog(f:RE):RE == (u:=isExpt(f,'exp)) case Record(var:Kernel RE,exponent:Integer) => rec := u::Record(var:Kernel RE,exponent:Integer) rec.exponent * first argument(rec.var) log f testkernel(var1:RE,y:S) : Boolean == var1:=eliminateRoot(var1,y) listvar1:=tableXkernels(var1,y) if (#listvar1 = 1) and ((listvar1(1) = (y::RE))@Boolean ) then true else if #listvar1 = 0 then true else false solveRetract(lexpr:L RE, lvar:L S):Union(L L EQ RE, "failed") == nlexpr : L Fraction Polynomial R := [] for expr in lexpr repeat rf:Union(Fraction Polynomial R, "failed") := retractIfCan(expr)$RE rf case "failed" => return "failed" nlexpr := cons(rf, nlexpr) radicalSolve(nlexpr, lvar)$RadicalSolvePackage(R) tryToTrans(lside: RE, x:S) : RE == if testTrig(lside,x) or testHTrig(lside,x) then convLside:=( simplify(lside) )::RE resultLside:=convLside listConvLside:=tableXkernels(convLside,x) if (#listConvLside) > 1 then NormConvLside:=normalize(convLside,x) NormConvLside:=( NormConvLside ) :: RE resultLside:=subsTan(NormConvLside , x) else if testLog(lside,x) then numlside:=numer(lside)::RE resultLside:=combineLog(numlside,x) else NormConvLside:=normalize(lside,x) NormConvLside:=( NormConvLside ) :: RE resultLside:=NormConvLside listConvLside:=tableXkernels(NormConvLside,x) if (#listConvLside) > 1 then cnormConvLside:=complexNormalize(lside,x) cnormConvLside:=cnormConvLside::RE resultLside:=cnormConvLside listcnorm:=tableXkernels(cnormConvLside,x) if (#listcnorm) > 1 then if testLog(cnormConvLside,x) then numlside:=numer(cnormConvLside)::RE resultLside:=combineLog(numlside,x) resultLside subsTan(exprvar:RE,y:S) : RE == Z:=new()@Symbol listofkern:=tableXkernels(exprvar,y) varkern:=(first listofkern)::RE Y:=(numer first argument first (kernels(varkern)))::RE test : Boolean := varkern=tan(((Y::RE)/(2::RE))::RE) if not( (#listofkern=1) and test) then return exprvar fZ:=eval(exprvar,varkern=(Z::RE)) fN:=(numer fZ)::RE f:=univariate(fN, first kernels(Z::RE)) secondfun:=(-2*(Y::RE)/((Y::RE)**2-1) )::RE g:=univariate(secondfun,first kernels(y::RE)) H:=(new()@Symbol)::RE newH:=univariate(H,first kernels(Z::RE)) result:=decomposeFunc(f,g,newH) if not ( result = f ) then result1:=result( H::RE ) resultnew:=eval(result1,H=(( tan((Y::RE))::RE ) )) else return exprvar eliminateKernRoot(var: RE, varkern: K) : RE == X:=new()@Symbol var1:=eval(var, (varkern::RE)=(X::RE) ) var2:=numer univariate(var1, first kernels(X::RE)) var3:= monomial(1, ( retract( second argument varkern)@I )::NNI)@SUP RE_ - monomial(first argument varkern, 0::NNI)@SUP RE resultvar:=resultant(var2, var3) eliminateRoot(var:RE, y:S) : RE == var1:=var while testRootk(var1,y) repeat varlistk1:=tableXkernels(var1,y) for i in varlistk1 repeat if is?(i, 'nthRoot) then var1:=eliminateKernRoot(var1,first kernels(i::RE)) var1 logsumtolog(var:RE) : RE == (listofexpr:=isPlus(var)) case "failed" => var listofexpr:= listofexpr ::L RE listforgcd:=[]::L R for i in listofexpr repeat exprcoeff:=leadingCoefficient(numer(i)) listforgcd:=cons(exprcoeff, listforgcd) gcdcoeff:=gcd(listforgcd)::RE newexpr:RE :=0 for i in listofexpr repeat exprlist:=splitExpr(i::RE) newexpr:=newexpr + logexpp(exprlist.2, exprlist.1/gcdcoeff) kernelofvar:=kernels(newexpr) var2:=1::RE for i in kernelofvar repeat var2:=var2*(first argument i) gcdcoeff * log(var2) testLog(expr:RE,Z:S) : Boolean == testList:=[log]::L S kernelofexpr:=tableXkernels(expr,Z) if #kernelofexpr = 0 then return false for i in kernelofexpr repeat if not member?(name operator (first kernels(i)),testList) or _ not testkernel( (first argument first kernels(i)) ,Z) then return false true splitExpr(expr:RE) : L RE == lcoeff:=leadingCoefficient((numer expr)) exprwcoeff:=expr listexpr:=isTimes(exprwcoeff) if listexpr case "failed" then [1::RE , expr] else listexpr:=remove!(lcoeff::RE , listexpr) cons(lcoeff::RE , listexpr) buildnexpr(expr:RE, Z:S) : L RE == nlist:=splitExpr(expr) n2list:=remove!(nlist.1, nlist) anscoeff:RE:=1 ansmant:RE:=0 for i in n2list repeat if freeOf?(i::RE,Z) then anscoeff:=(i::RE)*anscoeff else ansmant:=(i::RE) [anscoeff, ansmant * nlist.1 ] logexpp(expr1:RE, expr2:RE) : RE == log( (first argument first kernels(expr1))**expr2 ) combineLog(expr:RE,Y:S) : RE == exprtable:Table(RE,RE):=table() (isPlus(expr)) case "failed" => expr ans:RE:=0 while expr ~= 0 repeat loopexpr:RE:=leadingMonomial(numer(expr))::RE if testLog(loopexpr,Y) and (#tableXkernels(loopexpr,Y)=1) then exprr:=buildnexpr(loopexpr,Y) if search(exprr.1,exprtable) case "failed" then exprtable.(exprr.1):=0 exprtable.(exprr.1):= exprtable.(exprr.1) + exprr.2 else ans:=ans+loopexpr expr:=(reductum(numer expr))::RE ansexpr:RE:=0 for i in keys(exprtable) repeat ansexpr:=ansexpr + logsumtolog(exprtable.i) * (i::RE) ansexpr:=ansexpr + ans testRootk(varlistk:RE,y:S) : Boolean == testList:=[nthRoot]::L S kernelofeqnvar:=tableXkernels(varlistk,y) if #kernelofeqnvar = 0 then return false for i in kernelofeqnvar repeat if member?(name operator (first kernels(i)),testList) then return true false tableXkernels(evar:RE,Z:S) : L RE == kOfvar:=kernels(evar) listkOfvar:=[]::L RE for i in kOfvar repeat if not freeOf?(i::RE,Z) then listkOfvar:=cons(i::RE,listkOfvar) listkOfvar testTrig(eqnvar:RE,Z:S) : Boolean == testList:=[sin , cos , tan , cot , sec , csc]::L S kernelofeqnvar:=tableXkernels(eqnvar,Z) if #kernelofeqnvar = 0 then return false for i in kernelofeqnvar repeat if not member?(name operator (first kernels(i)),testList) or _ not testkernel( (first argument first kernels(i)) ,Z) then return false true testHTrig(eqnvar:RE,Z:S) : Boolean == testList:=[sinh , cosh , tanh , coth , sech , csch]::L S kernelofeqnvar:=tableXkernels(eqnvar,Z) if #kernelofeqnvar = 0 then return false for i in kernelofeqnvar repeat if not member?(name operator (first kernels(i)),testList) or _ not testkernel( (first argument first kernels(i)) ,Z) then return false true -- Auxiliary local function for use in funcinv. makeInterval(l:R):C INT F == if R has complex and R has ConvertibleTo(C F) then map(interval$INT(F),convert(l)$R)$ComplexFunctions2(F,INT F) else error "This should never happen" funcinv(k:RE,l:RE) : Union(RE,"failed") == is?(k,'sin) => asin(l) is?(k,'cos) => acos(l) is?(k,'tan) => atan(l) is?(k,'cot) => acot(l) is?(k,'sec) => l = 0 => "failed" asec(l) is?(k,'csc) => l = 0 => "failed" acsc(l) is?(k,'sinh) => asinh(l) is?(k,'cosh) => acosh(l) is?(k,'tanh) => atanh(l) is?(k,'coth) => acoth(l) is?(k,'sech) => asech(l) is?(k,'csch) => acsch(l) is?(k,'atan) => tan(l) is?(k,'acot) => l = 0 => "failed" cot(l) is?(k,'asin) => sin(l) is?(k,'acos) => cos(l) is?(k,'asec) => sec(l) is?(k,'acsc) => l = 0 => "failed" csc(l) is?(k,'asinh) => sinh(l) is?(k,'acosh) => cosh(l) is?(k,'atanh) => tanh(l) is?(k,'acoth) => l = 0 => "failed" coth(l) is?(k,'asech) => sech(l) is?(k,'acsch) => l = 0 => "failed" csch(l) is?(k,'exp) => l = 0 => "failed" simplifyingLog l is?(k,'log) => if R has complex and R has ConvertibleTo(C F) then -- We will check to see if the imaginary part lies in [-Pi,Pi) ze : Expression C INT F ze := map(makeInterval,l)$ExpressionFunctions2(R,C INT F) z : Union(C INT F,"failed") := retractIfCan ze z case "failed" => exp l im := imag z fpi : Float := pi() (-fpi < inf(im)) and (sup(im) <= fpi) => exp l "failed" else -- R not Complex or something which doesn't map to Complex Floats exp l is?(k,'%power) => (t:=normalize(l)) = 0 => "failed" log t l import SystemSolvePackage(RE) ker2Poly(k:Kernel RE, lvar:L S):Polynomial RE == member?(nm:=name operator k, lvar) => nm :: Polynomial RE k :: RE :: Polynomial RE smp2Poly(pol:SMP(R,Kernel RE), lvar:L S):Polynomial RE == map(ker2Poly(#1, lvar), #1::RE::Polynomial RE, pol)$PolynomialCategoryLifting( IndexedExponents Kernel RE, Kernel RE, R, SMP(R, Kernel RE), Polynomial RE) makeFracPoly(expr:RE, lvar:L S):Fraction Polynomial RE == smp2Poly(numer expr, lvar) / smp2Poly(denom expr, lvar) makeREpol(pol:Polynomial RE):RE == lvar := variables pol lval : List RE := [v::RE for v in lvar] ground eval(pol,lvar,lval) makeRE(frac:Fraction Polynomial RE):RE == makeREpol(numer frac)/makeREpol(denom frac) solve1Pol(pol:Polynomial RE, var: S, sol:L EQ RE):L L EQ RE == repol := eval(makeREpol pol, sol) vsols := solve(repol, var) [cons(vsol, sol) for vsol in vsols] solve1Sys(plist:L Polynomial RE, lvar:L S):L L EQ RE == rplist := reverse plist rlvar := reverse lvar sols : L L EQ RE := list(empty()) for p in rplist for v in rlvar repeat sols := "append"/[solve1Pol(p,v,sol) for sol in sols] sols @ The input \begin{verbatim} solve(sinh(z)=cosh(z),z) \end{verbatim} generates the error (reported as bug \# 102): \begin{verbatim} >> Error detected within library code: No identity element for reduce of empty list using operation append \end{verbatim} <>= solveList(lexpr:L RE, lvar:L S):L L EQ RE == ans1 := solveRetract(lexpr, lvar) not(ans1 case "failed") => ans1 :: L L EQ RE lfrac:L Fraction Polynomial RE := [makeFracPoly(expr, lvar) for expr in lexpr] trianglist := triangularSystems(lfrac, lvar) -- "append"/[solve1Sys(plist, lvar) for plist in trianglist] l: L L L EQ RE := [solve1Sys(plist, lvar) for plist in trianglist] reduce(append, l, []) solve(leqs:L EQ RE, lvar:L S):L L EQ RE == lexpr:L RE := [lhs(eq)-rhs(eq) for eq in leqs] solveList(lexpr, lvar) -- solve(leqs:L EQ RE, lker:L Kernel RE):L L EQ RE == -- lexpr:L RE := [lhs(eq)-rhs(eq) for eq in leqs] -- lvar :L S := [new()$S for k in lker] -- lval :L RE := [kernel v for v in lvar] -- nlexpr := [eval(expr,lker,lval) for expr in lexpr] -- ans := solveList(nlexpr, lvar) -- lker2 :L Kernel RE := [v::Kernel(RE) for v in lvar] -- lval2 := [k::RE for k in lker] -- [[map(eval(#1,lker2,lval2), neq) for neq in sol] for sol in ans] @ \section{package SOLVESER TransSolvePackageService} <>= )abbrev package SOLVESER TransSolvePackageService ++ Author: W. Wiwianka ++ Date Created: Summer 1991 ++ Change History: 9/91 ++ Basic Operations: decomposeFunc, unvectorise ++ Related Constructors: ++ Keywords: ++ Description: This package finds the function func3 where func1 and func2 ++ are given and func1 = func3(func2) . If there is no solution then ++ function func1 will be returned. ++ An example would be \spad{func1:= 8*X**3+32*X**2-14*X ::EXPR INT} and ++ \spad{func2:=2*X ::EXPR INT} convert them via univariate ++ to FRAC SUP EXPR INT and then the solution is \spad{func3:=X**3+X**2-X} ++ of type FRAC SUP EXPR INT TransSolvePackageService(R) : Exports == Implementation where R : IntegralDomain RE ==> Expression R EQ ==> Equation S ==> Symbol V ==> Variable L ==> List SUP ==> SparseUnivariatePolynomial ACF ==> AlgebraicallyClosedField() Exports == with decomposeFunc : ( Fraction SUP RE , Fraction SUP RE, Fraction SUP RE ) -> Fraction SUP RE ++ decomposeFunc(func1, func2, newvar) returns a function func3 where ++ func1 = func3(func2) and expresses it in the new variable newvar. ++ If there is no solution then func1 will be returned. unvectorise : ( Vector RE , Fraction SUP RE , Integer ) -> Fraction SUP RE ++ unvectorise(vect, var, n) returns ++ \spad{vect(1) + vect(2)*var + ... + vect(n+1)*var**(n)} where ++ vect is the vector of the coefficients of the polynomail , var ++ the new variable and n the degree. Implementation == add import ACF import TranscendentalManipulations(R, RE) import ElementaryFunctionStructurePackage(R, RE) import SparseUnivariatePolynomial(R) import LinearSystemMatrixPackage(RE,Vector RE,Vector RE,Matrix RE) import HomogeneousAggregate(R) ---- Local Function Declarations ---- subsSolve : ( SUP RE, NonNegativeInteger, SUP RE, SUP RE, Integer, Fraction SUP RE) -> Union(SUP RE , "failed" ) --++ subsSolve(f, degf, g1, g2, m, h) -- exported functions unvectorise(vect:Vector RE, var:Fraction SUP RE,n:Integer) : Fraction SUP RE == Z:=new()@Symbol polyvar: Fraction SUP RE :=0 for i in 1..((n+1)::Integer) repeat vecti:=univariate(vect( i ),first kernels(Z::RE)) polyvar:=polyvar + ( vecti )*( var )**( (n-i+1)::NonNegativeInteger ) polyvar decomposeFunc(exprf:Fraction SUP RE , exprg:Fraction SUP RE, newH:Fraction SUP RE ) : Fraction SUP RE == X:=new()@Symbol f1:=numer(exprf) f2:=denom(exprf) g1:=numer(exprg) g2:=denom(exprg) degF:=max(degree(numer(exprf)),degree(denom(exprf))) degG:=max(degree(g1),degree(g2)) newF1,newF2 : Union(SUP RE, "failed") N:= degF exquo degG if not ( N case "failed" ) then m:=N::Integer newF1:=subsSolve(f1,degF,g1,g2,m,newH) if f2 = 1 then newF2:= 1 :: SUP RE else newF2:=subsSolve(f2,degF,g1,g2,m,newH) if ( not ( newF1 case "failed" ) ) and ( not ( newF2 case "failed" ) ) then newF:=newF1/newF2 else return exprf else return exprf -- local functions subsSolve(F:SUP RE, DegF:NonNegativeInteger, G1:SUP RE, G2:SUP RE, M:Integer, HH: Fraction SUP RE) : Union(SUP RE , "failed" ) == coeffmat:=new((DegF+1),1,0)@Matrix RE for i in 0..M repeat coeffmat:=horizConcat(coeffmat, (vectorise( ( ( G1**((M-i)::NonNegativeInteger) )*G2**i ), (DegF+1) )::Matrix RE) ) vec:= vectorise(F,DegF+1) coeffma:=subMatrix(coeffmat,1,(DegF+1),2,(M+2)) solvar:=solve(coeffma,vec) if not ( solvar.particular case "failed" ) then solvevarlist:=(solvar.particular)::Vector RE resul:= numer(unvectorise(solvevarlist,( HH ),M)) resul else return "failed" @ \section{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. @ <<*>>= <> <> <> @ \eject \begin{thebibliography}{99} \bibitem{1} nothing \end{thebibliography} \end{document}