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[Maxima-commits] CVS: maxima/doc/info/pt_BR contrib_ode.texi, 1.1, 1.2 grobner.texi, 1.1, 1.2 maxima.texi, 1.48, 1.49
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From: Jorge B. <fic...@us...> - 2007-06-10 19:06:03
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Update of /cvsroot/maxima/maxima/doc/info/pt_BR In directory sc8-pr-cvs16.sourceforge.net:/tmp/cvs-serv27129 Modified Files: contrib_ode.texi grobner.texi maxima.texi Log Message: Brazilian Portuguese translation. contrib_ode.texi -> first version translated (English cvs version 1.5) grobner.texi -> first version translated (English cvs version 1.3) maxima.texi -> reflecting the chages above. Index: contrib_ode.texi =================================================================== RCS file: /cvsroot/maxima/maxima/doc/info/pt_BR/contrib_ode.texi,v retrieving revision 1.1 retrieving revision 1.2 diff -u -d -r1.1 -r1.2 --- contrib_ode.texi 9 Jun 2007 14:47:43 -0000 1.1 +++ contrib_ode.texi 10 Jun 2007 19:05:56 -0000 1.2 @@ -1,59 +1,61 @@ +@c Language: Brazilian Portuguese, Encoding: iso-8859-1 +@c /contrib_ode.texi/1.5/Sat Jun 2 00:13:11 2007// @menu -* Introduction to contrib_ode:: -* Functions and Variables for contrib_ode:: -* Possible improvements to contrib_ode:: -* Test cases for contrib_ode:: -* References for contrib_ode:: +* Introdu@value{cedilha}@~{a}o a contrib_ode:: +* Fun@value{cedilha}@~{o}es e Vari@'{a}veis Definidas para contrib_ode:: +* Possibilidades de melhorias em contrib_ode:: +* Casos de teste para contrib_ode:: +* Refer@^{e}ncias bibliogr@'{a}ficas para contrib_ode:: @end menu -@node Introduction to contrib_ode, Functions and Variables for contrib_ode, contrib_ode, contrib_ode +@node Introdu@value{cedilha}@~{a}o a contrib_ode, Fun@value{cedilha}@~{o}es e Vari@'{a}veis Definidas para contrib_ode, contrib_ode, contrib_ode -@section Introduction to contrib_ode +@section Introdu@value{cedilha}@~{a}o a contrib_ode -MAXIMA's ordinary differential equation (ODE) solver @code{ode2} solves -elementary linear ODEs of first and second order. The function -@code{contrib_ode} extends @code{ode2} with additional methods for linear -and non-linear first order ODEs and linear homogeneous second order ODEs. -The code is still under development and the calling sequence may change -in future releases. Once the code has stabilized it may be -moved from the contib directory and integrated into MAXIMA. +O resolvedor de equa@value{cedilha}@~{o}es diferenciais ordin@'{a}rias (EDO) do MAXIMA, o @code{ode2}, resolve +EDO's elementares de primeira e segunda ordem. A fun@value{cedilha}@~{a}o +@code{contrib_ode} extende @code{ode2} com m@'{e}todos adicionais para EDO's lineares +e EDO's n@~{a}o lineares de primeira ordem e EDO's lineares homog@^{e}neas de segunda ordem. +O c@'{o}digo est@'{a} ainda em desenvolvimemto e a seq@"{u}@^{e}ncia de chamada da fun@value{cedilha}@~{a}o pode mudar +em futuras vers@~{o}es. Uma vez que o c@'{o}digo estiver estabilizado essa fun@value{cedilha}@~{a}o pode ser +movida do diret@'{o}rio contrib e integrada dentro do MAXIMA. -This package must be loaded with the command @code{load('contrib_ode)} -before use. +Esse pacote deve torna-se dispon@'{i}vel para uso com o comando @code{load('contrib_ode)} +em primeiro lugar. -The calling convention for @code{contrib_ode} is identical to @code{ode2}. -It takes -three arguments: an ODE (only the left hand side need be given if the -right hand side is 0), the dependent variable, and the independent -variable. When successful, it returns a list of solutions. +A conven@value{cedilha}@~{a}o de chamada para @code{contrib_ode} @'{e} id@^{e}ntica a @code{ode2}. +Toma +tr@^{e}s argumentos: uma EDO (somente o lado esquerdo precisa ser fornecido se o +lado direito for 0), a vari@'{a}vel dependente, e a vari@'{a}vel +independente. Quando @code{contrib_ode} obtiver sucesso, retorna uma lista de solu@value{cedilha}@~{o}es. -The form of the solution differs from @code{ode2}. -As non-linear equations can have multiple solutions, -@code{contrib_ode} returns a list of solutions. Each solution can -have a number of forms: +A forma de retorno da lista de solu@value{cedilha}@~{a}o difere de @code{ode2}. +Como equa@value{cedilha}@~{o}es n@~{a}o lineares podem ter m@'{u}ltiplas solu@value{cedilha}@~{o}es, +@code{contrib_ode} retorna uma lista de solu@value{cedilha}@~{o}es. Cada solu@value{cedilha}@~{a}o pode +ter v@'{a}rias formas: @itemize @bullet @item -an explicit solution for the dependent variable, +uma solu@value{cedilha}@~{a}o expl@'{i}cita para a vari@'{a}vel dependente, @item -an implicit solution for the dependent variable, +uma solu@value{cedilha}@~{a}o impl@'{i}cita para a vari@'{a}vel dependente, @item -a parametric solution in terms of variable %t, or +uma solu@value{cedilha}@~{a}o param@'{e}trica em termos de vari@'{a}vel %t, ou @item -a tranformation into another ODE in variable %u. +uma transfrma@value{cedilha}@~{a}o em outra EDO na vari@'{a}vel %u. @end itemize -@code{%c} is used to represent the constant of integration for first order equations. -@code{%k1} and @code{%k2} are the constants for second order equations. -If contrib_ode -cannot obtain a solution for whatever reason, it returns false, after -perhaps printing out an error message. +@code{%c} @'{e} usado para representar a constante de integra@value{cedilha}@~{a}o para equa@value{cedilha}@~{o}es de primeira ordem. +@code{%k1} e @code{%k2} s@~{a}o constantes para equa@value{cedilha}@~{o}es de segunda ordem. +Se @code{contrib_ode} +n@~{a}o puder obter uma solu@value{cedilha}@~{a}o por qualquer raz@~{a}o, @code{false} @'{e} retornado, ap@'{o}s +talvez mostrar uma mensagem de erro. -It is necessary to return a list of solutions, as even first order -non-linear ODEs can have multiple solutions. For example: +Isso @'{e} necess@'{a}rio para retornar uma lista de solu@value{cedilha}@~{o}es, como mesmo EDO's de primeira +ordem n@~{a}o lineares podem ter solu@value{cedilha}@~{o}es multiplas. Por exemplo: @c ===beg=== @c load('contrib_ode)$ @@ -78,8 +80,8 @@ (%o4) factor @end example -Nonlinear odes can have singular solutions without constants of -integration, as in the second solution of the following example: +EDO's n@~{a}o lineares podem ter solu@value{cedilha}@~{o}es singulares sem constantes de +integra@value{cedilha}@~{a}o, como na segunda solu@value{cedilha}@~{a}o do seguinte exemplo: @c ===beg=== @c load('contrib_ode)$ @@ -107,9 +109,9 @@ @end example -The following ode has two parametric solutions in terms of the dummy -variable %t. In this case the parametric solutions can be manipulated -to give explicit solutions. +A seguinte EDO possui duas solu@value{cedilha}@~{o}es param@'{e}tricas em termos da vari@'{a}vel +fict@'{i}cia %t. Nesse caso as solu@value{cedilha}@~{o}es param@'{e}tricaspodem ser manipuladas +para fornecer solu@value{cedilha}@~{o}es expl@'{i}citas. @c ===beg=== @c load('contrib_ode)$ @@ -134,7 +136,7 @@ (%o4) lagrange @end example -The following example (Kamke 1.112) demonstrates an implicit solution. +O seguinte exemplo (Kamke 1.112) demonstra uma solu@value{cedilha}@~{a}o impl@'{i}cita. @c ===beg=== @c load('contrib_ode)$ @@ -166,9 +168,9 @@ -The following Riccati equation is transformed into a linear -second order ODE in the variable %u. MAXIMA is unable to -solve the new ODE, so it is returned unevaluated. +A seguinte equa@value{cedilha}@~{a}o de Riccati @'{e} transformada em uma EDO linear +de segunda ordem na vari@'{a}vel %u. MAXIMA n@~{a}o est@'{a} apto a +resolver a nova EDO, de forma que essa nova EDO @'{e} retornada sem avalia@value{cedilha}@~{a}o. @c ===beg=== @c load('contrib_ode)$ @c eqn:x^2*'diff(y,x)=a+b*x^n+c*x^2*y^2; @@ -197,37 +199,37 @@ @end example -For first order ODEs contrib_ode calls @code{ode2}. It then tries the -following methods: factorization, Clairault, Lagrange, Riccati, -Abel and Lie symmetry methods. The Lie method is not attempted -on Abel equations if the Abel method fails, but it is tried -if the Riccati method returns an unsolved second order ODE. +Para EDO's de primeira ordem @code{contrib_ode} chama @code{ode2}. @code{ode2} tenta ent@~{a}o os +seguintes m@'{e}todos: fatora@value{cedilha}@~{a}o, Clairault, Lagrange, Riccati, +Abel e os m@'{e}todos de simetria de Lie. O m@'{e}todo de Lie n@~{a}o @'{e} tentado +sobre equa@value{cedilha}@~{o}es de Abel se o m@'{e}todo de Abel falhar, mas @'{e} tendado +se o m@'{e}todo de Riccati uma EDO de segunda ordem n@~{a}o resolvida. -For second order ODEs contrib_ode calls @code{ode2} then @code{odelin}. +Para EDO's de segunda ordem @code{contrib_ode} chama @code{ode2} e em seguida @code{odelin}. -Extensive debugging traces and messages are displayed if the command -put('contrib_ode,true,'verbose) is executed. +Rastros extensivos de depura@value{cedilha}@~{a}o mensagens s@~{a}o mostradas se o comando +@code{put('contrib_ode,true,'verbose)} for executado. -@node Functions and Variables for contrib_ode, Possible improvements to contrib_ode, Introduction to contrib_ode, contrib_ode -@section Functions and Variables for contrib_ode +@node Fun@value{cedilha}@~{o}es e Vari@'{a}veis Definidas para contrib_ode, Possibilidades de melhorias em contrib_ode, Introdu@value{cedilha}@~{a}o a contrib_ode, contrib_ode +@section Fun@value{cedilha}@~{o}es e Vari@'{a}veis Definidas para contrib_ode -@deffn {Function} contrib_ode (@var{eqn}, @var{y}, @var{x}) +@deffn {Fun@value{cedilha}@~{a}o} contrib_ode (@var{eqn}, @var{y}, @var{x}) -Returns a list of solutions of the ODE @var{eqn} with -independent variable @var{x} and dependent variable @var{y}. +Retorna uma lista de solu@value{cedilha}@~{o}es da EDO @var{eqn} com +vari@'{a}vel independente @var{x} e vari@'{a}vel dependente @var{y}. @end deffn -@deffn {Function} odelin (@var{eqn}, @var{y}, @var{x}) +@deffn {Fun@value{cedilha}@~{a}o} odelin (@var{eqn}, @var{y}, @var{x}) -@code{odelin} solves linear homogeneous ODEs of first and -second order with -independent variable @var{x} and dependent variable @var{y}. -It returns a fundamental solution set of the ODE. +@code{odelin} resolve EDO's lineares homog@^{e}neas de primeira e +segunda ordem com +vari@'{a}vel independente @var{x} e vari@'{a}vel dependente @var{y}. +@code{odelin} retorna um conjunto solu@value{cedilha}@~{a}o fundamental da EDO. -For second order ODEs, @code{odelin} uses a method, due to Bronstein -and Lafaille, that searches for solutions in terms of given -special functions. +para EDO's de segunda ordem, @code{odelin} usa um m@'{e}todo, devido a Bronstein +e Lafaille, que busca por solu@value{cedilha}@~{o}es em termos de fun@value{cedilha}@~{o}es +especiais dadas. @c ===beg=== @c load('contrib_ode)$ @@ -258,11 +260,11 @@ @end deffn -@deffn {Function} ode_check (@var{eqn}, @var{soln}) +@deffn {Fun@value{cedilha}@~{a}o} ode_check (@var{eqn}, @var{sol}) -Returns the value of ODE @var{eqn} after substituting a -possible solution @var{soln}. The value is equivalent to -zero if @var{soln} is a solution of @var{eqn}. +Retorna o valor da EDO @var{eqn} ap@'{o}s substituir uma +poss@'{i}vel solu@value{cedilha}@~{a}o @var{sol}. O valor @'{e} igual a +zero se @var{sol} for uma solu@value{cedilha}@~{a}o of @var{eqn}. @c ===beg=== @c load('contrib_ode)$ @@ -299,155 +301,155 @@ @end deffn -@defvr {Global variable} @code{method} +@defvr {Vari@'{a}vel global} @code{method} -The variable @code{method} is set to the successful solution -method. +A vari@'{a}vel @code{method} @'{e} escolhida para o m@'{e}todo que resolver com sucesso +uma dada EDO. @end defvr -@defvr {variable} @code{%c} +@defvr {Vari@'{a}vel} @code{%c} -@code{%c} is the integration constant for first order ODEs +@code{%c} @'{e} a constante de integra@value{cedilha}@~{a}o para EDO's de primeira ordem. @end defvr -@defvr {variable} @code{%k1} +@defvr {Vari@'{a}vel} @code{%k1} -@code{%k1} is the first integration constant for second order ODEs. +@code{%k1} @'{e} a primeira constante de integra@value{cedilha}@~{a}o para EDO's de segunda ordem. @end defvr -@defvr {variable} @code{%k2} +@defvr {Vari@'{a}vel} @code{%k2} -@code{%k2} is the second integration constant for second order ODEs. +@code{%k2} @'{e} a segunda constante de integra@value{cedilha}@~{a}o para EDO's de segunda ordem. @end defvr -@deffn {Function} gauss_a (@var{a}, @var{b}, @var{c}, @var{x}) +@deffn {Fun@value{cedilha}@~{a}o} gauss_a (@var{a}, @var{b}, @var{c}, @var{x}) -@code{gauss_a(a,b,c,x)} and @code{gauss_b(a,b,c,x)} are 2F1 -geometric functions. They represent any two independent -solutions of the hypergeometric differential equation +@code{gauss_a(a,b,c,x)} e @code{gauss_b(a,b,c,x)} s@~{a}o fun@value{cedilha}@~{o}es +hipergeom@'{e}tricas 2F1. Elas represetnam quaisquer duas solu@value{cedilha}@~{o}es +independentes da equa@value{cedilha}@~{a}o diferencial hipergeom@'{e}trica @code{x(1-x) diff(y,x,2) + [c-(a+b+1)x diff(y,x) - aby = 0} (A&S 15.5.1). -The only use of these functions is in solutions of ODEs returned by -@code{odelin} and @code{contrib_ode}. The definition and use of these -functions may change in future releases of maxima. +O @'{u}nico uso dessas fun@value{cedilha}@~{o}es @'{e} em solu@value{cedilha}@~{o}es de EDO's retornadas por +@code{odelin} e @code{contrib_ode}. A defini@value{cedilha}@~{a}o e o uso dessas +fun@value{cedilha}@~{o}es pode mudar em futuras vers@~{o}es do maxima. -See also @code{gauss_b}, @code{dgauss_a} and @code{gauss_b}. +Veja tamb@'{e}m @code{gauss_b}, @code{dgauss_a} e @code{gauss_b}. @end deffn -@deffn {Function} gauss_b (@var{a}, @var{b}, @var{c}, @var{x}) -See @code{gauss_a}. +@deffn {Fun@value{cedilha}@~{a}o} gauss_b (@var{a}, @var{b}, @var{c}, @var{x}) +Veja @code{gauss_a}. @end deffn -@deffn {Function} dgauss_a (@var{a}, @var{b}, @var{c}, @var{x}) -The derivative with respect to x of @code{gauss_a(a,b,c,x)}. +@deffn {Fun@value{cedilha}@~{a}o} dgauss_a (@var{a}, @var{b}, @var{c}, @var{x}) +A derivada em rela@value{cedilha}@~{a}o a x de @code{gauss_a(a,b,c,x)}. @end deffn -@deffn {Function} dgauss_b (@var{a}, @var{b}, @var{c}, @var{x}) -The derivative with respect to x of @code{gauss_b(a,b,c,x)}. +@deffn {Fun@value{cedilha}@~{a}o} dgauss_b (@var{a}, @var{b}, @var{c}, @var{x}) +A derivada em rela@value{cedilha}@~{a}o a x de @code{gauss_b(a,b,c,x)}. @end deffn -@deffn {Function} kummer_m (@var{a}, @var{b}, @var{x}) +@deffn {Fun@value{cedilha}@~{a}o} kummer_m (@var{a}, @var{b}, @var{x}) -Kummer's M function, as defined in Abramowitz and Stegun, +A fun@value{cedilha}@~{a}o M de Kummer, como definida em Abramowitz e Stegun, @i{Handbook of Mathematical Functions}, Section 13.1.2. -The only use of this function is in solutions of ODEs returned by -@code{odelin} and @code{contrib_ode}. The definition and use of this -function may change in future releases of maxima. +O @'{u}nico uso dessas fun@value{cedilha}@~{o}es @'{e} em solu@value{cedilha}@~{o}es de EDO's retornadas por +@code{odelin} e @code{contrib_ode}. A defini@value{cedilha}@~{a}o e o uso dessas +fun@value{cedilha}@~{o}es pode mudar em futuras vers@~{o}es do maxima. -See also @code{kummer_u}, @code{dkummer_m} and @code{dkummer_u}. +Veja tamb@'{e}m @code{kummer_u}, @code{dkummer_m} e @code{dkummer_u}. @end deffn -@deffn {Function} kummer_u (@var{a}, @var{b}, @var{x}) +@deffn {Fun@value{cedilha}@~{a}o} kummer_u (@var{a}, @var{b}, @var{x}) -Kummer's U function, as defined in Abramowitz and Stegun, +A fun@value{cedilha}@~{a}o U de Kummer, como definida em Abramowitz e Stegun, @i{Handbook of Mathematical Functions}, Section 13.1.3. -See @code{kummer_m}. +Veja @code{kummer_m}. @end deffn -@deffn {Function} dkummer_m (@var{a}, @var{b}, @var{x}) -The derivative with respect to x of @code{kummer_m(a,b,x)}. +@deffn {Fun@value{cedilha}@~{a}o} dkummer_m (@var{a}, @var{b}, @var{x}) +A derivada com rela@value{cedilha}@~{a}o a x de @code{kummer_m(a,b,x)}. @end deffn -@deffn {Function} dkummer_u (@var{a}, @var{b}, @var{x}) -The derivative with respect to x of @code{kummer_u(a,b,x)}. +@deffn {Fun@value{cedilha}@~{a}o} dkummer_u (@var{a}, @var{b}, @var{x}) +A derivada com rela@value{cedilha}@~{a}o a x de @code{kummer_u(a,b,x)}. @end deffn -@node Possible improvements to contrib_ode, Test cases for contrib_ode, Functions and Variables for contrib_ode, contrib_ode -@section Possible improvements to contrib_ode +@node Possibilidades de melhorias em contrib_ode, Casos de teste para contrib_ode, Fun@value{cedilha}@~{o}es e Vari@'{a}veis Definidas para contrib_ode, contrib_ode +@section Possibilidades de melhorias em contrib_ode -These routines are work in progress. I still need to: +Essas rotinas aida est@~{a}o sendo aperfei@value{cedilha}oadas. @'{E} necess@'{a}rio ainda: @itemize @bullet @item -Extend the FACTOR method @code{ode1_factor} to work for multiple roots. +Extender o m@'{e}todo FACTOR @code{ode1_factor} para trabalhar com ra@'{i}zes multiplas. @item -Extend the FACTOR method @code{ode1_factor} to attempt to solve higher - order factors. At present it only attemps to solve linear factors. +Extender o m@'{e}todo FACTOR @code{ode1_factor} para tentar resolver fatores + de mais alta ordem. Atualmente somente tenta resolver fatores lineares. @item -Fix the LAGRANGE routine @code{ode1_lagrange} to prefer real roots over - complex roots. +Corrigir a rotina de LAGRANGE @code{ode1_lagrange} para preferira@'{i}zes reais a + ra@'{i}zes complexas. @item -Add additional methods for Riccati equations. +Aumentar a quantidade de m@'{e}todos adicionais para equa@value{cedilha}@~{o}es de Riccati. @item -Improve the detection of Abel equations of second kind. The exisiting - pattern matching is weak. +Melhorar a detec@value{cedilha}@~{a}o de equa@value{cedilha}@~{o}es de Abel do segundo tipo. O modelo + existente de coincid@^{e}ncia @'{e} fraco. @item -Work on the Lie symmetry group routine @code{ode1_lie}. There are quite a - few problems with it: some parts are unimplemented; some test cases - seem to run forever; other test cases crash; yet others return very - complex "solutions". I wonder if it really ready for release yet. +Trabalho sobre a rotina do grupo de simetria de Lie @code{ode1_lie}. Existem poucos por@'{e}m + grandes problemas com essa rotina: algumas partes precisam de implementa@value{cedilha}@~{a}o; alguns casos de teste + parecem executar indefinidamente; outros casos de teste abortam inesplicavelmente; outros ainda retorna "solu@value{cedilha}@~{o}es" + muito complexas. Seria surpreendente se estivesse pronto para se liberar uma vers@~{a}o est@'{a}vel. @item -Add more test cases. +Adicionar mais casos de teste. @end itemize -@node Test cases for contrib_ode, References for contrib_ode, Possible improvements to contrib_ode, contrib_ode -@section Test cases for contrib_ode +@node Casos de teste para contrib_ode, Refer@^{e}ncias bibliogr@'{a}ficas para contrib_ode, Possibilidades de melhorias em contrib_ode, contrib_ode +@section Casos de teste para contrib_ode -The routines have been tested on a approximately one thousand test cases -from Murphy, -Kamke, Zwillinger and elsewhere. These are included in the tests subdirectory. +Asrotinas foram tesadas sobre aproximadamente mil casos de teste +por Murphy, +Kamke, Zwillinger e outros. Esses testes est@~{a}o inclu@'{i}dos no subdiret@'{o}rio de testes. @itemize @bullet @item -The Clairault routine @code{ode1_clairault} finds all known solutions, - including singular solutions, of the Clairault equations in Murphy and +A rotina de Clairault @code{ode1_clairault} encontra todas as solu@value{cedilha}@~{o}es conhecidas, + incluindo solu@value{cedilha}@~{o}es singulares, das equa@value{cedilha}@~{o}es de Clairault em Murphy e Kamke. @item -The other routines often return a single solution when multiple - solutions exist. +As outras rotinas muitas vezes retornam uma solu@value{cedilha}@~{a}o simples quando existem + multiplas solu@value{cedilha}@~{o}es. @item -Some of the "solutions" from ode1_lie are overly complex and - impossible to check. +Algumas das "solu@value{cedilha}@~{o}es" de @code{ode1_lie} s@~{a}o extremamente complexas e + imposs@'{i}veis de verificar. @item -There are some crashes. +Existe algumas interrup@value{cedilha}@~{o}es inexplic@'{a}vies de execu@value{cedilha}@~{a}o. @end itemize -@node References for contrib_ode, ,Test cases for contrib_ode, contrib_ode -@section References for contrib_ode +@node Refer@^{e}ncias bibliogr@'{a}ficas para contrib_ode, ,Casos de teste para contrib_ode, contrib_ode +@section Refer@^{e}ncias bibliogr@'{a}ficas para contrib_ode @enumerate @@ -473,11 +475,11 @@ @item E. S. Cheb-Terrab, A. D. Roche, Symmetries and First Order - ODE Patterns, Computer Physics Communications 113 (1998), p 239. + EDO Patterns, Computer Physics Communications 113 (1998), p 239. (http://lie.uwaterloo.ca/papers/ode_vii.pdf) @item -E. S. Cheb-Terrab, T. Koloknikov, First Order ODEs, +E. S. Cheb-Terrab, T. Koloknikov, First Order EDO's, Symmetries and Linear Transformations, European Journal of Applied Mathematics, Vol. 14, No. 2, pp. 231-246 (2003). (http://arxiv.org/abs/math-ph/0007023) @@ -489,7 +491,7 @@ @item M Bronstein, S Lafaille, -Solutions of linear ordinary differential equations in terms +Solutions of linear ordinary equa@value{cedilha}@~{o}es diferenciais in terms of special functions, Proceedings of ISSAC 2002, Lille, ACM Press, 23-28. (http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac2002.pdf) Index: grobner.texi =================================================================== RCS file: /cvsroot/maxima/maxima/doc/info/pt_BR/grobner.texi,v retrieving revision 1.1 retrieving revision 1.2 diff -u -d -r1.1 -r1.2 --- grobner.texi 9 Jun 2007 14:47:43 -0000 1.1 +++ grobner.texi 10 Jun 2007 19:05:56 -0000 1.2 @@ -1,45 +1,48 @@ +/grobner.texi/1.3/Sat Jun 2 00:13:21 2007// +@c Language: Brazilian Portuguese, Encoding: iso-8859-1 +@c /grobner.texi/1.3/Sat Jun 2 00:13:21 2007// @menu -* Introduction to grobner :: -* Functions and Variables for grobner :: +* Introdu@value{cedilha}@~{a}o a grobner:: +* Fun@value{cedilha}@~{o}es e Vari@'{a}veis Definidas para grobner:: @end menu -@node Introduction to grobner, Functions and Variables for grobner, Top, Top -@section Introduction to grobner +@node Introdu@value{cedilha}@~{a}o a grobner, Fun@value{cedilha}@~{o}es e Vari@'{a}veis Definidas para grobner, Top, Top +@section Introdu@value{cedilha}@~{a}o a grobner -@code{grobner} is a package for working with Groebner bases in Maxima. +@code{grobner} @'{e} um pacote para trabalhos com bases de Groebner no Maxima. @noindent -A tutorial on @emph{Groebner Bases} can be found at +Um tutorial sobre @emph{Bases de Groebner} pode ser encontrado em @noindent @url{http://www.geocities.com/CapeCanaveral/Hall/3131/} @noindent -To use the following functions you must load the @file{grobner.lisp} package. +Para usar as seguintes fun@value{cedilha}@~{o}es voc@^{e} deve primeiramente tornar o pacote @file{grobner.lisp} dispon@'{i}vel para uso: @example load(grobner); @end example @noindent -A demo can be started by +Uma demonstra@value{cedilha}@~{a}o de uso pode ser iniciada com @example demo("grobner.demo"); @end example @noindent -or +ou com @example batch("grobner.demo") @end example @noindent -Some of the calculation in the demo will take a lot of time -therefore the output @file{grobner-demo.output} of the demo can -be found in the same directory as the demo file. +Alguns dos c@'{a}lculos no arquivo de demonstra@value{cedilha}@~{a}o ir@~{a}o tomar um pouco de tempo +portanto a sa@'{i}da @file{grobner-demo.output} do arquivo de demonstra@value{cedilha}@~{a}o pode +ser encontrada no mesmo diret@'{o}rio que o arquivo de demonstra@value{cedilha}@~{a}o. -@subsection Notes on the grobner package -The package was written by +@subsection Notas sobre o pacote grobner +O pacote foi escrito por @noindent Marek Rychlik @@ -48,16 +51,16 @@ @url{http://alamos.math.arizona.edu} @noindent -and is released 2002-05-24 under the terms of the General Public License(GPL) (see file @file{grobner.lisp}. -This documentation was extracted from the files +e foi liberado em 24/05/2002 nos termos da Licen@value{cedilha}a P@'{u}blica Geral (GPL/GNU/FSF) (veja o arquivo @file{grobner.lisp}. +Essa documenta@value{cedilha}@~{a}o foi extra@'{i}da dos arquivos @flushleft @file{README}, @file{grobner.lisp}, @file{grobner.demo}, @file{grobner-demo.output} @end flushleft @noindent -by G@"unter Nowak. Suggestions for improvement of the documentation can -be discussed at the @emph{maxima}-mailing-list @email{maxima@@math.utexas.edu}. -The code is a little bit out of date now. Modern implementation use the fast @emph{F4} algorithm described in +por G@"unter Nowak. Sugest@~{o}es de melhorias da documenta@value{cedilha}@~{a}o podem +ser discutidas em @emph{maxima}-mailing-list @email{maxima@@math.utexas.edu}. +O c@'{o}digo est@'{a} um pouco desatualizado atualmente. Implementa@value{cedilha}@~{o}es modernas utilizam o algor@'{i}tmo r@'{a}pido @emph{F4} descrito em @smallformat A new efficient algorithm for computing Gr@"obner bases (F4) Jean-Charles Faug@`ere @@ -65,114 +68,114 @@ January 20, 1999 @end smallformat -@subsection Implementations of admissible monomial orders in grobner +@subsection Implementa@value{cedilha}@~{o}es de ordem monomial admiss@'{i}vel em grobner @itemize @bullet @item @code{lex} -pure lexicographic, -default order for monomial comparisons +puramente lexicogr@'{a}fica, +ordena@value{cedilha}@~{a}o padr@~{a}o para compara@value{cedilha}@~{o}es monomiais @item @code{grlex} -total degree order, ties broken by lexicographic +ordena@value{cedilha}@~{a}o total de grau, quando houver empate @'{e} quebrada pela ordem lexicogr@'{a}fica @item @code{grevlex} -total degree, ties broken by reverse lexicographic +grau total, quando houver empate @'{e} quebrada pela ordem lexicogr@'{a}fica reversa @item @code{invlex} -inverse lexicographic order +ordena@value{cedilha}@~{a}o lexicogr@'{a}fica reversa @end itemize -@node Functions and Variables for grobner, , Introduction to grobner, Top -@section Functions and Variables for grobner +@node Fun@value{cedilha}@~{o}es e Vari@'{a}veis Definidas para grobner, , Introdu@value{cedilha}@~{a}o a grobner, Top +@section Fun@value{cedilha}@~{o}es e Vari@'{a}veis Definidas para grobner -@subsection Global switches for grobner +@subsection Comutadores globais para grobner -@defvr {Option variable} poly_monomial_order -Default value: @code{lex} +@defvr {Vari@'{a}vel de op@value{cedilha}@~ao} poly_monomial_order +Valor padr@~{a}o: @code{lex} -This global switch controls which monomial order is used in polynomial and Groebner Bases calculations. If not set, @code{lex} will be used. +Esse comutador globalcontrola qual a ordena@value{cedilha}@~{a}o monomial @'{e} usada em polinomio e em c@'{a}lculos com Bases de Groebner. Se n@~{a}o for escolhidat, @code{lex} ser@'{a} usada. @end defvr -@defvr {Option variable} poly_coefficient_ring -Default value: @code{expression_ring} +@defvr {Vari@'{a}vel de op@value{cedilha}@~ao} poly_coefficient_ring +Valor padr@~{a}o: @code{expression_ring} -This switch indicates the coefficient ring of the polynomials that -will be used in grobner calculations. If not set, @emph{maxima's} general -expression ring will be used. This variable may be set to -@code{ring_of_integers} if desired. +Esse comutador indica o anel de coeficiente dos polin@^{o}mios que +ir@'{a} ser usado em c@'{a}lculos de grobner. Se n@~{a}o for escolhido, o anel de express@~{a}o +geral do @emph{maxima's} ir@'{a} ser usado. Essa vari@'{a}vel pode ser escolhida para +@code{ring_of_integers} se for desejado. @end defvr -@defvr {Option variable} poly_primary_elimination_order -Default value: @code{false} +@defvr {Vari@'{a}vel de op@value{cedilha}@~ao} poly_primary_elimination_order +Valor padr@~{a}o: @code{false} -Name of the default order for eliminated variables in -elimination-based functions. If not set, @code{lex} will be used. +Nome da ordem padr@~{a}o de elimina@value{cedilha}@~{a}o de vari@'{a}veis em +fun@value{cedilha}@~{o}es de elimina@value{cedilha}@~{a}o. Se n@~{a}o for escolhida, @code{lex} ir@'{a} ser usada. @end defvr -@defvr {Option variable} poly_secondary_elimination_order -Default value: @code{false} +@defvr {Vari@'{a}vel de op@value{cedilha}@~ao} poly_secondary_elimination_order +Valor padr@~{a}o: @code{false} -Name of the default order for kept variables in elimination-based functions. If not set, @code{lex} will be used. +Nome da ordem padr@~{a}o para manter vari@'{a}veis em fun@value{cedilha}@~{o}es de elimina@value{cedilha}@~{a}o. Se n@~{a}o for escolhida, @code{lex} ir@'{a} ser usada. @end defvr -@defvr {Option variable} poly_elimination_order -Default value: @code{false} +@defvr {Vari@'{a}vel de op@value{cedilha}@~ao} poly_elimination_order +Valor padr@~{a}o: @code{false} -Name of the default elimination order used in elimination -calculations. If set, it overrides the settings in variables -@code{poly_primary_elimination_order} and @code{poly_secondary_elimination_order}. -The user must ensure that this is a true elimination order valid -for the number of eliminated variables. +Nome da ordem padr@~{a}o de fun@value{cedilha}@~{o}es de +elimina@value{cedilha}@~{a}o. Se escolhida, ir@'{a} sobrescrever as escolhas nas vari@'{a}veis +@code{poly_primary_elimination_order} e @code{poly_secondary_elimination_order}. +O usu@'{a}rio deve garantir que essa @'{e} uma ordem de elimina@value{cedilha}@~{a}o verdadeira v@'{a}lida +para o n@'{u}mero de vari@'{a}veis eliminadas. @end defvr -@defvr {Option variable} poly_return_term_list -Default value: @code{false} +@defvr {Vari@'{a}vel de op@value{cedilha}@~ao} poly_return_term_list +Valor padr@~{a}o: @code{false} -If set to @code{true}, all functions in this package will return each -polynomial as a list of terms in the current monomial order rather -than a @emph{maxima} general expression. +Se escolhida para @code{true}, todas as fun@value{cedilha}@~{o}es no pacote @code{grobner} ir@~{a}o retornar cada +polin@^{o}mio como uma lista de termos na ordem monomial corrente em lugar de +retornar uma express@~{a}o geral do @emph{maxima}. @end defvr -@defvr {Option variable} poly_grobner_debug -Default value: @code{false} +@defvr {Vari@'{a}vel de op@value{cedilha}@~ao} poly_grobner_debug +Valor padr@~{a}o: @code{false} -If set to @code{true}, produce debugging and tracing output. +Se escolhida para @code{true}, produz sa@'{i}da de depura@value{cedilha}@~{a}o e rastros. @end defvr -@defvr {Option variable} poly_grobner_algorithm -Default value: @code{buchberger} +@defvr {Vari@'{a}vel de op@value{cedilha}@~ao} poly_grobner_algorithm +Valor padr@~{a}o: @code{buchberger} -Possible values: +Valores poss@'{i}veis: @itemize @item @code{buchberger} @item @code{parallel_buchberger} @item @code{gebauer_moeller} @end itemize -The name of the algorithm used to find the Groebner Bases. +O nome do algor@'{i}tmo usado para encontrar as bases de Groebner. @end defvr -@defvr {Option variable} poly_top_reduction_only -Default value: @code{false} +@defvr {Vari@'{a}vel de op@value{cedilha}@~ao} poly_top_reduction_only +Valor padr@~{a}o: @code{false} -If not @code{false}, use top reduction only whenever possible. Top -reduction means that division algorithm stops after the first -reduction. +Se n@~{a}o for @code{false}, usa redu@value{cedilha}@~{a}o de topo somente se for poss@'{i}vel. Redu@value{cedilha}@~{a}o de +topo significa que o algor@'{i}tmo de divis@~{a}o para ap@'{o}s a primeira +redu@value{cedilha}@~{a}o. @end defvr -@subsection Simple operators in grobner -@code{poly_add}, @code{poly_subtract}, @code{poly_multiply} and @code{poly_expt} -are the arithmetical operations on polynomials. -These are performed using the internal representation, but the results are converted back to the -@emph{maxima} general form. +@subsection Operadores simples em grobner +@code{poly_add}, @code{poly_subtract}, @code{poly_multiply} e @code{poly_expt} +s@~{a}o as opera@value{cedilha}@~{o}es aritm@'{e}ticas sobre polin@^{o}mios. +Elas s@~{a}o executadas usando representa@value{cedilha}@~{a}o interna, mas os resultados s@~{a}o convertidos de volta @`{a} +forma geral do @emph{maxima}. -@deffn {Function} poly_add (@var{poly1}, @var{poly2}, @var{varlist}) -Adds two polynomials @var{poly1} and @var{poly2}. +@deffn {Fun@value{cedilha}@~ao} poly_add (@var{poli1}, @var{poli2}, @var{varlist}) +Adiciona dois polin@^{o}mios @var{poli1} e @var{poli2}. @example (%i1) poly_add(z+x^2*y,x-z,[x,y,z]); @@ -182,8 +185,8 @@ @end deffn -@deffn {Function} poly_subtract (@var{poly1}, @var{poly2}, @var{varlist}) -Subtracts a polynomial @var{poly2} from @var{poly1}. +@deffn {Fun@value{cedilha}@~ao} poly_subtract (@var{poli1}, @var{poli2}, @var{varlist}) +Subtrai o polin@^{o}mio @var{poli2} do polin@^{o}mio @var{poli1}. @example (%i1) poly_subtract(z+x^2*y,x-z,[x,y,z]); @@ -192,8 +195,8 @@ @end example @end deffn -@deffn {Function} poly_multiply (@var{poly1}, @var{poly2}, @var{varlist}) -Returns the product of polynomials @var{poly1} and @var{poly2}. +@deffn {Fun@value{cedilha}@~ao} poly_multiply (@var{poli1}, @var{poli2}, @var{varlist}) +Retorna o produto dos polin@^{o}mios @var{poli1} e @var{poli2}. @example (%i2) poly_multiply(z+x^2*y,x-z,[x,y,z])-(z+x^2*y)*(x-z),expand; @@ -201,12 +204,12 @@ @end example @end deffn -@deffn {Function} poly_s_polynomial (@var{poly1}, @var{poly2}, @var{varlist}) -Returns the @emph{syzygy polynomial} (@emph{S-polynomial}) of two polynomials @var{poly1} and @var{poly2}. +@deffn {Fun@value{cedilha}@~ao} poly_s_polynomial (@var{poli1}, @var{poli2}, @var{varlist}) +Retorna o @emph{polin@^{o}mio syzygy} (@emph{S-polinomial}) de dois polin@^{o}mios @var{poli1} e @var{poli2}. @end deffn -@deffn {Function} poly_primitive_part (@var{poly1}, @var{varlist}) -Returns the polynomial @var{poly} divided by the GCD of its coefficients. +@deffn {Fun@value{cedilha}@~ao} poly_primitive_part (@var{poli1}, @var{varlist}) +Retorna o polin@^{o}mio @var{poli} dividido pelo MDC entre seus coeficientes. @example (%i1) poly_primitive_part(35*y+21*x,[x,y]); @@ -214,22 +217,22 @@ @end example @end deffn -@deffn {Function} poly_normalize (@var{poly}, @var{varlist}) -Returns the polynomial @var{poly} divided by the leading coefficient. -It assumes that the division is possible, which may not always be the -case in rings which are not fields. +@deffn {Fun@value{cedilha}@~ao} poly_normalize (@var{poli}, @var{varlist}) +Retorna o polin@^{o}mio @var{poli} dividido pelo coeficiente lider. +@code{poly_normalize} assume que a divis@~{a}o @'{e} poss@'{i}vel, o que nem sempre ocorre +em an@'{e}is que n@~{a}o s@~{a}o corpos (fields). @end deffn -@subsection Other functions in grobner +@subsection Outras fun@value{cedilha}@~{o}es em grobner -@deffn {Function} poly_expand (@var{poly}, @var{varlist}) -This function parses polynomials to internal form and back. It -is equivalent to @code{expand(poly)} if @var{poly} parses correctly to -a polynomial. If the representation is not compatible with a -polynomial in variables @var{varlist}, the result is an error. -It can be used to test whether an expression correctly parses to the -internal representation. The following examples illustrate that -indexed and transcendental function variables are allowed. +@deffn {Fun@value{cedilha}@~ao} poly_expand (@var{poli}, @var{varlist}) +Essa fun@value{cedilha}@~{a}o transforma polin@^{o}mios para a forma interna e da forma interna para a forma geral. @code{poly_expand} +@'{e} equivalente a @code{expand(poly)} se @var{poli} passa corretamente para +um polin@^{o}mio. Se a representa@value{cedilha}@~{a}o n@~{a}o for compat@'{i}vel com um +polin@^{o}mio nas vari@'{a}veis @var{varlist}, o resultado @'{e} um erro. +Esse resultado em erro pode ser usado para testar se uma express@~{a}o transforma-se corretamente para a +representa@value{cedilha}@~{a}o interna. Os seguintes exemplos ilustra que +vari@'{a}veis de fun@value{cedilha}@~{o}es indexadas e transcendentes s@~{a}o permitidas. @example (%i1) poly_expand((x-y)*(y+x),[x,y]); @@ -254,8 +257,8 @@ @end example @end deffn -@deffn {Function} poly_expt (@var{poly}, @var{number}, @var{varlist}) -exponentitates @var{poly} by a positive integer @var{number}. If @var{number} is not a positive integer number an error will be raised. +@deffn {Fun@value{cedilha}@~ao} poly_expt (@var{poli}, @var{n@'{u}mero}, @var{varlist}) +eleva @var{poli} a um inteiro positivo @var{n@'{u}mero}. If @var{n@'{u}mero} n@~{a}o for um inteiro positivo um erro ir@'{a} ser mostrado. @example (%i1) poly_expt(x-y,3,[x,y])-(x-y)^3,expand; @@ -263,8 +266,8 @@ @end example @end deffn -@deffn {Function} poly_content (@var{poly}. @var{varlist}) -@code{poly_content} extracts the GCD of its coefficients +@deffn {Fun@value{cedilha}@~ao} poly_content (@var{poli}. @var{varlist}) +@code{poly_content} extrai o MDC entre seus coeficientes @example (%i1) poly_content(35*y+21*x,[x,y]); @@ -272,14 +275,14 @@ @end example @end deffn -@deffn {Function} poly_pseudo_divide (@var{poly}, @var{polylist}, @var{varlist}) -Pseudo-divide a polynomial @var{poly} by the list of @math{n} polynomials @var{polylist}. Return -multiple values. The first value is a list of quotients @math{a}. The -second value is the remainder @math{r}. The third argument is a scalar -coefficient @math{c}, such that @math{c*poly} can be divided by @var{polylist} within the ring -of coefficients, which is not necessarily a field. Finally, the -fourth value is an integer count of the number of reductions -performed. The resulting objects satisfy the equation: +@deffn {Fun@value{cedilha}@~ao} poly_pseudo_divide (@var{poli}, @var{polilist}, @var{varlist}) +Realiza a divis@~{a}o falsa do polin@^{o}mio @var{poli} pela lista de @math{n} polin@^{o}mios @var{polilist}. Retorna +multiplos valores. O primeiro valor @'{e} uma lista de quocientes @math{a}. O +segundo valor @'{e} o resto @math{r}. O terceiro argumento @'{e} um coeficiente +escalar @math{c}, tal que @math{c*poli} pode ser dividido por@var{polilist} dentro do anel +dos coeficientes, que n@~{a}o @'{e} necess@'{a}riamente corpo. Finalmente, o +quarto valor @'{e} um contador inteiro do n@'{u}mero de redu@value{cedilha}@~{o}es +realizadas. O objetos resultantes satisfazem @`{a} equa@value{cedilha}@~{a}o: @iftex @tex @@ -291,141 +294,141 @@ @end ifnottex @end deffn -@deffn {Function} poly_exact_divide (@var{poly1}, @var{poly2}, @var{varlist}) -Divide a polynomial @var{poly1} by another polynomial @var{poly2}. Assumes that exact -division with no remainder is possible. Returns the quotient. +@deffn {Fun@value{cedilha}@~ao} poly_exact_divide (@var{poli1}, @var{poli2}, @var{varlist}) +Divide um polin@^{o}mio @var{poli1} por outro polin@^{o}mio @var{poli2}. Assume que a divis@~{a}o +exata (sem resto) @'{e} poss@'{i}vel. Retorna o quociente. @end deffn -@deffn {Function} poly_normal_form (@var{poly}, @var{polylist}, @var{varlist}) -@code{poly_normal_form} finds the normal form of a polynomial @var{poly} with respect -to a set of polynomials @var{polylist}. +@deffn {Fun@value{cedilha}@~ao} poly_normal_form (@var{poli}, @var{polilist}, @var{varlist}) +@code{poly_normal_form} encontra a forma normal de um polin@^{o}mio @var{poli} com rela@value{cedilha}@~{a}o a +um conjunto de polin@^{o}mios @var{polilist}. @end deffn -@deffn {Function} poly_buchberger_criterion (@var{polylist}, @var{varlist}) -Returns @code{true} if @var{polylist} is a Groebner basis with respect to the current term +@deffn {Fun@value{cedilha}@~ao} poly_buchberger_criterion (@var{polilist}, @var{varlist}) +Returns @code{true} if @var{polilist} is a Groebner basis with respect to the current term order, by using the Buchberger -criterion: for every two polynomials @math{h1} and @math{h2} in @var{polylist} the -S-polynomial @math{S(h1,h2)} reduces to 0 @math{modulo} @var{polylist}. +criterion: for every two polynomials @math{h1} and @math{h2} in @var{polilist} the +S-polynomial @math{S(h1,h2)} reduces to 0 @math{modulo} @var{polilist}. @end deffn -@deffn {Function} poly_buchberger (@var{polylist_fl} @var{varlist}) -@code{poly_buchberger} performs the Buchberger algorithm on a list of -polynomials and returns the resulting Groebner basis. +@deffn {Fun@value{cedilha}@~ao} poly_buchberger (@var{polilist_fl} @var{varlist}) +@code{poly_buchberger} realiza o algor@'{i}tmo de Buchberger sobre uma lista de +polin@^{o}mios e retorna a base de Grobner resultante. @end deffn -@subsection Standard postprocessing of Groebner Bases +@subsection P@'{o}sprocessamento pad@~{a}o de bases de Groebner @iftex @tex -The \emph{k-th elimination ideal} $I_k$ of an ideal $I$ over -$K [ x_1, ...,x_1 ]$ is $I \cap K [ x_{k + 1}, ..., x_n ]$. +O \emph{k-@'{e}simo ideal de elimina@value{cedilha}@~{a}o} $I_k$ de um ideal $I$ sobre +$K [ x_1, ...,x_1 ]$ @'{e} $I \cap K [ x_{k + 1}, ..., x_n ]$. \noindent -The \emph{colon ideal} $I : J$ is the ideal $\{ h|\forall w \in J : wh \in +O \emph{ideal quociente} $I : J$ @'{e} o ideal $\{ h|\forall w in J : wh \in I \}$.@* \noindent -The ideal $I : p^{\infty}$ is the ideal $\{ h|\exists n \in N : p^n h \in I \}$.@* +O ideal $I : p^{\infty}$ @'{e} o ideal $\{ h|\exists n \in N : p^n h \in I \}$.@* \noindent -The ideal $I : J^{\infty}$ is the ideal $\{ h|\exists n \in N, \exists p \in J: p^n h \in I \}$.@* +O ideal $I : J^{\infty}$ @'{e} o ideal $\{ h|\exists n \in N, \exists p \in J: p^n h \in I \}$.@* \noindent -The \emph{radical ideal} $\sqrt{I}$ is the ideal $\{ h| \exists n \in N : +O \emph{ideal radical} $\sqrt{I}$ @'{e} o ideal $\{ h| \exists n \in N : h^n \in I \}$.@* @end tex @end iftex @ifnottex -The @emph{k-th elimination Ideal} @math{I_k} of an Ideal @math{I} over @math{K[ x[1],...,x[n] ]} is the ideal @math{intersect(I, K[ x[k+1],...,x[n] ])}.@* +O @emph{k-@'{e}simo ideal de elimina@value{cedilha}@~{a}o} @math{I_k} de uma Ideal @math{I} sobre @math{K[ x[1],...,x[n] ]} @'{e} o ideal @math{intersec@value{cedilha}@~{a}o(I, K[ x[k+1],...,x[n] ])}.@* @noindent -The @emph{colon ideal} @math{I:J} is the ideal @math{@{h|for all w in J: w*h in I@}}.@* +O @emph{ideal quociente} @math{I:J} @'{e} o ideal @math{@{h|for all w em J: w*h em I@}}.@* @noindent -The ideal @math{I:p^inf} is the ideal @math{@{h| there is a n in N: p^n*h in I@}}.@* +O ideal @math{I:p^inf} @'{e} o ideal @math{@{h| existe um n em N: p^n*h em I@}}.@* @noindent -The ideal @math{I:J^inf} is the ideal @math{@{h| there is a n in N and a p in J: p^n*h in I@}}.@* +O ideal @math{I:J^inf} @'{e} o ideal @math{@{h| existe um n em N \and a p em J: p^n*h em I@}}.@* @noindent -The @emph{radical ideal} @math{sqrt(I)} is the ideal -@math{@{h| there is a n in N : h^n in I @}}. +O @emph{ideal radical} @math{sqrt(I)} @'{e} o ideal +@math{@{h| existe um n em N : h^n em I @}}. @end ifnottex @noindent -@deffn {Function} poly_reduction (@var{polylist}, @var{varlist}) -@code{poly_reduction} reduces a list of polynomials @var{polylist}, so that -each polynomial is fully reduced with respect to the other polynomials. +@deffn {Fun@value{cedilha}@~ao} poly_reduction (@var{polilist}, @var{varlist}) +@code{poly_reduction} reduz uma lista de polin@^{o}mios @var{polilist}, de forma que +cada poin@^{o}mio @'{e} completametne reduzido com rela@value{cedilha}@~{a}o a outros polin@^{o}mios. @end deffn -@deffn {Function} poly_minimization (@var{polylist}, @var{varlist}) -Returns a sublist of the polynomial list @var{polylist} spanning the same -monomial ideal as @var{polylist} but minimal, i.e. no leading monomial -of a polynomial in the sublist divides the leading monomial -of another polynomial. +@deffn {Fun@value{cedilha}@~ao} poly_minimization (@var{polilist}, @var{varlist}) +Retorna uma sublista da lista de polin@^{o}mios @var{polilist} gerando o mesmo +ideal de mon@^{o}mio que @var{polilist} mas minimo, i.e. nenhum mon@^{o}mio l@'{i}der +de um polin@^{o}mio na sublista divide o mon@^{o}mio l@'{i}der +de outro polin@^{o}mio. @end deffn -@deffn {Function} poly_normalize_list (@var{polylist}, @var{varlist}) -@code{poly_normalize_list} applies @code{poly_normalize} to each polynomial in the list. -That means it divides every polynomial in a list @var{polylist} by its leading coefficient. +@deffn {Fun@value{cedilha}@~ao} poly_normalize_list (@var{polilist}, @var{varlist}) +@code{poly_normalize_list} aplica @code{poly_normalize} a cada polin@^{o}mio na lista. +Que significa que @code{poly_normalize_list} divide todo polin@^{o}mio em uma lista @var{polilist} por seu coeficiente l@'{i}der. @end deffn -@deffn {Function} poly_grobner (@var{polylist}, @var{varlist}) -Returns a Groebner basis of the ideal span by the polynomials @var{polylist}. Affected by the global flags. +@deffn {Fun@value{cedilha}@~ao} poly_grobner (@var{polilist}, @var{varlist}) +Retorna uma base de Groebner do ideal gerado pelos polin@^{o}mios @var{polilist}. Afetado pelos sinalizadores globais. @end deffn -@deffn {Function} poly_reduced_grobner (@var{polylist}, @var{varlist}) -Returns a reduced Groebner basis of the ideal span by the polynomials @var{polylist}. Affected by the global flags. +@deffn {Fun@value{cedilha}@~ao} poly_reduced_grobner (@var{polilist}, @var{varlist}) +Retorna uma base de Groebner reduzida do ideal gerado pelos polin@^{o}mios @var{polilist}. Afetado pelos sinalizadores globais. @end deffn -@deffn {Function} poly_depends_p (@var{poly}, @var{var}, @var{varlist}) -@code{poly_depends} tests whether a polynomial depends on a variable var. +@deffn {Fun@value{cedilha}@~ao} poly_depends_p (@var{poli}, @var{var}, @var{varlist}) +@code{poly_depends} testa se um polin@^{o}mio depende da vari@'{a}vel @var{var}. @end deffn -@deffn {Function} poly_elimination_ideal (@var{polylist}, @var{number}, @var{varlist}) +@deffn {Fun@value{cedilha}@~ao} poly_elimination_ideal (@var{polilist}, @var{num}, @var{varlist}) -@code{poly_elimination_ideal} returns the grobner basis of the @math{number}-th elimination ideal of an -ideal specified as a list of generating polynomials (not necessarily Groebner basis +@code{poly_elimination_ideal} retorna a base de grobner do @math{num}-@'{e}simo ideal de elimina@value{cedilha}@~{a}o de um +ideal especificado como uma lista de polin@^{o}mios geradores (n@~{a}o necess@'{a}riamente base de Groebner) @end deffn -@deffn {Function} poly_colon_ideal (@var{polylist1}, @var{polylist2}, @var{varlist}) +@deffn {Fun@value{cedilha}@~ao} poly_colon_ideal (@var{polilist1}, @var{polilist2}, @var{varlist}) -Returns the reduced Groebner basis of the colon ideal +Retorna a base reduzida de Groebner do ideal quociente -@math{I(polylist1):I(polylist2)} +@math{I(polilist1):I(polilist2)} @noindent -where @math{polylist1} and @math{polylist2} are two lists of polynomials. +onde @math{polilist1} e @math{polilist2} s@~{a}o duas listas de polin@^{o}mios. @end deffn -@deffn {Function} poly_ideal_intersection (@var{polylist1}, @var{polylist2}, @var{varlist}) +@deffn {Fun@value{cedilha}@~ao} poly_ideal_intersection (@var{polilist1}, @var{polilist2}, @var{varlist}) -@code{poly_ideal_intersection} returns the intersection of two ideals. +@code{poly_ideal_intersection} retorna a intersec@value{cedilha}@~{a}o entre dois ideais. @end deffn -@deffn {Function} poly_lcm (@var{poly1}, @var{poly2}, @var{varlist}) -Returns the lowest common multiple of @var{poly1} and @var{poly2}. +@deffn {Fun@value{cedilha}@~ao} poly_lcm (@var{poli1}, @var{poli2}, @var{varlist}) +Retorna o m@'{i}nimo m@'{u}ltiplo comum entre @var{poli1} e @var{poli2}. @end deffn -@deffn {Function} poly_gcd (@var{poly1}, @var{poly2}, @var{varlist}) -Returns the greatest common divisor of @var{poly1} and @var{poly2}. +@deffn {Fun@value{cedilha}@~ao} poly_gcd (@var{poli1}, @var{poli2}, @var{varlist}) +Retorna m@'{a}ximo divisor comum de @var{poli1} e @var{poli2}. @end deffn -@deffn {Function} poly_grobner_equal (@var{polylist1}, @var{polylist2}, @var{varlist}) -@code{poly_grobner_equal} tests whether two Groebner Bases generate the same ideal. -Returns @code{true} if two lists of polynomials @var{polylist1} and @var{polylist2}, assumed to be Groebner Bases, -generate the same ideal, and @code{false} otherwise. -This is equivalent to checking that every polynomial of the first basis reduces to 0 -modulo the second basis and vice versa. Note that in the example below the -first list is not a Groebner basis, and thus the result is @code{false}. +@deffn {Fun@value{cedilha}@~ao} poly_grobner_equal (@var{polilist1}, @var{polilist2}, @var{varlist}) +@code{poly_grobner_equal} testa se duas bases de Groebner geram o mesmo ideal. +Retorna @code{true} se as duas listas de polin@^{o}mios @var{polilist1} e @var{polilist2}, assumidas serem bases de Groebner, +geram o mesmo ideal, e @code{false} de outra forma. +Isso @'{e} equivalente a verificar que todo polin@^{o}mio da primeira base @'{e} reduzido a 0 +m@'{o}dulo a segunda base e vice-versa. Note que no exemplo abaixo a +primeira lista n@~{a}o @'{e} uma base de Groebner, e dessa forma o resultado @'{e} @code{false}. @example (%i1) poly_grobner_equal([y+x,x-y],[x,y],[x,y]); @@ -434,24 +437,24 @@ @end deffn -@deffn {Function} poly_grobner_subsetp (@var{polylist1}, @var{polylist2}, @var{varlist}) +@deffn {Fun@value{cedilha}@~ao} poly_grobner_subsetp (@var{polilist1}, @var{polilist2}, @var{varlist}) -@code{poly_grobner_subsetp} tests whether an ideal generated by @var{polylist1} -is contained in the ideal generated by @var{polylist2}. For this test to always succeed, -@var{polylist2} must be a Groebner basis. +@code{poly_grobner_subsetp} testa se um ideal gerado pela @var{polilist1} +est@'{a} contido em um ideal gerado pela @var{polilist2}. Para esse teste sempre tenha sucesso, +@var{polilist2} deve ser uma base de Groebner. @end deffn -@deffn {Function} poly_grobner_member (@var{poly}, @var{polylist}, @var{varlist}) +@deffn {Fun@value{cedilha}@~ao} poly_grobner_member (@var{poli}, @var{polilist}, @var{varlist}) -Returns @code{true} if a polynomial @var{poly} belongs to the ideal generated by the -polynomial list @var{polylist}, which is assumed to be a Groebner basis. Returns @code{false} otherwise. +Retorna @code{true} se um polin@^{o}mio @var{poli} pertence ao ideal gerado pela +lista polinomial @var{polilist}, que @'{e} assumida como sendouma base de Groebner. Retorna @code{false} de outra forma. -@code{poly_grobner_member} tests whether a polynomial belongs to an ideal generated by a list of polynomials, -which is assumed to be a Groebner basis. Equivalent to @code{normal_form} being 0. +@code{poly_grobner_member} testa se um polin@^{o}mio pertence a um ideal gerado por uma lista de polin@^{o}mios, +que @'{e} assumida ser uma base de Groebner. Equivale a @code{normal_form} sendo 0. @end deffn -@deffn {Function} poly_ideal_saturation1 (@var{polylist}, @var{poly}, @var{varlist}) -Returns the reduced Groebner basis of the saturation of the ideal +@deffn {Fun@value{cedilha}@~ao} poly_ideal_saturation1 (@var{polilist}, @var{poli}, @var{varlist}) +Retorna abase de Groebner reduzida da satura@value{cedilha}@~{a}o do ideal @iftex @tex $$I(polylist):poly^\infty$$ @@ -463,13 +466,13 @@ @end ifnottex @noindent -Geometrically, over an algebraically closed field, this is the set -of polynomials in the ideal generated by @var{polylist} which do not identically -vanish on the variety of @var{poly}. +Geometricamente, sobre um corpo algebricamente fechado, esse @'{e} um conjunto +de polinmios no ideal gerado por @var{polilist} que n@~{a}o tende identicamente a +zero sobre a varia@value{cedilha}@~{a}o de @var{poli}. @end deffn -@deffn {Function} poly_ideal_saturation (@var{polylist1}, @var{polylist2}, @var{varlist}) -Returns the reduced Groebner basis of the saturation of the ideal +@deffn {Fun@value{cedilha}@~ao} poly_ideal_saturation (@var{polilist1}, @var{polilist2}, @var{varlist}) +Retorna a base de Groebner reduzida da satura@value{cedilha}@~{a}o do ideal @iftex @tex $$I(polylist1):I(polylist2)^\infty$$ @@ -481,14 +484,15 @@ @end ifnottex @noindent -Geometrically, over an algebraically closed field, this is the set of polynomials in -the ideal generated by @var{polylist1} which do not identically vanish on the -variety of @var{polylist2}. +Geometricamente, sobre um corpo algebricamente fechado, esse @'{e} um conjunto +de polinmios no ideal gerado por @var{polilist1} que n@~{a}o tende identicamente a +zero sobre a varia@value{cedilha}@~{a}o de @var{polilist2}. + @end deffn -@deffn {Function} poly_ideal_polysaturation1 (@var{polylist1}, @var{polylist2}, @var{varlist}) -@var{polylist2} ist a list of n polynomials @code{[poly1,...,polyn]}. -Returns the reduced Groebner basis of the ideal +@deffn {Fun@value{cedilha}@~ao} poly_ideal_polysaturation1 (@var{polilist1}, @var{polilist2}, @var{varlist}) +@var{polilist2} ist a list of n polynomials @code{[poly1,...,polyn]}. +Retorna a base de Groebner reduzida do ideal @iftex @tex $$I(polylist):poly1^\infty:...:polyn^\infty$$ @@ -500,15 +504,15 @@ @end ifnottex @noindent -obtained by a -sequence of successive saturations in the polynomials -of the polynomial list @var{polylist2} of the ideal generated by the -polynomial list @var{polylist1}. +obtido por uma +seq@"{u}@^{e}ncia de sucessivas satura@value{cedilha}@~{o}es nos polin@^{o}mios +da lista polin@^{o}mial @var{polilist2} do ideal gerado pela +lista polinomial @var{polilist1}. @end deffn -@deffn {Function} poly_ideal_polysaturation (@var{polylist}, @var{polylistlist}, @var{varlist}) -@var{polylistlist} is a list of n list of polynomials @code{[polylist1,...,polylistn]}. -Returns the reduced Groebner basis of the saturation of the ideal +@deffn {Fun@value{cedilha}@~ao} poly_ideal_polysaturation (@var{polilist}, @var{polilistlist}, @var{varlist}) +@var{polilistlist} is a list of n list of polynomials @code{[polylist1,...,polylistn]}. +Retorna a base reduzida de Groebner da satura@value{cedilha}@~{a}o do ideal @iftex @tex $$I(polylist):I(polylist_1)^\infty:...:I(polylist_n)^\infty$$ @@ -520,11 +524,8 @@ @end ifnottex @end deffn -@deffn {Function} poly_saturation_extension (@var{poly}, @var{polylist}, @var{varlist1}, @var{varlist2}) - -@code{poly_saturation_extension} implements the famous Rabinowitz trick. -@end deffn +@deffn {Fun@value{cedilha}@~ao} poly_saturation_extension (@var{poli}, @var{polilist}, @var{varlist1}, @var{varlist2}) -@deffn {Function} poly_polysaturation_extension (@var{poly}, @var{polylist}, @var{varlist1}, @var{varlist2}) +@code{poly_saturation_extension} implementa o famoso artif@'{i}cio de Rabinowitz. @end deffn Index: maxima.texi =================================================================== RCS file: /cvsroot/maxima/maxima/doc/info/pt_BR/maxima.texi,v retrieving revision 1.48 retrieving revision 1.49 diff -u -d -r1.48 -r1.49 --- maxima.texi 9 Jun 2007 14:45:04 -0000 1.48 +++ maxima.texi 10 Jun 2007 19:05:57 -0000 1.49 @@ -448,11 +448,11 @@ contrib_ode -* Introduction to contrib_ode:: -* Functions and Variables for contrib_ode:: -* Possible improvements to contrib_ode:: -* Test cases for contrib_ode:: -* References for contrib_ode:: +* Introdu@value{cedilha}@~{a}o a contrib_ode:: +* Fun@value{cedilha}@~{o}es e Vari@'{a}veis Definidas para contrib_ode:: +* Possibilidades de melhorias em contrib_ode:: +* Casos de teste para contrib_ode:: +* Refer@^{e}ncias bibliogr@'{a}ficas para contrib_ode:: descriptive @@ -496,8 +496,8 @@ grobner -* Introduction to grobner:: -* Functions and Variables for grobner:: +* Introdu@value{cedilha}@~{a}o a grobner:: +* Fun@value{cedilha}@~{o}es e Vari@'{a}veis Definidas para grobner:: impdiff |