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\NeedsTeXFormat{LaTeX2e}[1995/12/01] \ProvidesPackage{tablor}[09/05/2010 v4.07 la machine a creer des tableaux de signes et variations] % \copyleft Connan le Barbare (aka Guillaume Connan) \copyright % This work may be distributed and/or mofified under the conditions % or the LaTeX Project Public Licence, either v1.3 or (at your option) % any later version. The latest version is in % http://www.latex-project.org/lppl/ % This work consists of the files tablor.sty, tablor-xetex.sty, tablor.cfg, tablor.tex, % tablor.pdf and tablor.html %% Cree 16 environnements : %% tableau de signes de 2 facteurs affines % \begin{TSa} % TSa(-2,3,-1,5,\tv); % \end{TSa} % %%%%%% Pour des tableaux de plus de 2 facteurs % % \begin{TS} % TS("P",[-2*x+3,x^2-1,x^2+1,x-1,x^2-2],[a,b],n,\tv); % \end{TS} % % pour les tableaux de signes avec quotient %\begin{TSq} %TSq("Q",[-2*x+3,-4*x+5],[x^2-16,x-2],[a,b],n,\tv) %%\end{TSq} % un tableau de variation : % % pour les tableaux de signes à une seule ligne % \begin{TSc} % TSc((x+10)/((x-5)*(x-2)),[-10,5],[2,5],n,0) % \end{TSc} % % % \begin{TV} % TV([0,+infinity],[0],"h","x",ln(x)-(ln(x))^2,1,n,\tv) % \end{TV} % % tableau de variation avec liste de valeurs % \begin{TVS} % TVS([1,2,3,4],[-1,-infinity,+infinity,2,9],[2],"f","x",\tv) % \end{TVS} % % % tableau de variation avec zones interdites % % \begin{TVZ} % TVZ([-infinity,+infinity],[],[[-1,1]],"f","x",sqrt(x^2-1),1,n,\tv) % \end{TVZ} % % % tableau avec valeurs intermediares %\begin{TVI} %TVI([-1,+infinity],[-1],"f","x",x2/sqrt(x+1)-1,1,2,n,\tv) %\end{TVI} %%% % tableau avec valeurs intermediares et racines exactes %\begin{TVIex} %TVIex([-1,+infinity],[-1],"f","x",x2/sqrt(x+1)-1,1,2,n,\tv) %\end{TVIex} %%% % % % tableau de variations avec f' sans zero formel %\begin{TVapp} % TVapp([0,+infinity],[0],"g","x",ln(x)-x*exp(2-x),1,\tv) % \end{TVapp} % % % tableau de variations avec f' sans zero formel %\begin{TVIapp} % TVIapp([0,+infinity],[0],"g","x",ln(x)-x*exp(2-x),1,0,\tv) % \end{TVIapp} % % %%% % et leurs pendants etoiles qui permettent l'affichage intermediaire du % fichier metapost pour le modifier % % % Courbes parametrees % \begin{TVP} % TVP([-infinity,+infinity],[[-1,2],[-1]],["x","y"],"t",[t^2/((t+1)*(t-2)),t^2*(t+2)/(t+1)],1,n,\tv) % \end{TVP} % % % \begin{TVP} % TVP([0,pi/2],[[],[]],["x","y"],"t",[2*cos(t),sin(2*t)],1,t,\tv) % \end{TVP} % % % Fonctions prolongeables par continuité % TVP([intervalles d'étude],[valeurs prolongeables],[valeurs interdites pour f'],"g","t",e^(-1/x^2),1,n,\tv); % \begin{TVPC} % TVPC([-infinity,+infinity],[0],[0],"g","t",e^(-1/x^2),1,n,\tv); % \end{TVPC} %% extensions requises %% Il faudra rajouter dans le preambule \usepackage{graphicx} si vous %% ne l'avez pas de base \RequirePackage{filecontents} \RequirePackage{ifthen} \RequirePackage{fancyvrb} \RequirePackage{ifpdf} \fvset{gobble=0} % option xcas present \newboolean{xcas}\setboolean{xcas}{false} \DeclareOption{xcas}{\setboolean{xcas}{true}} %% Initialisation du choix d'OS \newboolean{windows}\setboolean{windows}{false} \DeclareOption{windows}{\setboolean{windows}{true}} \ProcessOptions\relax %% on configure tablor dans un fichier exterieur pour la plateforme %% et l'editeur \IfFileExists{tablor.cfg}{\input{tablor.cfg}}%\typeout{pas de fichier tablor.cfg}} %% Definit des commandes disque selon l'OS utilise \ifthenelse{\boolean{windows}}% {\newcommand{\rem}{DEL } \newcommand{\cat}{TYPE } \newcommand{\cp}{COPY } \newcommand{\echod}{ECHO } \newcommand{\echof}{ }}% {\newcommand{\rem}{rm }\newcommand{\cat}{cat } \newcommand{\cp}{cp } \newcommand{\echod}{echo "} \newcommand{\echof}{"}} %% pour ceux compilant via pdflatex \ifpdf \DeclareGraphicsRule{*}{mps}{*}{} \fi %% pour nettoyer les fichiers auxiliaires \AtEndDocument{\immediate\write18{\rem *.user XCas* Xcas* *.mpx} } %% Pour clore les fichiers metapost \begin{VerbatimOut}{queue.mp} end \end{VerbatimOut} %% Nettoie les fichiers log dont le nom depend du choix de l'utilisateur %% Par defaut, c'est le nom du fichier tex courant (\jobname) %% Clôt le fichier metapost contenant le recapitulatif de tous les tableaux \newcommand{\nettoyer}[1][\jobname]% {\immediate\write18{\rem #1.Tab.log queue.mp enteteMP.cfg session.tex config.cxx} } %% Donne comme prefixe aux tableaux le prefixe courant %% Peut-être modifie par \initablor \newcommand{\nomtravail}{\jobname} %% initialise les compteurs \newcounter{TVn} \newcommand{\tv}{\theTVn} \newcounter{TVnbis} \newcommand{\tvbis}{\theTVnbis} %% permet de donner un prefixe aux tableaux produits (\jobname par defaut) %% effectue quelques verifications : \newcommand{\initablor}[1][\jobname]{% \renewcommand{\nomtravail}{#1}% Arret du nom des tableaux \setcounter{TVn}{0}% Initialisation du compteur de tableaux. \ifthenelse{\boolean{xcas}}% Avec l'option XCas {\IfFileExists{\nomtravail_Tab.mp}% Si Tableaux.mp est present... {\immediate\write18{\rem \nomtravail_Tab.mp}}% le detruire {}% \immediate\write18{\cp enteteMP.cfg \nomtravail_Tab.mp}% Reconstituer l'entête de Tableaux.mp } {\IfFileExists{\nomtravail_Tab.mp}% Sans l'option XCas, si % Tableaux.mp existe {\immediate\write18{mpost -interaction=batchmode \nomtravail_Tab}}% l'executer pour reconstituer les figures {\PackageWarning{tablor}{Pas de source metapost pour creer les tableaux.}}% sinon message d'erreur % (mais pas d'arret car les tableaux % peuvent être presents ) }}% %% commande pour lancer giac selon l'OS \makeatletter \newcommand{\executGiacmp}[1]{% \ifthenelse{\boolean{windows}}% {\immediate\write18{giac #1 }}% {\immediate\write18{giac <#1 }}} \makeatother %%% % %%% LES SCRIPTS GIAC/XCAS % %%% %% %% Code giac/Xcas pour les Tableaux de Variations %% \begin{VerbatimOut}{XcasTV.cxx} TV(L,F,nom,nomv,f,ftt,trigo,nmr):={ nl:=size(L); f:=unapply(f,x); fp:=function_diff(f); Z:=concat(L,F); S:=[]; if(trigo==t){ all_trig_solutions:=1; reset_solve_counter(-1,-1); SS:=solve(factor(simplify(fp(x))),x); ns:=size(SS); for(k:=0;k<ns;k++){ m:=0; while(evalf(simplify(subst(SS[k],n_1=m)))<=evalf(L[nl-1])){ S:=concat(S,simplify(subst(SS[k],n_1=m)));m:=m+1; };m:=-1; while(evalf(subst(SS[k],n_1=m))>=L[0]){ S:=concat(S,simplify(subst(SS[k],n_1=m)));m:=m-1; } } }else{ S:=solve(factor(simplify(fp(x))),x); } si size(S)>0 alors pour j de 0 jusque size(S)-1 faire qq:=member(simplify(S[j]),Z)==0; kk:=(evalf(S[j])>=evalf(L[0])) and (evalf(S[j])<=evalf(L[nl-1])); if(kk==1){if(qq==1){Z:=append(Z,simplify(S[j]))}}; fpour fsi; Z:=sort(Z); nz:=size(Z); tantque evalf(Z[0])==evalf(Z[1]) faire Z:=Z[1..nz-1];nz:=size(Z); ftantque; si size(S)>0 alors pour j de 0 jusque size(S)-1 faire kk:=(evalf(S[j])>=evalf(L[0])) and (evalf(S[j])<=evalf(L[nl-1])); if(kk==1){Z:=append(Z,simplify(S[j]))}; fpour fsi; Z:=sort([op(set[op(Z)])]); nz:=size(Z); l0:=" newLigneVariables(btex $"+nomv+"$ etex);";lp:=" "; lf:=" ";lsp:=" "; pour m de 0 jusque nz-1 faire l0:=l0+"val(btex $"+latex(Z[m])+"$ etex); ";fpour; k0:= evalf(limit(f(x),x=Z[0],1))> evalf(limit(f(x),x=Z[1],-1)); kz:=evalf(limit(f(x),x=Z[nz-1],-1))> evalf(limit(f(x),x=Z[nz-2],1)); lsi:=lsic+nom+"'("+nomv+")}$ etex);"+ if(Z[0]==-infinity){if(sign(evalf(fp(if(Z[1]==+infinity){0}else{Z[1]-10^(-5)})))==1.0){"plus;"}else{"moins;"}}else{if(member(Z[0],F)==0){ if(fp(Z[0])==0){"valBarre(btex 0 etex);"}else{" "}+ if(evalf(sign(fp(Z[0]+10^(-5))))==1.0){"plus;"}else{"moins;"}}else{"nonDefBarre;"+ if(evalf(sign(fp((Z[0]+10^(-5)))))==1.0){"plus;"}else{"moins;"} }} if(nz>2){ for(r:=1; r<=nz-2;r++){ ksp:=evalf(fp(Z[r]+0.1))>0; lsp:=lsp+if(member(Z[r],F)==0){"valBarre(btex 0 etex);"}else{"nonDefBarre;"}+ if(ksp==1){"plus;"}else{"moins;"} }; } lsf:=if(member(Z[nz-1],F)==0){""}else{"nonDefBarre; "} lm0:=limit(f(x),x=Z[0],1)==-infinity; li:=lvic+nom+"}$ etex);"+ if(member(Z[0],F)==0){"valPos(btex $"+if(lm0==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[0],1)))}+"$ etex,"} else{"nonDefBarre;limDroite(btex $"+if(lm0==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[0],1)))}+"$ etex,"}+ if(k0==1){"1"}else{"0"}+ ");"; if(nz>2){ for(r:=1; r<=nz-2;r++){ krm:=evalf(limit(f(x),x=Z[r-1],1))< evalf(limit(f(x),x=Z[r],-1)); krp:=evalf(limit(f(x),x=Z[r],1))> evalf(limit(f(x),x=Z[r+1],-1)) ; lmrm:=limit(f(x),x=Z[r],-1)==-infinity;lmrp:=limit(f(x),x=Z[r],1)==-infinity; lp:=lp+if(member(Z[r],F)){ "limGauche(btex $"+if(lmrm==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[r],-1)))}+"$ etex,"+if(krm==1){"1);"}else{"0);"}+"nonDefBarre;limDroite(btex $"+if(lmrp==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[r],1)))}+"$ etex,"+if(krp==1){"1);"}else{"0);"}} else{"valPos(btex $"+latex(simplify(f(Z[r])))+"$ etex,"+if(sign(evalf(fp(Z[r]-0.01)))==sign(fp(Z[r]+0.01))){"0.5);"}else{if(krp==1){"1);"}else{"0); "}}} }; } lnz:=limit(f(x),x=Z[nz-1],-1)==-infinity; lf:=if(member(Z[nz-1],F)==0){"valPos(btex $"+ if(lnz==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[nz-1],-1)))}+"$ etex,"+ if(kz==1){"1);"}else{"0);"}} else{"limGauche(btex $"+ if(lnz==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[nz-1],-1)))}+"$ etex,"+ if(kz==1){"1);nonDefBarre;"}else{"0);nonDefBarre;"}}; MetaLfc:=if(ftt==2){if(nz>2){" beginTableau("+nmr+")"+ l0+lsi+lsp+lsf+" endTableau; ";}else{ "beginTableau("+nmr+")"+ l0+ lsi+lsf+" endTableau; "; } }else{ if(ftt==0){if(nz>2){"beginTableau("+nmr+")"+ l0+ li+ lp+ lf +" endTableau; ";}else{"beginTableau("+nmr+")"+ l0+ li+ lf +" endTableau; ";}}else{ if(nz>2){"beginTableau("+nmr+")"+ l0+ lsi+lsp+lsf+ li+ lp+ lf +" endTableau; ";}else{"beginTableau("+nmr+")"+ l0+ lsi+lsf+ li+ lf +" endTableau; ";} }} sortie:=fopen("XCasmpfc.mp"); fprint(sortie,Unquoted,MetaLfc); fclose(sortie); }:; \end{VerbatimOut} %%%% % % % TVPC : pour les fonctions prolongeables par continuité. %% %% \begin{VerbatimOut}{XcasTVPC.cxx} TVPC(L,F,FP,nom,nomv,f,ftt,trigo,nmr):={ nl:=size(L); f:=unapply(f,x); fp:=function_diff(f); Z:=concat(L,F); Z:=concat(Z,FP); S:=[]; if(trigo==t){ all_trig_solutions:=1; reset_solve_counter(-1,-1); SS:=solve(factor(simplify(fp(x))),x); ns:=size(SS); for(k:=0;k<ns;k++){ m:=0; while(evalf(simplify(subst(SS[k],n_1=m)))<=evalf(L[nl-1])){ S:=concat(S,simplify(subst(SS[k],n_1=m)));m:=m+1; };m:=-1; while(evalf(subst(SS[k],n_1=m))>=L[0]){ S:=concat(S,simplify(subst(SS[k],n_1=m)));m:=m-1; } } }else{ S:=solve(factor(simplify(fp(x))),x); } si size(S)>0 alors pour j de 0 jusque size(S)-1 faire qq:=member(simplify(S[j]),Z)==0; kk:=(evalf(S[j])>=evalf(L[0])) and (evalf(S[j])<=evalf(L[nl-1])); if(kk==1){if(qq==1){Z:=append(Z,simplify(S[j]))}}; fpour fsi; Z:=sort(Z); nz:=size(Z); tantque evalf(Z[0])==evalf(Z[1]) faire Z:=Z[1..nz-1];nz:=size(Z); ftantque; si size(S)>0 alors pour j de 0 jusque size(S)-1 faire kk:=(evalf(S[j])>=evalf(L[0])) and (evalf(S[j])<=evalf(L[nl-1])); if(kk==1){Z:=append(Z,simplify(S[j]))}; fpour fsi; Z:=sort([op(set[op(Z)])]); nz:=size(Z); l0:=" newLigneVariables(btex $"+nomv+"$ etex);";lp:=" "; lf:=" ";lsp:=" "; pour m de 0 jusque nz-1 faire l0:=l0+"val(btex $"+latex(Z[m])+"$ etex); ";fpour; k0:= evalf(limit(f(x),x=Z[0],1))> evalf(limit(f(x),x=Z[1],-1)); kz:=evalf(limit(f(x),x=Z[nz-1],-1))> evalf(limit(f(x),x=Z[nz-2],1)); lsi:=lsic+nom+"'("+nomv+")}$ etex);"+ if(Z[0]==-infinity){if(sign(evalf(fp(if(Z[1]==+infinity){0}else{Z[1]-10^(-5)})))==1.0){"plus;"}else{"moins;"}}else{if(member(Z[0],FP)==0){ if(fp(Z[0])==0){"valBarre(btex 0 etex);"}else{" "}+ if(evalf(sign(fp(Z[0]+10^(-5))))==1.0){"plus;"}else{"moins;"}}else{"nonDefBarre;"+ if(evalf(sign(fp((Z[0]+10^(-5)))))==1.0){"plus;"}else{"moins;"} }} if(nz>2){ for(r:=1; r<=nz-2;r++){ ksp:=evalf(fp(Z[r]+0.1))>0; lsp:=lsp+if(member(Z[r],FP)==0){"valBarre(btex 0 etex);"}else{"nonDefBarre;"}+ if(ksp==1){"plus;"}else{"moins;"} }; } lsf:=if(member(Z[nz-1],FP)==0){""}else{"nonDefBarre; "} lm0:=limit(f(x),x=Z[0],1)==-infinity; li:=lvic+nom+"}$ etex);"+ if(member(Z[0],F)==0){"valPos(btex $"+if(lm0==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[0],1)))}+"$ etex,"} else{"limDroite(btex $"+if(lm0==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[0],1)))}+"$ etex,"}+ if(k0==1){"1"}else{"0"}+ ");"; if(nz>2){ for(r:=1; r<=nz-2;r++){ krm:=evalf(limit(f(x),x=Z[r-1],1))< evalf(limit(f(x),x=Z[r],-1)); krp:=evalf(limit(f(x),x=Z[r],1))> evalf(limit(f(x),x=Z[r+1],-1)) ; lmrm:=limit(f(x),x=Z[r],-1)==-infinity;lmrp:=limit(f(x),x=Z[r],1)==-infinity; lp:=lp+if(member(Z[r],F)){ "valPos(btex $"+if(lmrm==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[r],-1)))}+"$ etex,"+if(krm==1){"1);"}else{"0);"} } else{"valPos(btex $"+latex(simplify(f(Z[r])))+"$ etex,"+if(sign(evalf(fp(Z[r]-0.01)))==sign(fp(Z[r]+0.01))){"0.5);"}else{if(krp==1){"1);"}else{"0); "}}} }; } lnz:=limit(f(x),x=Z[nz-1],-1)==-infinity; lf:=if(member(Z[nz-1],F)==0){"valPos(btex $"+ if(lnz==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[nz-1],-1)))}+"$ etex,"+ if(kz==1){"1);"}else{"0);"}} else{"limGauche(btex $"+ if(lnz==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[nz-1],-1)))}+"$ etex,"+ if(kz==1){"1);"}else{"0);"}}; MetaLfc:=if(ftt==2){if(nz>2){" beginTableau("+nmr+")"+ l0+lsi+lsp+lsf+" endTableau; ";}else{ "beginTableau("+nmr+")"+ l0+ lsi+lsf+" endTableau; "; } }else{ if(ftt==0){if(nz>2){"beginTableau("+nmr+")"+ l0+ li+ lp+ lf +" endTableau; ";}else{"beginTableau("+nmr+")"+ l0+ li+ lf +" endTableau; ";}}else{ if(nz>2){"beginTableau("+nmr+")"+ l0+ lsi+lsp+lsf+ li+ lp+ lf +" endTableau; ";}else{"beginTableau("+nmr+")"+ l0+ lsi+lsf+ li+ lf +" endTableau; ";} }} sortie:=fopen("XCasmpfc.mp"); fprint(sortie,Unquoted,MetaLfc); fclose(sortie); }:; \end{VerbatimOut} %% %% %% TV avec une zone interdite : on rajoute comme argument la liste des intervalles interdits %% par exemple, pour sqrt(x^2-1) : TVZ([-infinity,100],[],[[-1,1]],"f","x",sqrt(x^2-1),1,1) %% \begin{VerbatimOut}{XcasTVZ.cxx} TVZ(L,F,FF,nom,nomv,f,ftt,trigo,nmr):={ nl:=size(L); nf:=size(FF); Ff:=NULL;IMIN:=NULL;IMAX:=NULL; for(k:=0;k<nf;k++){ if(FF[k][0]>L[0]){Imin[k]:=FF[k][0];LL:=L}else{Imin[k]:=L[0];LL:=[L[1]]}; if(FF[k][1]<L[1]){Imax[k]:=FF[k][1];LL:=L}else{Imax[k]:=L[1];LL:=[L[0]]}; Ff:=Ff,[Imin[k],Imax[k]]; IMIN:=IMIN,Imin[k]; IMAX:=IMAX,Imax[k]; } FF:=[Ff]; IMIN:=[IMIN]; IMAX:=[IMAX]; f:=unapply(f,x); fp:=function_diff(f); Z:=concat(LL,F); for(k:=0;k<nf;k++){ Z:=concat(Z,FF[k]); } S:=[]; if(trigo==t){ all_trig_solutions:=1; reset_solve_counter(-1,-1); SS:=solve(factor(simplify(fp(x))),x); ns:=size(SS); for(k:=0;k<ns;k++){ m:=0; while(evalf(simplify(subst(SS[k],n_1=m)))<=evalf(L[nl-1])){ S:=concat(S,simplify(subst(SS[k],n_1=m)));m:=m+1; };m:=-1; while(evalf(subst(SS[k],n_1=m))>=L[0]){ S:=concat(S,simplify(subst(SS[k],n_1=m)));m:=m-1; } } }else{ S:=solve(factor(simplify(fp(x))),x); } si size(S)>0 alors pour j de 0 jusque size(S)-1 faire for(k:=0;k<nf;k++){ kk:=(evalf(S[j])>=evalf(L[0])) and (evalf(S[j])<=evalf(L[nl-1])); kK:=(evalf(S[j])<evalf(Imin[k])) or (evalf(S[j])>evalf(Imax[k])); Kk:=(kk) and kK; if(Kk==1){Z:=append(Z,simplify(S[j]))}; } fpour fsi; Z:=sort([op(set[op(Z)])]); nz:=size(Z); for(j:=0;j<nf;j++){ for(k:=1;k<nz;k++){ if ((Z[k]>Imin[j])and(Z[k]<Imax[j])){Z:=augment(Z[0..k-1],Z[k+1..nz-1]);nz:=nz-1; } } } nz:=size(Z); l0:=" newLigneVariables(btex $"+nomv+"$ etex);";lp:=" "; lf:=" ";lsp:=" "; pour m de 0 jusque nz-1 faire l0:=l0+"val(btex $"+latex(Z[m])+"$ etex); ";fpour; k0:= evalf(limit(f(x),x=Z[0],1))> evalf(limit(f(x),x=Z[1],-1)); kz:=evalf(limit(f(x),x=Z[nz-1],-1))> evalf(limit(f(x),x=Z[nz-2],1)); lsi:=lsic+nom+"'("+nomv+")}$ etex);"+ if(member(Z[0],IMIN)!=0){if((member(Z[0],F)==0) and (fp(Z[0])!=undef)){"debutNonDef;"}else{"debutNonDefStrict;"}}else{if(Z[0]==-infinity){if(sign(evalf(fp(if(Z[1]==+infinity){0}else{Z[1]-10^(-5)})))==1.0){"plus;"}else{"moins;"}}else{if(member(Z[0],F)==0){ if(fp(Z[0])==0){"valBarre(btex 0 etex);"}else{" "}+ if(evalf(sign(fp(Z[0]+10^(-5))))==1.0){"plus;"}else{"moins;"}}else{"nonDefBarre;"+ if(evalf(sign(fp((Z[0]+10^(-5)))))==1.0){"plus;"}else{"moins;"} }}} // modif 3 avril 2010 if(nz>2){ for(r:=1; r<=nz-2;r++){ lsp:=lsp+ if(member(Z[r],IMIN)!=0){ if((member(Z[r],F)==0) and (fp(Z[r])!=undef)){"debutNonDef;"} else{"debutNonDefStrict;"}} else{if(member(Z[r],IMAX)!=0){if((member(Z[r],F)==0) and (fp(Z[r])!=undef)){"finNonDef;"} else{"finNonDefStrict;"}+ if(evalf(fp(Z[r]+0.01))>0){"plus;"} else{"moins;"}} else{if(member(Z[r],F)==0){"valBarre(btex 0 etex);"} else{"nonDefBarre;"}+ if(evalf(fp(Z[r]+0.01))>0){"plus;"} else{"moins;"} }} }}; // fin modif lsf:=if(member(Z[nz-1],IMAX)!=0){if((member(Z[nz-1],F)==0) and (fp(Z[nz-1])!=undef)){"finNonDef;"}else{"finNonDefStrict;"}}else{if(member(Z[nz-1],F)==0){""}else{"nonDefBarre; "}} lm0:=limit(f(x),x=Z[0],1)==-infinity; li:=lvic +nom+"}$ etex);"+ if(member(Z[0],F)==0){"valPos(btex $"+if(lm0==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[0],1)))}+"$ etex,"} else{"nonDefBarre;limDroite(btex $"+if(lm0==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[0],1)))}+"$ etex,"}+ if(k0==1){"1"}else{"0"}+ ");"; if(nz>2){ for(r:=1; r<=nz-2;r++){ krm:=evalf(limit(f(x),x=Z[r-1],1))< evalf(limit(f(x),x=Z[r],-1)); krp:=evalf(limit(f(x),x=Z[r],1))> evalf(limit(f(x),x=Z[r+1],-1)) ; lmrm:=limit(f(x),x=Z[r],-1)==-infinity;lmrp:=limit(f(x),x=Z[r],1)==-infinity; lp:=lp+if(member(Z[r],IMIN)!=0){"limGauche(btex $"+if(lmrm==1){ "-\\infty"}else{ latex(simplify(limit(f(x),x=Z[r],-1)))} +"$ etex,"+if(krm==1){ "1);"}else{"0);"} +if(member(Z[r],F)==0){"debutNonDef;"}else{"debutNonDefStrict;"} }//fsi Zr=Imin else{ if (member(Z[r],IMAX)!=0){if(member(Z[r],F)==0){"finNonDef;"}else{"finNonDefStrict;"}+"limDroite(btex $"+if(lmrp==1){ "-\\infty"}else{latex(simplify(limit(f(x),x=Z[r],1)))} +"$ etex,"+if(krp==1){ "1);"}else{"0);"} }else{ if(member(Z[r],F)){ "limGauche(btex $"+if(lmrm==1){ "-\\infty"}else{ latex(simplify(limit(f(x),x=Z[r],-1)))} +"$ etex,"+if(krm==1){ "1);"}else{"0);"} +"nonDefBarre;limDroite(btex $"+if(lmrp==1){ "-\\infty"}else{latex(simplify(limit(f(x),x=Z[r],1)))} +"$ etex,"+if(krp==1){ "1);"}else{"0);"} }//fsi (member Zr F) else{"valPos(btex$"+latex(simplify(f(Z[r])))+"$etex,"+ if(sign(evalf(fp(Z[r]-0.01)))==sign(fp(Z[r]+0.01))){ "0.5);"}else{ if(krp==1){ "1);"}else{"0);"}//felse(krp) }//felse(valpos) }//felse(member Zr F) } //felse(Zr=Imax) }//felse(Zr=Imin) };//ffor }//fsi nz lnz:=limit(f(x),x=Z[nz-1],-1)==-infinity; lf:=if(member(Z[nz-1],F)==0){"valPos(btex $"+ if(lnz==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[nz-1],-1)))}+"$ etex,"+ if(kz==1){"1);"}else{"0);"}} else{"limGauche(btex $"+ if(lnz==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[nz-1],-1)))}+"$ etex,"+ if(kz==1){"1);nonDefBarre;"}else{"0);nonDefBarre;"}}; MetaLfc:=if(ftt==2){if(nz>2){" beginTableau("+nmr+")"+ l0+lsi+lsp+lsf+" endTableau; ";}else{ "beginTableau("+nmr+")"+ l0+ lsi+lsf+" endTableau; "; } }else{ if(ftt==0){if(nz>2){"beginTableau("+nmr+")"+ l0+ li+ lp+ lf +" endTableau; ";}else{"beginTableau("+nmr+")"+ l0+ li+ lf +" endTableau; ";}}else{ if(nz>2){"beginTableau("+nmr+")"+ l0+ lsi+lsp+lsf+ li+ lp+ lf +" endTableau; ";}else{"beginTableau("+nmr+")"+ l0+ lsi+lsf+ li+ lf +" endTableau; ";} }} sortie:=fopen("XCasmpfc.mp"); fprint(sortie,Unquoted,MetaLfc); fclose(sortie); } :; \end{VerbatimOut} %% %% %% Quand les solutions formelles de f'(x)=0 ne sont pas calculables %% \begin{VerbatimOut}{XcasTVapp.cxx} TVapp(L,F,nom,nomv,f,ftt,nmr):={ local nl,fp,z0,z,nz,S,k,j,m,kk,kok,Z,l0,lp,lf,lsp,k0,kz,lsi,r,ksp,lsf,lm0,li,krm,krp,lmrm,lmrp,lnz; nl:=size(L); f:=unapply(f,x); fp:=function_diff(f); z0:=concat(L,F);z:=sort(z0); nz:=size(z); S:=NULL; if(L==[-infinity,+infinity]){j:=[seq(-50+2*k,k=0..50)]minus F; for k in j do for(m:=-5;m<=5;m++){S:=S,resoudre_numerique(fp(y),y,k+m*0.1,k+(m+1)*0.1,bisection_solver)};end_for} else{if(L[0]==-infinity){j:=[seq(2*k,k=-25..0.5*floor(L[1]))] minus F; for k in j do for(m:=-5;m<=5;m++){ S:=S,resoudre_numerique(fp(y),y,k+m*0.1,k+(m+1)*0.1,bisection_solver)};end_for} else{if(L[1]==+infinity){ j:=[seq(2*k,k=floor(0.5*L[0])..0.5*50)] minus F; for k in j do for(m:=-5;m<=5;m++){ S:=S,resoudre_numerique(fp(y),y,k+m*0.1,k+(m+1)*0.1,bisection_solver)};end_for } else{ j:=[seq(2*k,k=0.5*floor(z[0])..0.5*floor(z[nz-1]))] minus F; for k in j do for(m:=-5;m<=5;m++){S:=S,resoudre_numerique(fp(y),y,k+m*0.1,k+(m+1)*0.1,bisection_solver)};end_for } }}; si size(S)>0 alors pour j de 0 jusque size(S)-1 faire kk:=(re(S[j])==S[j]);kok:=(evalf(S[j])>=L[0]) and (evalf(S[j])<=L[1]); if(kk==1){if(kok==1){z:=append(z,simplify(S[j]))}}; fpour; fsi; S:=NULL; S:=S,z[0]; for(j:=1;j<size(z);j++){ if(z[j]!=undef and (abs(z[j])>1e-15 or z[j]==0)){ S:=S,z[j]}; } z:=[S]; Z:=sort(z); nz:=size(Z); S:=NULL; S:=S,Z[0]; for(j:=1;j<nz;j++){ if(Z[j]!=S[size(S)-1]){ S:=S,Z[j]}; } Z:=[S]; nz:=size(Z); l0:=" newLigneVariables(btex $"+nomv+"$ etex);";lp:=" "; lf:=" ";lsp:=" "; pour m de 0 jusque nz-1 faire l0:=l0+"val(btex $"+latex(Z[m])+"$ etex); ";fpour; k0:= evalf(limit(f(x),x=Z[0],1))> evalf(limit(f(x),x=Z[1],-1)); kz:=evalf(limit(f(x),x=Z[nz-1],-1))> evalf(limit(f(x),x=Z[nz-2],1)); lsi:=lsic+nom+"'("+nomv+")}$ etex);"+ if(Z[0]==-infinity){if(sign(evalf(fp(if(Z[1]==+infinity){0}else{Z[1]-10^(-5)})))==1.0){"plus;"}else{"moins;"}}else{if(member(Z[0],F)==0){ if(fp(Z[0])==0){"valBarre(btex 0 etex);"}else{" "}+ if(evalf(sign(fp(Z[0]+10^(-5))))==1.0){"plus;"}else{"moins;"}}else{"nonDefBarre;"+ if(evalf(sign(fp((Z[0]+10^(-5)))))==1.0){"plus;"}else{"moins;"} }} if(nz>2){ for(r:=1; r<=nz-2;r++){ ksp:=evalf(fp(Z[r]+0.01))>0; lsp:=lsp+if(member(Z[r],F)==0){"valBarre(btex 0 etex);"}else{"nonDefBarre;"}+ if(ksp==1){"plus;"}else{"moins;"} }; } lsf:=if(member(Z[nz-1],F)==0){""}else{"nonDefBarre; "} lm0:=limit(f(x),x=Z[0],1)==-infinity; li:=lvic+nom+"}$ etex);"+ if(member(Z[0],F)==0){"valPos(btex $"+if(lm0==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[0],1)))}+"$ etex,"} else{"nonDefBarre;limDroite(btex $"+if(lm0==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[0],1)))}+"$ etex,"}+ if(k0==1){"1"}else{"0"}+ ");"; if(nz>2){ for(r:=1; r<=nz-2;r++){ krm:=evalf(limit(f(x),x=Z[r-1],1))< evalf(limit(f(x),x=Z[r],-1)); krp:=evalf(limit(f(x),x=Z[r],1))> evalf(limit(f(x),x=Z[r+1],-1)) ; lmrm:=limit(f(x),x=Z[r],-1)==-infinity;lmrp:=limit(f(x),x=Z[r],1)==-infinity; lp:=lp+if(member(Z[r],F)){ "limGauche(btex $"+if(lmrm==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[r],-1)))}+"$ etex,"+if(krm==1){"1);"}else{"0);"}+"nonDefBarre;limDroite(btex $"+if(lmrp==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[r],1)))}+"$ etex,"+if(krp==1){"1);"}else{"0);"}} else{"valPos(btex $"+latex(simplify(f(Z[r])))+"$ etex,"+if(sign(evalf(fp(Z[r]-0.01)))==sign(fp(Z[r]+0.01))){"0.5);"}else{if(krp==1){"1);"}else{"0); "}}} }; } lnz:=limit(f(x),x=Z[nz-1],-1)==-infinity; lf:=if(member(Z[nz-1],F)==0){"valPos(btex $"+ if(lnz==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[nz-1],-1)))}+"$ etex,"+ if(kz==1){"1);"}else{"0);"}} else{"limGauche(btex $"+ if(lnz==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[nz-1],-1)))}+"$ etex,"+ if(kz==1){"1);nonDefBarre;"}else{"0);nonDefBarre;"}}; MetaLfc:=if(ftt==2){if(nz>2){" beginTableau("+nmr+")"+ l0+lsi+lsp+lsf+" endTableau; ";}else{ "beginTableau("+nmr+")"+ l0+ lsi+lsf+" endTableau; "; } }else{ if(ftt==0){if(nz>2){"beginTableau("+nmr+")"+ l0+ li+ lp+ lf +" endTableau; ";}else{"beginTableau("+nmr+")"+ l0+ li+ lf +" endTableau; ";}}else{ if(nz>2){"beginTableau("+nmr+")"+ l0+ lsi+lsp+lsf+ li+ lp+ lf +" endTableau; ";}else{"beginTableau("+nmr+")"+ l0+ lsi+lsf+ li+ lf +" endTableau; ";} }} sortie:=fopen("XCasmpfc.mp"); fprint(sortie,Unquoted,MetaLfc); fclose(sortie); }:; \end{VerbatimOut} %% %% Code giac/Xcas pour les Tableaux de Variations avec %% Valeurs intermediaires %% \begin{VerbatimOut}{XcasTVI.cxx} TVI(L,F,nom,nomv,f,ftt,ao,trigo,nmr):={ nl:=size(L); f:=unapply(f,x); fp:=function_diff(f); Z:=concat(L,F); S:=[]; if(trigo==t){ all_trig_solutions:=1; reset_solve_counter(-1,-1); SS:=solve(factor(simplify(fp(x))),x); ns:=size(SS); for(k:=0;k<ns;k++){ m:=0; while(evalf(simplify(subst(SS[k],n_1=m)))<=evalf(L[nl-1])){ S:=concat(S,simplify(subst(SS[k],n_1=m)));m:=m+1; };m:=-1; while(evalf(subst(SS[k],n_1=m))>=L[0]){ S:=concat(S,simplify(subst(SS[k],n_1=m)));m:=m-1; } } }else{ S:=solve(factor(simplify(fp(x))),x); } si size(S)>0 alors pour j de 0 jusque size(S)-1 faire kk:=(evalf(S[j])>=evalf(L[0])) and (evalf(S[j])<=evalf(L[nl-1])); if(kk==1){Z:=append(Z,simplify(S[j]))}; fpour fsi; Z:=sort([op(set[op(Z)])]); nz:=size(Z); l0:=" newLigneVariables(btex $"+nomv+"$ etex);";lp:=" "; lf:=" ";lsp:=" "; LI:=limit(f(x),x,Z[0],1); LF:=limit(f(x),x,Z[nz-1],-1); LP:=NULL; PB:=1; if(nz>2){ for(r:=1; r<=nz-2;r++){LP:=LP,if(member(Z[r],F)){limit(f(x),x,Z[r],-1),limit(f(x),x,Z[r],1)}else{f(Z[r])}; if(member(Z[r],F)){PB:=PB,0,1}else{PB:=PB,1}; } }; if(nz>2){ LL:=[LI,LP,LF]; PB:=[PB,1]}else{LL:=[LI,LF];PB:=[1,1]}; NL:=size(LL); A:=NULL;aa:=1; kk:=0; if(NL==nz){for(k:=0;k<nz-1;k++){TestS:=(evalf(sign(LL[k]-ao))==evalf(sign(LL[k+1]-ao))) or (evalf(sign(LL[k]-ao))==0.0)or (evalf(sign(LL[k+1]-ao))==0.0); if(TestS==0){A:=A,aa;l0:=l0+"val(btex $"+latex(Z[k])+"$ etex);"+"val(btex $\\alpha_"+aa+"$ etex);";aa:=aa+1;}else{l0:=l0+"val(btex $"+latex(Z[k])+"$ etex);"}} l0:=l0+"val(btex $"+latex(Z[nz-1])+"$ etex);"}; //chgmt NL->nz if(NL>nz){for(k:=0;k<NL-1;k++){TestS:=(evalf(sign(LL[k]-ao))==evalf(sign(LL[k+1]-ao))) or (evalf(sign(LL[k]-ao))==0.0)or (evalf(sign(LL[k+1]-ao))==0.0); if(PB[k]==1){if(TestS==0){ A:=A,aa;l0:=l0+"val(btex $"+latex(Z[kk])+"$ etex);"+"val(btex $\\alpha_"+aa+"$ etex);";aa:=aa+1;kk:=kk+1} else{l0:=l0+"val(btex $"+latex(Z[kk])+"$ etex);";kk:=kk+1}}; } l0:=l0+"val(btex $"+latex(Z[nz-1])+"$ etex);" }; TestS:=(evalf(sign(LL[0]-ao))==evalf(sign(LL[1]-ao))) or (evalf(sign(LL[0]-ao))==0.0) or (evalf(sign(LL[1]-ao))==0.0); k0:= evalf(limit(f(x),x=Z[0],1))> evalf(limit(f(x),x=Z[1],-1)); kz:=evalf(limit(f(x),x=Z[nz-1],-1))> evalf(limit(f(x),x=Z[nz-2],1)); lsi:=lsic+nom+"'("+nomv+")}$ etex);"+ if(Z[0]==-infinity){if(evalf(sign(fp(if(Z[1]==+infinity){0}else{Z[1]-10^(-5)})))==1.0){"plus;"}else{"moins;"}}else{if(member(Z[0],F)==0){ if(fp(Z[0])==0){"valBarre(btex 0 etex);"}else{" "}+ if(evalf(sign(fp((Z[0]+10^(-3)))))==1.0){"plus;"}else{"moins;"}}else{"nonDefBarre;"+ if(evalf(sign(fp(10^(-3)+Z[0])))==1.0){"plus;"}else{"moins;"} }}+if(TestS==0){"valBarre(btex$ $ etex);"+ if(evalf(sign(fp(10^(-3)+ifte(Z[0]==-infinity,ifte(Z[1]==+infinity,ifte(member(0,F)==0,0,0.01),ifte(member(Z[1]-1,F)==0,Z[1]-1,Z[1]-1.1)),Z[0]))))==1.0){"plus;"}else{"moins;"}}else{" "}; if(nz>2){rr:=1; if(nz==NL){for(r:=1; r<=NL-2;r++){ TestS:=(evalf(sign(LL[r]-ao))==evalf(sign(LL[r+1]-ao))) or (evalf(sign(LL[r]-ao))==0.0)or (evalf(sign(LL[r+1]-ao))==0.0); ksp:=evalf(fp(Z[r]+0.01))>0; TestL:=(abs(LL[r])==abs(LL[r+1])); lsp:=lsp+if(member(Z[r],F)==0){"valBarre(btex 0 etex);"}else{"nonDefBarre;"}+ if(ksp==1){"plus;"}else{"moins;"}+if(TestS==0){"valBarre(btex $ $ etex);"}else{" "}+if(TestS==0){if(ksp==1){"plus;"}else{"moins;"}}else{" "}; }} else{for(r:=1; r<=NL-2;r++){kspp:=evalf(fp(Z[rr]+0.01))>0;TestS:=(evalf(sign(LL[r]-ao))==evalf(sign(LL[r+1]-ao))) or (evalf(sign(LL[r]-ao))==0.0)or (evalf(sign(LL[r+1]-ao))==0.0); if(PB[r]==1){if(TestS==0){lsp:=lsp+if(member(Z[rr],F)==0){"valBarre(btex 0 etex);"}else{"nonDefBarre;"}+ if(kspp==1){"plus;"}else{"moins;"}+"valBarre(btex $ $ etex);"+if(kspp==1){"plus;"}else{"moins;"};rr:=rr+1;} else{lsp:=lsp+if(member(Z[rr],F)==0){"valBarre(btex 0 etex);"}else{"nonDefBarre;"}+ if(kspp==1){"plus;"}else{"moins;"};rr:=rr+1;} }}}}; lsf:=if(member(Z[nz-1],F)==0){" "}else{"nonDefBarre;"} lm0:=limit(f(x),x=Z[0],1)==-infinity; TestS:=(evalf(sign(LL[0]-ao))==evalf(sign(LL[1]-ao))) or (evalf(sign(LL[0]-ao))==0.0) or (evalf(sign(LL[1]-ao))==0.0); li:=lvic+nom+"}$ etex); "+ if(member(Z[0],F)==0){"valPos(btex $"+if(lm0==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[0],1)))}+"$ etex,"} else{"nonDefBarre;limDroite(btex $"+if(lm0==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[0],1)))}+"$ etex,"}+ if(k0==1){"1);"}else{"0);"}+if(TestS==0){"valPos(btex $ "+ao+" $ etex,0.5);"}else{" "}; if(nz>2){if(nz==NL){for(r:=1; r<=nz-2;r++){TestS:=(evalf(sign(LL[r]-ao))==evalf(sign(LL[r+1]-ao))) or (evalf(sign(LL[r]-ao))==0.0)or (evalf(sign(LL[r+1]-ao))==0.0); krm:=evalf(limit(f(x),x=Z[r-1],1))< evalf(limit(f(x),x=Z[r],-1)); krp:=evalf(limit(f(x),x=Z[r],1))> evalf(limit(f(x),x=Z[r+1],-1)) ; lmrm:=limit(f(x),x=Z[r],-1)==-infinity;lmrp:=limit(f(x),x=Z[r],1)==-infinity; lp:=lp+if(member(Z[r],F)) { "limGauche(btex $"+if(lmrm==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[r],-1)))}+"$ etex,"+if(krm==1){"1);"}else{"0);"}+"nonDefBarre;limDroite(btex $"+if(lmrp==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[r],1)))}+"$ etex,"+if(krp==1){"1);"}else{"0);"}} else{"valPos(btex $"+latex(simplify(f(Z[r])))+"$ etex,"+if(evalf(sign(fp(Z[r]-0.01)))==sign(fp(Z[r]+0.01))){"0.5);"}else{if(krp==1){"1);"}else{"0);"}}}+if(TestS==0){"valPos(btex $ "+ao+" $ etex,0.5);" }else{" "}; };//for }else{rr:=1;for(r:=1; r<=NL-2;r++){TestS:=(evalf(sign(LL[r]-ao))==evalf(sign(LL[r+1]-ao))) or (evalf(sign(LL[r]-ao))==0.0)or (evalf(sign(LL[r+1]-ao))==0.0); krm:=evalf(limit(f(x),x=Z[rr-1],1))< evalf(limit(f(x),x=Z[rr],-1)); krp:=evalf(limit(f(x),x=Z[rr],1))> evalf(limit(f(x),x=Z[rr+1],-1)) ; lmrm:=limit(f(x),x=Z[rr],-1)==-infinity;lmrp:=limit(f(x),x=Z[rr],1)==-infinity; TestL:=(abs(LL[r])==abs(LL[r+1])); if(PB[r]==1){if(TestS==0){lp:=lp+if(member(Z[rr],F)){ "limGauche(btex $"+if(lmrm==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[rr],-1)))}+"$ etex,"+if(krm==1){"1);"}else{"0);"}+"nonDefBarre;limDroite(btex $"+if(lmrp==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[rr],1)))}+"$ etex,"+if(krp==1){"1);"}else{"0);"}} else{"valPos(btex $"+latex(simplify(f(Z[rr])))+"$ etex,"+if(evalf(sign(fp(Z[rr]-0.01)))==sign(fp(Z[rr]+0.01))){"0.5);"}else{if(krp==1){"1);"}else{"0);"}}}+"valPos(btex $ "+ao+" $ etex,0.5); ";rr:=rr+1; }// testS==0 else{lp:=lp+if(member(Z[rr],F)){ "limGauche(btex $"+if(lmrm==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[rr],-1)))}+"$ etex,"+if(krm==1){"1);"}else{"0);"}+"nonDefBarre;limDroite(btex $"+if(lmrp==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[rr],1)))}+"$ etex,"+if(krp==1){"1);"}else{"0);"}} else{"valPos(btex $"+latex(simplify(f(Z[rr])))+"$ etex,"+if(evalf(sign(fp(Z[rr]-0.01)))==sign(fp(Z[rr]+0.01))){"0.5);"}else{if(krp==1){"1);"}else{"0); "}}};rr:=rr+1; }//else testS==0 }//PB[r]==1 }//for nz<NL }// else nz<NL //if nz=NL };//if nz>2 lnz:=limit(f(x),x=Z[nz-1],-1)==-infinity; lf:=if(member(Z[nz-1],F)==0){"valPos(btex $"+ if(lnz==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[nz-1],-1)))}+"$ etex,"+ if(kz==1){"1);"}else{"0);"}} else{"limGauche(btex $"+ if(lnz==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[nz-1],-1)))}+"$ etex,"+ if(kz==1){"1);nonDefBarre;"}else{"0);nonDefBarre;"}}; MetaLfc:= if(ftt==2){if(nz>2){"beginTableau("+nmr+")"+ l0+lsi+lsp+lsf+" endTableau; ";}else{ "beginTableau("+nmr+")"+ l0+ lsi+lsf+" endTableau; "; } }else{ if(ftt==0){if(nz>2){"beginTableau("+nmr+")"+ l0+ li+ lp+ lf +" endTableau; ";}else{"beginTableau("+nmr+")"+ l0+ li+ lf +" endTableau; ";}}else{ if(nz>2){"beginTableau("+nmr+")"+ l0+ lsi+lsp+lsf+ li+ lp+ lf +" endTableau; ";}else{"beginTableau("+nmr+")"+ l0+ lsi+lsf+ li+ lf +" endTableau; ";} }}; sortie:=fopen("XCasmpfc.mp"); fprint(sortie,Unquoted,MetaLfc); fclose(sortie); }:; \end{VerbatimOut} %% % % % % Pour avoir les racines sous forme exacte.... quand c'est possible ! % % % %%%%%%%%% \begin{VerbatimOut}{XcasTVIex.cxx} TVIex(L,F,nom,nomv,f,ftt,ao,trigo,nmr):={ nl:=size(L); f:=unapply(f,x); fp:=function_diff(f); Z:=concat(L,F); S:=[]; Sex:=NULL; Zex:=solve(f(x)=ao); Zex:=sort(Zex); for(j:=0;j<size(Zex);j++){ if((evalf(Zex[j])>=evalf(L[0])) and (evalf(Zex[j])<=evalf(L[nl-1]))){Sex:=Sex,Zex[j]}; }; Sex:=[Sex]; if(trigo==t){ all_trig_solutions:=1; reset_solve_counter(-1,-1); SS:=solve(factor(simplify(fp(x))),x); ns:=size(SS); for(k:=0;k<ns;k++){ m:=0; while(evalf(simplify(subst(SS[k],n_1=m)))<=evalf(L[nl-1])){ S:=concat(S,simplify(subst(SS[k],n_1=m)));m:=m+1; };m:=-1; while(evalf(subst(SS[k],n_1=m))>=L[0]){ S:=concat(S,simplify(subst(SS[k],n_1=m)));m:=m-1; } } }else{ S:=solve(factor(simplify(fp(x))),x); } si size(S)>0 alors pour j de 0 jusque size(S)-1 faire kk:=(evalf(S[j])>=evalf(L[0])) and (evalf(S[j])<=evalf(L[nl-1])); if(kk==1){Z:=append(Z,simplify(S[j]))}; fpour fsi; Z:=sort([op(set[op(Z)])]); nz:=size(Z); l0:=" newLigneVariables(btex $"+nomv+"$ etex);";lp:=" "; lf:=" ";lsp:=" "; LI:=limit(f(x),x,Z[0],1); LF:=limit(f(x),x,Z[nz-1],-1); LP:=NULL; PB:=1; if(nz>2){ for(r:=1; r<=nz-2;r++){LP:=LP,if(member(Z[r],F)){limit(f(x),x,Z[r],-1),limit(f(x),x,Z[r],1)}else{f(Z[r])}; if(member(Z[r],F)){PB:=PB,0,1}else{PB:=PB,1}; } }; if(nz>2){ LL:=[LI,LP,LF]; PB:=[PB,1]}else{LL:=[LI,LF];PB:=[1,1]}; NL:=size(LL); A:=NULL;aa:=0; kk:=0; if(NL==nz){for(k:=0;k<nz-1;k++){TestS:=(evalf(sign(LL[k]-ao))==evalf(sign(LL[k+1]-ao))) or (evalf(sign(LL[k]-ao))==0.0)or (evalf(sign(LL[k+1]-ao))==0.0); if(TestS==0){A:=A,aa;l0:=l0+"val(btex $"+latex(Z[k])+"$ etex);"+"val(btex $"+latex(simplify(Sex[aa]))+"$ etex);";aa:=aa+1;}else{l0:=l0+"val(btex $"+latex(Z[k])+"$ etex);"}} l0:=l0+"val(btex $"+latex(Z[nz-1])+"$ etex);"}; //chgmt NL->nz if(NL>nz){for(k:=0;k<NL-1;k++){TestS:=(evalf(sign(LL[k]-ao))==evalf(sign(LL[k+1]-ao))) or (evalf(sign(LL[k]-ao))==0.0)or (evalf(sign(LL[k+1]-ao))==0.0); if(PB[k]==1){if(TestS==0){ A:=A,aa;l0:=l0+"val(btex $"+latex(Z[kk])+"$ etex);"+"val(btex $"+latex(simplify(Sex[aa]))+"$ etex);";aa:=aa+1;kk:=kk+1} else{l0:=l0+"val(btex $"+latex(Z[kk])+"$ etex);";kk:=kk+1}}; } l0:=l0+"val(btex $"+latex(Z[nz-1])+"$ etex);" }; TestS:=(evalf(sign(LL[0]-ao))==evalf(sign(LL[1]-ao))) or (evalf(sign(LL[0]-ao))==0.0) or (evalf(sign(LL[1]-ao))==0.0); k0:= evalf(limit(f(x),x=Z[0],1))> evalf(limit(f(x),x=Z[1],-1)); kz:=evalf(limit(f(x),x=Z[nz-1],-1))> evalf(limit(f(x),x=Z[nz-2],1)); lsi:=lsic+nom+"'("+nomv+")}$ etex);"+ if(Z[0]==-infinity){if(evalf(sign(fp(if(Z[1]==+infinity){0}else{Z[1]-10^(-5)})))==1){"plus;"}else{"moins;"}}else{if(member(Z[0],F)==0){ if(fp(Z[0])==0){"valBarre(btex 0 etex);"}else{" "}+ if(sign(fp((Z[0]+10^(-3))))==1){"plus;"}else{"moins;"}}else{"nonDefBarre;"+ if(sign(fp(10^(-3)+Z[0]))==1){"plus;"}else{"moins;"} }}+if(TestS==0){"valBarre(btex$ $ etex);"+ if(evalf(sign(fp(10^(-3)+ifte(Z[0]==-infinity,ifte(Z[1]==+infinity,ifte(member(0,F)==0,0,0.01),ifte(member(Z[1]-1,F)==0,Z[1]-1,Z[1]-1.1)),Z[0]))))==1.0){"plus;"}else{"moins;"}}else{" "}; if(nz>2){rr:=1; if(nz==NL){for(r:=1; r<=NL-2;r++){ TestS:=(evalf(sign(LL[r]-ao))==evalf(sign(LL[r+1]-ao))) or (evalf(sign(LL[r]-ao))==0.0)or (evalf(sign(LL[r+1]-ao))==0.0); ksp:=evalf(fp(Z[r]+0.01))>0; TestL:=(abs(LL[r])==abs(LL[r+1])); lsp:=lsp+if(member(Z[r],F)==0){"valBarre(btex 0 etex);"}else{"nonDefBarre;"}+ if(ksp==1){"plus;"}else{"moins;"}+if(TestS==0){"valBarre(btex $ $ etex);"}else{" "}+if(TestS==0){if(ksp==1){"plus;"}else{"moins;"}}else{" "}; }} else{for(r:=1; r<=NL-2;r++){kspp:=evalf(fp(Z[rr]+0.01))>0;TestS:=(evalf(sign(LL[r]-ao))==evalf(sign(LL[r+1]-ao))) or (evalf(sign(LL[r]-ao))==0.0)or (evalf(sign(LL[r+1]-ao))==0.0); if(PB[r]==1){if(TestS==0){lsp:=lsp+if(member(Z[rr],F)==0){"valBarre(btex 0 etex);"}else{"nonDefBarre;"}+ if(kspp==1){"plus;"}else{"moins;"}+"valBarre(btex $ $ etex);"+if(kspp==1){"plus;"}else{"moins;"};rr:=rr+1;} else{lsp:=lsp+if(member(Z[rr],F)==0){"valBarre(btex 0 etex);"}else{"nonDefBarre;"}+ if(kspp==1){"plus;"}else{"moins;"};rr:=rr+1;} }}}}; lsf:=if(member(Z[nz-1],F)==0){" "}else{"nonDefBarre;"} lm0:=limit(f(x),x=Z[0],1)==-infinity; TestS:=(evalf(sign(LL[0]-ao))==evalf(sign(LL[1]-ao))) or (evalf(sign(LL[0]-ao))==0.0) or (evalf(sign(LL[1]-ao))==0.0); li:=lvic+nom+"}$ etex); "+ if(member(Z[0],F)==0){"valPos(btex $"+if(lm0==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[0],1)))}+"$ etex,"} else{"nonDefBarre;limDroite(btex $"+if(lm0==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[0],1)))}+"$ etex,"}+ if(k0==1){"1);"}else{"0);"}+if(TestS==0){"valPos(btex $ "+ao+" $ etex,0.5);"}else{" "}; if(nz>2){if(nz==NL){for(r:=1; r<=nz-2;r++){TestS:=(evalf(sign(LL[r]-ao))==evalf(sign(LL[r+1]-ao))) or (evalf(sign(LL[r]-ao))==0.0)or (evalf(sign(LL[r+1]-ao))==0.0); krm:=evalf(limit(f(x),x=Z[r-1],1))< evalf(limit(f(x),x=Z[r],-1)); krp:=evalf(limit(f(x),x=Z[r],1))> evalf(limit(f(x),x=Z[r+1],-1)) ; lmrm:=limit(f(x),x=Z[r],-1)==-infinity;lmrp:=limit(f(x),x=Z[r],1)==-infinity; lp:=lp+if(member(Z[r],F)) { "limGauche(btex $"+if(lmrm==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[r],-1)))}+"$ etex,"+if(krm==1){"1);"}else{"0);"}+"nonDefBarre;limDroite(btex $"+if(lmrp==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[r],1)))}+"$ etex,"+if(krp==1){"1);"}else{"0);"}} else{"valPos(btex $"+latex(simplify(f(Z[r])))+"$ etex,"+if(evalf(sign(fp(Z[r]-0.01)))==sign(fp(Z[r]+0.01))){"0.5);"}else{if(krp==1){"1);"}else{"0);"}}}+if(TestS==0){"valPos(btex $ "+ao+" $ etex,0.5);" }else{" "}; };//for }else{rr:=1;for(r:=1; r<=NL-2;r++){TestS:=(evalf(sign(LL[r]-ao))==evalf(sign(LL[r+1]-ao))) or (evalf(sign(LL[r]-ao))==0.0)or (evalf(sign(LL[r+1]-ao))==0.0); krm:=evalf(limit(f(x),x=Z[rr-1],1))< evalf(limit(f(x),x=Z[rr],-1)); krp:=evalf(limit(f(x),x=Z[rr],1))> evalf(limit(f(x),x=Z[rr+1],-1)) ; lmrm:=limit(f(x),x=Z[rr],-1)==-infinity;lmrp:=limit(f(x),x=Z[rr],1)==-infinity; TestL:=(abs(LL[r])==abs(LL[r+1])); if(PB[r]==1){if(TestS==0){lp:=lp+if(member(Z[rr],F)){ "limGauche(btex $"+if(lmrm==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[rr],-1)))}+"$ etex,"+if(krm==1){"1);"}else{"0);"}+"nonDefBarre;limDroite(btex $"+if(lmrp==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[rr],1)))}+"$ etex,"+if(krp==1){"1);"}else{"0);"}} else{"valPos(btex $"+latex(simplify(f(Z[rr])))+"$ etex,"+if(evalf(sign(fp(Z[rr]-0.01)))==sign(fp(Z[rr]+0.01))){"0.5);"}else{if(krp==1){"1);"}else{"0);"}}}+"valPos(btex $ "+ao+" $ etex,0.5); ";rr:=rr+1; }// testS==0 else{lp:=lp+if(member(Z[rr],F)){ "limGauche(btex $"+if(lmrm==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[rr],-1)))}+"$ etex,"+if(krm==1){"1);"}else{"0);"}+"nonDefBarre;limDroite(btex $"+if(lmrp==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[rr],1)))}+"$ etex,"+if(krp==1){"1);"}else{"0);"}} else{"valPos(btex $"+latex(simplify(f(Z[rr])))+"$ etex,"+if(evalf(sign(fp(Z[rr]-0.01)))==sign(fp(Z[rr]+0.01))){"0.5);"}else{if(krp==1){"1);"}else{"0); "}}};rr:=rr+1; }//else testS==0 }//PB[r]==1 }//for nz<NL }// else nz<NL //if nz=NL };//if nz>2 lnz:=limit(f(x),x=Z[nz-1],-1)==-infinity; lf:=if(member(Z[nz-1],F)==0){"valPos(btex $"+ if(lnz==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[nz-1],-1)))}+"$ etex,"+ if(kz==1){"1);"}else{"0);"}} else{"limGauche(btex $"+ if(lnz==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[nz-1],-1)))}+"$ etex,"+ if(kz==1){"1);nonDefBarre;"}else{"0);nonDefBarre;"}}; MetaLfc:= if(ftt==2){if(nz>2){"beginTableau("+nmr+")"+ l0+lsi+lsp+lsf+" endTableau; ";}else{ "beginTableau("+nmr+")"+ l0+ lsi+lsf+" endTableau; "; } }else{ if(ftt==0){if(nz>2){"beginTableau("+nmr+")"+ l0+ li+ lp+ lf +" endTableau; ";}else{"beginTableau("+nmr+")"+ l0+ li+ lf +" endTableau; ";}}else{ if(nz>2){"beginTableau("+nmr+")"+ l0+ lsi+lsp+lsf+ li+ lp+ lf +" endTableau; ";}else{"beginTableau("+nmr+")"+ l0+ lsi+lsf+ li+ lf +" endTableau; ";} }}; sortie:=fopen("XCasmpfc.mp"); fprint(sortie,Unquoted,MetaLfc); fclose(sortie); }:; \end{VerbatimOut} %% %% %% Quand les solutions de f'(x)=0 ne sont pas formellement calculables %% \begin{VerbatimOut}{XcasTVIapp.cxx} TVIapp(L,F,nom,nomv,f,ftt,ao,nmr):={ nl:=size(L); f:=unapply(f,x); fp:=function_diff(f); z0:=concat(L,F);z:=sort(z0); nz:=size(z); S:=op(fsolve(fp(x),x)); if(L==[-infinity,+infinity]){j:=[seq(-50+2*k,k=0..50)]minus F; for k in j do for(m:=-5;m<=5;m++){S:=S,resoudre_numerique(fp(y),y,k+m*0.1,k+(m+1)*0.1,bisection_solver)};end_for} else{if(L[0]==-infinity){j:=[seq(2*k,k=-25..0.5*floor(L[1]))] minus F; for k in j do for(m:=-5;m<=5;m++){ S:=S,resoudre_numerique(fp(y),y,k+m*0.1,k+(m+1)*0.1,bisection_solver)};end_for} else{if(L[1]==+infinity){ j:=[seq(2*k,k=floor(0.5*L[0])..0.5*50)] minus F; for k in j do for(m:=-5;m<=5;m++){ S:=S,resoudre_numerique(fp(y),y,k+m*0.1,k+(m+1)*0.1,bisection_solver)};end_for } else{ j:=[seq(2*k,k=0.5*floor(z[0])..0.5*floor(z[nz-1]))] minus F; for k in j do for(m:=-5;m<=5;m++){S:=S,resoudre_numerique(fp(y),y,k+m*0.1,k+(m+1)*0.1,bisection_solver)};end_for } }}; si size(S)>0 alors pour j de 0 jusque size(S)-1 faire kk:=(re(S[j])==S[j]);kok:=(evalf(S[j])>=L[0]) and (evalf(S[j])<=L[1]); if(kk==1){if(kok==1){z:=append(z,simplify(S[j]))}}; fpour; fsi; S:=NULL; S:=S,z[0]; for(j:=1;j<size(z);j++){ if(z[j]!=undef and (abs(z[j])>1e-15 or z[j]==0)){ S:=S,z[j]}; } z:=[S]; Z:=sort(z); nz:=size(Z); S:=NULL; S:=S,Z[0]; for(j:=1;j<nz;j++){ if(Z[j]!=S[size(S)-1]){ S:=S,Z[j]}; } Z:=[S]; nz:=size(Z); l0:=" newLigneVariables(btex $"+nomv+"$ etex);";lp:=" "; lf:=" ";lsp:=" "; LI:=limit(f(x),x,Z[0],1); LF:=limit(f(x),x,Z[nz-1],-1); LP:=NULL; PB:=1; if(nz>2){ for(r:=1; r<=nz-2;r++){LP:=LP,if(member(Z[r],F)){limit(f(x),x,Z[r],-1),limit(f(x),x,Z[r],1)}else{f(Z[r])}; if(member(Z[r],F)){PB:=PB,0,1}else{PB:=PB,1}; } }; if(nz>2){ LL:=[LI,LP,LF]; PB:=[PB,1]}else{LL:=[LI,LF];PB:=[1,1]}; NL:=size(LL); A:=NULL;aa:=1; kk:=0; if(NL==nz){for(k:=0;k<nz-1;k++){TestS:=(evalf(sign(LL[k]-ao))==evalf(sign(LL[k+1]-ao))) or (evalf(sign(LL[k]-ao))==0.0)or (evalf(sign(LL[k+1]-ao))==0.0); if(TestS==0){A:=A,aa;l0:=l0+"val(btex $"+latex(Z[k])+"$ etex);"+"val(btex $\\alpha_"+aa+"$ etex);";aa:=aa+1;}else{l0:=l0+"val(btex $"+latex(Z[k])+"$ etex);"}} l0:=l0+"val(btex $"+latex(Z[nz-1])+"$ etex);"}; //chgmt NL->nz if(NL>nz){for(k:=0;k<NL-1;k++){TestS:=(evalf(sign(LL[k]-ao))==evalf(sign(LL[k+1]-ao))) or (evalf(sign(LL[k]-ao))==0.0)or (evalf(sign(LL[k+1]-ao))==0.0); if(PB[k]==1){if(TestS==0){ A:=A,aa;l0:=l0+"val(btex $"+latex(Z[kk])+"$ etex);"+"val(btex $\\alpha_"+aa+"$ etex);";aa:=aa+1;kk:=kk+1} else{l0:=l0+"val(btex $"+latex(Z[kk])+"$ etex);";kk:=kk+1}}; } l0:=l0+"val(btex $"+latex(Z[nz-1])+"$ etex);" }; TestS:=(evalf(sign(LL[0]-ao))==evalf(sign(LL[1]-ao))) or (evalf(sign(LL[0]-ao))==0.0) or (evalf(sign(LL[1]-ao))==0.0); k0:= evalf(limit(f(x),x=Z[0],1))> evalf(limit(f(x),x=Z[1],-1)); kz:=evalf(limit(f(x),x=Z[nz-1],-1))> evalf(limit(f(x),x=Z[nz-2],1)); lsi:=lsic+nom+"'("+nomv+")}$ etex);"+ if(Z[0]==-infinity){if(evalf(sign(fp(if(Z[1]==+infinity){0}else{Z[1]-10^(-5)})))==1){"plus;"}else{"moins;"}}else{if(member(Z[0],F)==0){ if(fp(Z[0])==0){"valBarre(btex 0 etex);"}else{" "}+ if(sign(fp((Z[0]+10^(-3))))==1){"plus;"}else{"moins;"}}else{"nonDefBarre;"+ if(sign(fp(10^(-3)+Z[0]))==1){"plus;"}else{"moins;"} }}+if(TestS==0){"valBarre(btex$ $ etex);"+ if(evalf(sign(fp(10^(-3)+ifte(Z[0]==-infinity,ifte(Z[1]==+infinity,ifte(member(0,F)==0,0,0.01),ifte(member(Z[1]-1,F)==0,Z[1]-1,Z[1]-1.1)),Z[0]))))==1.0){"plus;"}else{"moins;"}}else{" "}; if(nz>2){rr:=1; if(nz==NL){for(r:=1; r<=NL-2;r++){ TestS:=(evalf(sign(LL[r]-ao))==evalf(sign(LL[r+1]-ao))) or (evalf(sign(LL[r]-ao))==0.0)or (evalf(sign(LL[r+1]-ao))==0.0); ksp:=evalf(fp(Z[r]+0.01))>0; TestL:=(abs(LL[r])==abs(LL[r+1])); lsp:=lsp+if(member(Z[r],F)==0){"valBarre(btex 0 etex);"}else{"nonDefBarre;"}+ if(ksp==1){"plus;"}else{"moins;"}+if(TestS==0){"valBarre(btex $ $ etex);"}else{" "}+if(TestS==0){if(ksp==1){"plus;"}else{"moins;"}}else{" "}; }} else{for(r:=1; r<=NL-2;r++){kspp:=evalf(fp(Z[rr]+0.01))>0;TestS:=(evalf(sign(LL[r]-ao))==evalf(sign(LL[r+1]-ao))) or (evalf(sign(LL[r]-ao))==0.0)or (evalf(sign(LL[r+1]-ao))==0.0); if(PB[r]==1){if(TestS==0){lsp:=lsp+if(member(Z[rr],F)==0){"valBarre(btex 0 etex);"}else{"nonDefBarre;"}+ if(kspp==1){"plus;"}else{"moins;"}+"valBarre(btex $ $ etex);"+if(kspp==1){"plus;"}else{"moins;"};rr:=rr+1;} else{lsp:=lsp+if(member(Z[rr],F)==0){"valBarre(btex 0 etex);"}else{"nonDefBarre;"}+ if(kspp==1){"plus;"}else{"moins;"};rr:=rr+1;} }}}}; lsf:=if(member(Z[nz-1],F)==0){" "}else{"nonDefBarre;"} lm0:=limit(f(x),x=Z[0],1)==-infinity; TestS:=(evalf(sign(LL[0]-ao))==evalf(sign(LL[1]-ao))) or (evalf(sign(LL[0]-ao))==0.0) or (evalf(sign(LL[1]-ao))==0.0); li:=lvic+nom+"}$ etex); "+ if(member(Z[0],F)==0){"valPos(btex $"+if(lm0==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[0],1)))}+"$ etex,"} else{"nonDefBarre;limDroite(btex $"+if(lm0==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[0],1)))}+"$ etex,"}+ if(k0==1){"1);"}else{"0);"}+if(TestS==0){"valPos(btex $ "+ao+" $ etex,0.5);"}else{" "}; if(nz>2){if(nz==NL){for(r:=1; r<=nz-2;r++){TestS:=(evalf(sign(LL[r]-ao))==evalf(sign(LL[r+1]-ao))) or (evalf(sign(LL[r]-ao))==0.0)or (evalf(sign(LL[r+1]-ao))==0.0); krm:=evalf(limit(f(x),x=Z[r-1],1))< evalf(limit(f(x),x=Z[r],-1)); krp:=evalf(limit(f(x),x=Z[r],1))> evalf(limit(f(x),x=Z[r+1],-1)) ; lmrm:=limit(f(x),x=Z[r],-1)==-infinity;lmrp:=limit(f(x),x=Z[r],1)==-infinity; lp:=lp+if(member(Z[r],F)) { "limGauche(btex $"+if(lmrm==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[r],-1)))}+"$ etex,"+if(krm==1){"1);"}else{"0);"}+"nonDefBarre;limDroite(btex $"+if(lmrp==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[r],1)))}+"$ etex,"+if(krp==1){"1);"}else{"0);"}} else{"valPos(btex $"+latex(simplify(f(Z[r])))+"$ etex,"+if(evalf(sign(fp(Z[r]-0.01)))==sign(fp(Z[r]+0.01))){"0.5);"}else{if(krp==1){"1);"}else{"0);"}}}+if(TestS==0){"valPos(btex $ "+ao+" $ etex,0.5);" }else{" "}; };//for }else{rr:=1;for(r:=1; r<=NL-2;r++){TestS:=(evalf(sign(LL[r]-ao))==evalf(sign(LL[r+1]-ao))) or (evalf(sign(LL[r]-ao))==0.0)or (evalf(sign(LL[r+1]-ao))==0.0); krm:=evalf(limit(f(x),x=Z[rr-1],1))< evalf(limit(f(x),x=Z[rr],-1)); krp:=evalf(limit(f(x),x=Z[rr],1))> evalf(limit(f(x),x=Z[rr+1],-1)) ; lmrm:=limit(f(x),x=Z[rr],-1)==-infinity;lmrp:=limit(f(x),x=Z[rr],1)==-infinity; TestL:=(abs(LL[r])==abs(LL[r+1])); if(PB[r]==1){if(TestS==0){lp:=lp+if(member(Z[rr],F)){ "limGauche(btex $"+if(lmrm==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[rr],-1)))}+"$ etex,"+if(krm==1){"1);"}else{"0);"}+"nonDefBarre;limDroite(btex $"+if(lmrp==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[rr],1)))}+"$ etex,"+if(krp==1){"1);"}else{"0);"}} else{"valPos(btex $"+latex(simplify(f(Z[rr])))+"$ etex,"+if(evalf(sign(fp(Z[rr]-0.01)))==sign(fp(Z[rr]+0.01))){"0.5);"}else{if(krp==1){"1);"}else{"0);"}}}+"valPos(btex $ "+ao+" $ etex,0.5); ";rr:=rr+1; }// testS==0 else{lp:=lp+if(member(Z[rr],F)){ "limGauche(btex $"+if(lmrm==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[rr],-1)))}+"$ etex,"+if(krm==1){"1);"}else{"0);"}+"nonDefBarre;limDroite(btex $"+if(lmrp==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[rr],1)))}+"$ etex,"+if(krp==1){"1);"}else{"0);"}} else{"valPos(btex $"+latex(simplify(f(Z[rr])))+"$ etex,"+if(evalf(sign(fp(Z[rr]-0.01)))==sign(fp(Z[rr]+0.01))){"0.5);"}else{if(krp==1){"1);"}else{"0); "}}};rr:=rr+1; }//else testS==0 }//PB[r]==1 }//for nz<NL }// else nz<NL //if nz=NL };//if nz>2 lnz:=limit(f(x),x=Z[nz-1],-1)==-infinity; lf:=if(member(Z[nz-1],F)==0){"valPos(btex $"+ if(lnz==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[nz-1],-1)))}+"$ etex,"+ if(kz==1){"1);"}else{"0);"}} else{"limGauche(btex $"+ if(lnz==1){"-\\infty"}else{latex(simplify(limit(f(x),x=Z[nz-1],-1)))}+"$ etex,"+ if(kz==1){"1);nonDefBarre;"}else{"0);nonDefBarre;"}}; MetaLfc:= if(ftt==2){if(nz>2){"beginTableau("+nmr+")"+ l0+lsi+lsp+lsf+" endTableau; ";}else{ "beginTableau("+nmr+")"+ l0+ lsi+lsf+" endTableau; "; } }else{ if(ftt==0){if(nz>2){"beginTableau("+nmr+")"+ l0+ li+ lp+ lf +" endTableau; ";}else{"beginTableau("+nmr+")"+ l0+ li+ lf +" endTableau; ";}}else{ if(nz>2){"beginTableau("+nmr+")"+ l0+ lsi+lsp+lsf+ li+ lp+ lf +" endTableau; ";}else{"beginTableau("+nmr+")"+ l0+ lsi+lsf+ li+ lf +" endTableau; ";} }}; sortie:=fopen("XCasmpfc.mp"); fprint(sortie,Unquoted,MetaLfc); fclose(sortie); }:; \end{VerbatimOut} %% %% Code giac/Xcas pour les Tableaux de variations de courbes parametrees %% \begin{VerbatimOut}{XcasTVP.cxx} TVP(L,F,nom,nomv,ff,ftt,trigo,nmr):={ nl:=size(L); fp:=[]; S:=[]; f:=[unapply(ff[0],t),unapply(ff[1],t)]; fp:=[function_diff(f[0]),function_diff(f[1])]; Z:=[]; LLL:=[]; all_trig_solutions:=1; reset_solve_counter(-1,-1); for(d:=0;d<=1;d++){ LLL:=concat(L,F[d]); Z:=LLL union Z; SS:=solve(factor(simplify(fp[d](t))),t); ns:=size(SS); for(k:=0;k<ns;k++){ if(trigo==t){ m:=0; while(evalf(simplify(subst(SS[k],n_1=m)))<=evalf(L[nl-1])){ S:=concat(S,simplify(subst(SS[k],n_1=m)));m:=m+1; };m:=-1; while(evalf(subst(SS[k],n_1=m))>=L[0]){ S:=concat(S,simplify(subst(SS[k],n_1=m)));m:=m-1; } }else{ S:=concat(S,SS); } } si size(S)>0 alors pour j de 0 jusque size(S)-1 faire qq:=member(simplify(S[j]),Z)==0; kk:=(evalf(S[j])>=evalf(L[0])) and (evalf(S[j])<=evalf(L[nl-1])); if(kk==1){if(qq==1){Z:=append(Z,simplify(S[j]))}}; fpour fsi; Z:=sort(Z); nz:=size(Z); tantque evalf(Z[0])==evalf(Z[1]) faire Z:=Z[1..nz-1];nz:=size(Z); ftantque; nz:=size(Z); u:=1; tantque (u<nz-2) and (nz>2) faire tantque evalf(Z[u])==evalf(Z[u+1]) faire Z:=augment(Z[0..u-1],Z[u+1..nz-1]);nz:=size(Z); ftantque; u:=u+1; ftantque; }; Z:=sort(Z); nz:=size(Z); l0:=" newLigneVariables(btex $"+nomv+"$ etex);";lp:="",""; lf:="","";lsp:="",""; pour m de 0 jusque nz-1 faire l0:=l0+"val(btex $"+latex(Z[m])+"$ etex); ";fpour; lsi:="",""; FFF:=[[],[]]; for(d:=0;d<=1;d++){ FFF[d]:=concat(F[d],[-infinity,+infinity]); k0:= evalf(limit(f[d](x),x=Z[0],1))> evalf(limit(f[d](x),x=Z[1],-1)); kz:=evalf(limit(f[d](x),x=Z[nz-1],-1))> evalf(limit(f[d](x),x=Z[nz-2],1)); //} //$ lsi[d]:=lsic+nom[d]+"'("+nomv+")}$ etex);"+if(member(Z[0],FFF[d])==0){"valBarre(btex $"+latex(simplify(fp[d](Z[0])))+"$ etex);"}else{if(Z[0]==-infinity){" "}else{"nonDefBarre; "}}+ if(Z[0]==-infinity){if(sign(evalf(fp[d](if(Z[1]==+infinity){0}else{Z[1]-10^(-5)})))==1){"plus;"}else{"moins;"}}else{if(member(Z[0],F[d])==0){ if(sign(fp[d](Z[0]+10^(-5)))==1){"plus;"}else{"moins;"}}else{ if(sign(fp[d]((Z[0]+10^(-5))))==1){"plus;"}else{"moins;"} }} if(nz>2){ for(r:=1; r<=nz-2;r++){ ksp:=evalf(fp[d](Z[r]+0.01))>0; lsp[d]:=lsp[d]+if(member(Z[r],F[d])==0){"valBarre(btex $"+latex(simplify(fp[d](Z[r])))+"$ etex);"}else{"nonDefBarre;"}+ if(ksp==1){"plus;"}else{"moins;"} }; } lsf[d]:=if(member(Z[nz-1],FFF[d])==0){"valBarre(btex $"+latex(simplify(fp[d](Z[nz-1])))+"$ etex);"}else{if(Z[nz-1]==+infinity){" "}else{"nonDefBarre;"}} } lm0:=1,2; li:=1,2; krm:=1,2; krp:=1,2; lmrm:=1,2; lmrp:=1,2; lp:="",""; lnz:=1,2; lf:=1,2; Kz:=1,2;K0:=1,2; for(d:=0;d<=1;d++){ K0[d]:= evalf(limit(f[d](x),x=Z[0],1))> evalf(limit(f[d](x),x=Z[1],-1)); Kz[d]:=evalf(limit(f[d](x),x,Z[nz-1],-1))> evalf(limit(f[d](x),x,Z[nz-2],1)); //{ //$ lm0[d]:=limit(f[d](x),x,Z[0],1)==-infinity; li[d]:=lvic+nom[d]+"}$ etex);"+ if(member(Z[0],F[d])==0){"valPos(btex $"+if(lm0[d]==1){"-\\infty"}else{latex(simplify(limit(f[d](x),x,Z[0],1)))}+"$ etex,"} else{"nonDefBarre;limDroite(btex $"+if(lm0[d]==1){"-\\infty"}else{latex(simplify(limit(f[d](x),x,Z[0],1)))}+"$ etex,"}+ if(K0[d]==1){"1"}else{"0"}+ ");"; if(nz>2){ for(r:=1; r<=nz-2;r++){ krm[d]:=evalf(limit(f[d](x),x=Z[r-1],1))< evalf(limit(f[d](x),x=Z[r],-1)); krp[d]:=evalf(limit(f[d](x),x=Z[r],1))> evalf(limit(f[d](x),x,Z[r+1],-1)) ; lmrm[d]:=limit(f[d](x),x,Z[r],-1)==-infinity;lmrp[d]:=limit(f[d](x),x,Z[r],1)==-infinity; lp[d]:=lp[d]+if(member(Z[r],F[d])){ "limGauche(btex $"+if(lmrm[d]==1){"-\\infty"}else{latex(simplify(limit(f[d](x),x,Z[r],-1)))}+"$ etex,"+if(krm[d]==1){"1);"}else{"0);"}+"nonDefBarre;limDroite(btex $"+if(lmrp[d]==1){"-\\infty"}else{latex(simplify(limit(f[d](x),x,Z[r],1)))}+"$ etex,"+if(krp[d]==1){"1);"}else{"0);"}} else{"valPos(btex $"+latex(simplify(f[d](Z[r])))+"$ etex,"+if(sign(evalf(fp[d](Z[r]-0.001)))==sign(evalf((fp[d](Z[r]+0.001))) )){"0.5);"}else{if(krp[d]==1){"1);"}else{"0); "}}} }; } lnz[d]:=limit(f[d](x),x=Z[nz-1],-1)==-infinity; lf[d]:=if(member(Z[nz-1],F[d])==0){"valPos(btex $"+ if(lnz[d]==1){"-\\infty"}else{latex(simplify(limit(f[d](x),x=Z[nz-1],-1)))}+"$ etex,"+ if(Kz[d]==1){"1);"}else{"0);"}} else{"limGauche(btex $"+ if(lnz[d]==1){"-\\infty"}else{latex(simplify(limit(f[d](x),x=Z[nz-1],-1)))}+"$ etex,"+ if(Kz[d]==1){"1);nonDefBarre;"}else{"0);nonDefBarre;"}}; } MetaLfc:=if(ftt==2){if(nz>2){" beginTableau("+nmr+")"+ l0+lsi[0]+lsp[0]+lsf[0]+lsi[1]+lsp[1]+lsf[1]+" endTableau; ";}else{ "beginTableau("+nmr+")"+ l0+ lsi[0]+lsf[0]+lsi[1]+lsf[1]+" endTableau; "; } }else{ if(ftt==0){if(nz>2){"beginTableau("+nmr+")"+ l0+ li[0]+ lp[0]+ lf[0]+ li[1]+ lp[1]+ lf[1] +" endTableau; ";}else{"beginTableau("+nmr+")"+ l0+ li[0]+ lf[0]+ li[1]+ lf[1] +" endTableau; ";}}else{ if(nz>2){"beginTableau("+nmr+")"+ l0+ lsi[0]+lsp[0]+lsf[0]+ li[0]+ lp[0]+ lf[0]+ lsi[1]+lsp[1]+lsf[1]+ li[1]+ lp[1]+ lf[1] +" endTableau; ";}else{"beginTableau("+nmr+")"+ l0+ lsi[0]+lsf[0]+ li[0]+ lf[0]+ lsi[1]+lsf[1]+ li[1]+ lf[1] +" endTableau; ";} } } sortie:=fopen("XCasmpfc.mp"); fprint(sortie,Unquoted,MetaLfc); fclose(sortie); }:; \end{VerbatimOut} %% %% Code giac/Xcas pour les Tableaux de signes de produits %% \begin{VerbatimOut}{XcasTabSignL.cxx} TS(nomf,L,D,trigo,nmr):={ L:=apply(f->unapply(f,x),L) n:=size(L); Z:=NULL; nl:=size(L); S:=[]; mini:=D[0]; maxi:=D[1]; pour k de 0 jusque n-1 faire if(trigo==t){ all_trig_solutions:=1; reset_solve_counter(-1,-1); SS:=solve(factor(simplify(L[k](x))),x); ns:=size(SS); for(j:=0;j<ns;j++){ m:=0; while(evalf(simplify(subst(SS[j],n_1=m)))<=evalf(maxi)){ S:=concat(S,simplify(subst(SS[j],n_1=m)));m:=m+1; };m:=-1; while(evalf(subst(SS[j],n_1=m))>=evalf(mini)){ S:=concat(S,simplify(subst(SS[j],n_1=m)));m:=m-1; } } }else{ S:=solve(L[k](x),x); } si size(S)>0 alors pour j de 0 jusque size(S)-1 faire if(S[j]>mini and S[j]<maxi){Z:=Z,simplify(S[j]);} fpour; fsi; fpour; Z:=sort(Z); nz:=size(Z); Z:=sort([op(set[(Z)])]); nz:=size(Z); if(nz==0){li:="";l0:="val(btex $"+latex(D[0])+"$ etex);val(btex $"+latex(D[1])+"$ etex);"; for(p:=0;p<=n-1;p++){li:=li+lsic+latex(L[p](x))+"}$ etex);"+ if(mini!=-infinity and L[p](mini)==0){" valBarre(btex 0 etex);"}else{" "}+ if(L[p]((mini+maxi)*0.5>0)){"plus;"}else{"moins;"}+if(maxi!=+infinity and L[p](maxi)==0){" valBarre(btex 0 etex);"}else{" "} } lf:=if(product(L[s]((mini+maxi)*.5),s,0,n-1)>0){"plus;"}else{"moins;"}; MetaLfc:=" beginTableau("+nmr+") newLigneVariables(btex $ {x}$ etex); "+l0+li+ lsic+nomf+"(x)}$ etex);"+ if(mini!=-infinity and product(L[s](mini),s,0,n-1)==0){" valBarre(btex 0 etex);"}else{" "}+ lf+ if(maxi!=+infinity and product(L[s](maxi),s,0,n-1)==0){" valBarre(btex 0 etex);"}else{" "}+" endTableau; " ; }else{ l0:="val(btex $"+latex(D[0])+"$ etex);";li:=" ";lr:=" "; pour m de 0 jusque nz-1 faire l0:=l0+"val(btex $"+latex(Z[m])+"$ etex); ";fpour; l0:=l0+"val(btex $"+latex(D[1])+"$ etex);"; for(p:=0;p<=n-1;p++){lp:=""; li:=li+lsic+latex(L[p](x))+"}$ etex);"+ if(mini!=-infinity and L[p](mini)==0){" valBarre(btex 0 etex);"}else{" "}+ if(L[p](Z[0]-0.01)>0){"plus;"}else{"moins;"}; for(r:=0; r<=nz-2;r++){ lp:=lp+if(simplify(L[p](Z[r]))==0){" valBarre(btex 0 etex);"}else{"barre; "}+ if(L[p]((Z[r]+Z[r+1])*.5)>0){"plus;"}else{"moins;"}}; li:=li+lp+ if(simplify(L[p](Z[nz-1]))==0){"valBarre(btex 0 etex);"}else{"barre;"}+ if(L[p](Z[nz-1]+1.0)>0){"plus;"}else{"moins; "}+if(maxi!=+infinity and L[p](maxi)==0){" valBarre(btex 0 etex);"}else{" "} }; pour t de 0 jusque nz-2 faire lr:=lr+if(product(L[s]((Z[t]+Z[t+1])*.5),s,0,n-1)>0){"plus;"}else{"moins;"}+"valBarre(btex 0 etex);" fpour MetaLfc:=" beginTableau("+nmr+") newLigneVariables(btex $ {x}$ etex); "+l0+ li + lsic+nomf+"(x)}$ etex);"+ if(mini!=-infinity and product(L[s](mini),s,0,n-1)==0){" valBarre(btex 0 etex);"}else{" "}+ if(product(L[s](Z[0]-0.01),s,0,n-1)>0){"plus;"}else{"moins;"}+"valBarre(btex 0 etex);"+ lr+ if(product(L[s](Z[nz-1]+0.01),s,0,n-1)>0){"plus;"}else{"moins;"}+ if(maxi!=+infinity and product(L[s](maxi),s,0,n-1)==0){" valBarre(btex 0 etex);"}else{" "}+" endTableau; "; } sortie:=fopen("XCasmpfc.mp"); fprint(sortie,Unquoted,MetaLfc); fclose(sortie); }:; \end{VerbatimOut} %% %% Code giac/Xcas pour les Tableaux de signes de quotients %% \begin{VerbatimOut}{XcasTabSignQ.cxx} TSq(nomf,L,Fo,D,trigo,nmr):={ L:=apply(f->unapply(f,x),L); Fo:=apply(f->unapply(f,x),Fo); L:=concat(L,Fo); n:=size(L); Z:=NULL; m:=size(Fo); F:=NULL;FF:=NULL; mini:=D[0]; maxi:=D[1]; S:=[]; SF:=[]; pour k de 0 jusque n-1 faire if(trigo==t){ all_trig_solutions:=1; reset_solve_counter(-1,-1); SS:=solve(factor(simplify(L[k](x))),x); ns:=size(SS); for(j:=0;j<ns;j++){ mm:=0; while(evalf(simplify(subst(SS[j],n_1=mm)))<=evalf(maxi)){ S:=concat(S,simplify(subst(SS[j],n_1=mm)));mm:=mm+1; };mm:=-1; while(evalf(subst(SS[j],n_1=mm))>=evalf(mini)){ S:=concat(S,simplify(subst(SS[j],n_1=mm)));mm:=mm-1; } } }else{ S:=concat(S,solve(L[k](x),x)); } si size(S)>0 alors pour j de 0 jusque size(S)-1 faire if(S[j]>mini and S[j]<maxi){Z:=Z,simplify(S[j]);} fpour; fsi; fpour; pour k de 0 jusque m-1 faire if(trigo==t){ all_trig_solutions:=1; reset_solve_counter(-1,-1); SSF:=solve(factor(simplify(Fo[k](x))),x); nsf:=size(SSF); for(j:=0;j<nsf;j++){ mm:=0; while(evalf(simplify(subst(SSF[j],n_1=mm)))<=evalf(maxi)){ SF:=concat(SF,simplify(subst(SSF[j],n_1=mm)));mm:=mm+1; };mm:=-1; while(evalf(subst(SSF[j],n_1=mm))>=evalf(mini)){ SF:=concat(SF,simplify(subst(SSF[j],n_1=mm)));mm:=mm-1; } } }else{ SF:=concat(SF,solve(Fo[k](x),x)); } si size(SF)>0 alors pour j de 0 jusque size(SF)-1 faire FF:=FF,simplify(SF[j]); if(SF[j]>mini and SF[j]<maxi){F:=F,simplify(SF[j]);} fpour; fsi; fpour; Z:=[Z,F]; Z:=sort([op(set[op(Z)])]); nz:=size(Z); if(nz==0){li:="";l0:="val(btex $"+latex(D[0])+"$ etex);val(btex $"+latex(D[1])+"$ etex);"; for(p:=0;p<=n-1;p++){li:=li+lsic+latex(L[p](x))+"}$ etex);"+ if(mini!=-infinity and L[p](mini)==0){" valBarre(btex 0 etex);"}else{" "}+ if(L[p]((mini+maxi)*0.5>0)){"plus;"}else{"moins;"}+if(maxi!=+infinity and L[p](maxi)==0){" valBarre(btex 0 etex);"}else{" "} } lf:=if(product(L[s]((mini+maxi)*.5),s,0,n-1)>0){"plus;"}else{"moins;"}; MetaLfc:=" beginTableau("+nmr+") newLigneVariables(btex $ {x}$ etex); "+l0+li+ lsic+nomf+"(x)}$ etex);"+ if(member(mini,FF)==0){if((mini!=-infinity) and (product(L[s](mini),s,0,n-1)==0)){" valBarre(btex 0 etex);"}else{" "}}else{"nonDefBarre;"}+ lf+ if(member(maxi,FF)==0){if((maxi!=+infinity) and (product(L[s](maxi),s,0,n-1)==0)){" valBarre(btex 0 etex);"}else{" "}}else{"nonDefBarre;"}+" endTableau; " ; }else{ l0:="val(btex $"+latex(D[0])+"$ etex);";li:=" ";lr:=" "; pour m de 0 jusque nz-1 faire l0:=l0+"val(btex $"+latex(Z[m])+"$ etex); ";fpour; l0:=l0+"val(btex $"+latex(D[1])+"$ etex);"; for(p:=0;p<=n-1;p++){lp:=""; li:=li+lsic+latex(L[p](x))+"}$ etex);"+ if(mini!=-infinity and L[p](mini)==0){" valBarre(btex 0 etex);"}else{" "}+ if(L[p](Z[0]-0.01)>0){"plus;"}else{"moins;"}; for(r:=0; r<=nz-2;r++){lp:=lp+if(simplify(L[p](Z[r]))==0){" valBarre(btex 0 etex);"}else{"barre;"}+ if(L[p]((Z[r]+Z[r+1])*.5)>0){"plus;"}else{"moins;"}}; li:=li+lp+ if(simplify(L[p](Z[nz-1]))==0){"valBarre(btex 0 etex);"}else{"barre;"}+ if(L[p](Z[nz-1]+1.0)>0){"plus;"}else{"moins; "}+if(maxi!=+infinity and L[p](maxi)==0){"valBarre(btex 0 etex);"}else{" "} }; pour t de 0 jusque nz-2 faire lr:=lr+if(product(L[s]((Z[t]+Z[t+1])*.5),s,0,n-1)>0){"plus;"}else{"moins;"}+ if(member(Z[t+1],FF)==0){"valBarre(btex 0 etex);"}else{ "nonDefBarre;"} fpour MetaLfc:=" beginTableau("+nmr+") newLigneVariables(btex $ {x}$ etex); "+l0+ li + lsic+nomf+"(x)}$ etex);"+ if(member(mini,FF)==0){if((mini!=-infinity) and (product(L[s](mini),s,0,n-1)==0)){" valBarre(btex 0 etex);"}else{" "}}else{"nonDefBarre;"}+ if(product(L[s](Z[0]-0.01),s,0,n-1)>0){"plus;"}else{"moins;"}+ if(member(Z[0],FF)==0){"valBarre(btex 0 etex);"}else{ "nonDefBarre;"}+ lr+ if(product(L[s](Z[nz-1]+0.01),s,0,n-1)>0){"plus;"}else{"moins;"}+ if(member(maxi,FF)==0){if((maxi!=+infinity) and (product(L[s](maxi),s,0,n-1)==0)){"valBarre(btex 0 etex);"}else{" "}}else{"nonDefBarre;"}+" endTableau; " } sortie:=fopen("XCasmpfc.mp"); fprint(sortie,Unquoted,MetaLfc); fclose(sortie); }:; \end{VerbatimOut} %% Code giac/Xcas pour les Tableaux de signes du produit %% de 2 facteurs affines %% \begin{VerbatimOut}{XcasTabSigna.cxx} TSa(a,b,c,d,nmr):={ zA:=solve(a*x+b=0,x)[0]; zB:=solve(c*x+d=0,x)[0]; zmin:=min(zA,zB); zmax:=max(zA,zB); Meta:= " beginTableau("+nmr+") newLigneVariables(btex $ {x}$ etex); val(btex $-\\infty$ etex);val(btex $"+latex(zmin)+"$ etex); val(btex $"+latex(zmax)+"$etex); val(btex $+\\infty$ etex); "+lsic+if(a==1){"x+"}else{if(a==-1){"-x+"}else{a+"x+"}}+b+"}$ etex);" + if(a>0){"moins;"}else{"plus;"}+ if(zmin==zA){"valBarre(btex 0 etex);"}else{"barre;"}+ if(zmin==zA){si a>0 alors "plus;"; sinon "moins;";fsi} else{si a>0 alors "moins;"; sinon "plus;"; fsi}+ if(zmin==zA){"barre;"}else{"valBarre(btex 0 etex);"}+ if(a>0){"plus;"}else{"moins;"} +lsic+if(c==1){"x+"}else{if(c==-1){"-x+"}else{c+"x+"}}+d+"}$ etex);" + if(c>0){"moins"}else{"plus"}+";"+ if(zmin==zB){"valBarre(btex 0 etex);"}else{"barre;"}+ if(zmin==zB){si c>0 alors "plus;"; sinon "moins;";fsi} else{si c>0 alors "moins;"; sinon "plus;"; fsi}+ if(zmin==zB){"barre;"}else{"valBarre(btex 0 etex);"}+ if(c>0){"plus;"}else{"moins;"} +lsic+"{("+if(a==1){"x+"}else{if(a==-1){"-x+"}else{a+"x+"}}+b+")("+if(c==1){"x+"}else{if(c==-1){"-x+"}else{c+"x+"}}+d+")}}$ etex);" + si a*c>0 alors plus; sinon moins;fsi+";"+ "valBarre(btex 0 etex);"+ si a*c>0 alors moins; sinon plus;fsi+";"+ "valBarre(btex 0 etex);"+ si a*c>0 alors plus; sinon moins;fsi+";"+" endTableau; " ; sortie:=fopen("XCasmpfc.mp"); fprint(sortie,Unquoted,Meta); fclose(sortie); }:; \end{VerbatimOut} %%$ %% Code giac/Xcas pour les Tableaux de Signes d'expression ne contenant %% qu'un seul terme \begin{VerbatimOut}{XcasTSc.cxx} TSc(g,D,F,trigo,nmr):={ f:=unapply(g,x); mini:=D[0]; maxi:=D[1];lm:=" "; Z:=mini,maxi; S:=[]; if(trigo==t){ all_trig_solutions:=1; reset_solve_counter(-1,-1); SS:=solve(factor(simplify(f(x))),x); ns:=size(SS); for(j:=0;j<ns;j++){ m:=0; while(evalf(simplify(subst(SS[j],n_1=m)))<=evalf(maxi)){ S:=concat(S,simplify(subst(SS[j],n_1=m)));m:=m+1; };m:=-1; while(evalf(subst(SS[j],n_1=m))>=evalf(mini)){ S:=concat(S,simplify(subst(SS[j],n_1=m)));m:=m-1; } } }else{ S:=solve(f(x),x); } if(size(S)==0 and size(F)==0){ l0:="val(btex $"+latex(D[0])+"$ etex);val(btex $"+latex(D[1])+"$ etex);"; li:=if(member(mini,F)!=0){"nonDefBarre;"}else{if(mini!=-infinity and f(mini)==0){" valBarre(btex 0 etex);"}else{" "}}+ if(mini!=-infinity or maxi!=+infinity){if(f((mini+maxi)*0.5>0)){"plus;"}else{"moins;"}}else{if(f(0.3145274774464545777744)>0){"plus;"}else{"moins;"}}; lf:=if(member(maxi,F)!=0){"nonDefBarre;"}else{if(maxi!=+infinity and f(maxi)==0){" valBarre(btex 0 etex);"}else{" "}}; }else{ if(size(S!=0)){pour j de 0 jusque size(S)-1 faire if(S[j]>mini and S[j]<maxi){Z:=Z,simplify(S[j])}; fpour} Z:=concat([Z],F); Z:=sort([op(set[op(Z)])]); nz:=size(Z); l0:=" ";li:=" ";lr:=" "; if(nz==2){l0:="val(btex $"+latex(D[0])+"$ etex);val(btex $"+latex(D[1])+"$ etex);"; li:=if(mini!=-infinity and f(mini)==0){" valBarre(btex 0 etex);"}else{if(member(mini,F)==0){" "}else{"nonDefBarre;"}}+ if(f((mini+maxi)*0.5)>0){"plus;"}else{"moins;"}; lf:=if(maxi!=+infinity and f(maxi)==0){" valBarre(btex 0 etex);"}else{if(member(maxi,F)==0){" "}else{"nonDefBarre;"} }; }else{ l0:="val(btex $"+latex(Z[0])+"$ etex);";li:=" "; pour m de 1 jusque nz-1 faire l0:=l0+"val(btex $"+latex(Z[m])+"$ etex); "; fpour; li:= if(mini!=-infinity and f(mini)==0){" valBarre(btex 0 etex);"}else{if(member(mini,F)==0){" "}else{"nonDefBarre;"} } lm:=if(nz>2){for(r:=0; r<nz-2;r++){lm:=lm+if(Z[r]==-infinity){ if(f((Z[r+1]-1))>0){"plus;"}else{"moins;"} }else{if(f((Z[r]+Z[r+1])*.5)>0){"plus;"}else{"moins;"}} + if(member(Z[r+1],F)==0){"valBarre(btex 0 etex);"}else{"nonDefBarre;"} }}else{" "}; lf:=if(f(Z[nz-2]+0.1)>0){"plus;"}else{"moins;"}+if(maxi!=+infinity and f(maxi)==0){"valBarre(btex 0 etex);"}else{if(member(maxi,F)==0){" "}else{"nonDefBarre;"} }; } }; MetaLfc:=" beginTableau("+nmr+") newLigneVariables(btex $\\displaystyle {x}$ etex); "+l0+lsic+latex(f(x))+"}$ etex);"+ li+lm+lf +" endTableau; " ; sortie:=fopen("XCasmpfc.mp"); fprint(sortie,Unquoted,MetaLfc); fclose(sortie); }:; \end{VerbatimOut} %%$ %%%% %%% %%% Pour des tableaux de variations simples sans fonctions %%% %%% \begin{VerbatimOut}{XcasTVS.cxx} TVS(La,Lo,F,nomf,nomv,nmr):={ na:=size(La); f:=F; if(member(La[0],F)){f:=f[1..size(f)-1]}; if(member(La[na-1],F)){f:=f[0..size(f)-2]}; Z:=sort(concat(La,f)); Zo:=sort([op(set[op(Z)])]); nz:=size(Z); nzo:=size(Zo); k0:= evalf(Lo[0])> evalf(Lo[1]); kz:=evalf(Lo[nz-1])> evalf(Lo[nz-2]); l0:=" newLigneVariables(btex $"+nomv+"$ etex);";lp:=" "; lf:=" "; for(m:=0;m<=nzo-1;m++){l0:=l0+"val(btex $"+latex(Zo[m])+"$ etex);"} li:=lvic+nomf+"}$ etex);"+ if(member(Z[0],F)==0){"valPos(btex $"+latex(Lo[0])+"$ etex,"} else{"nonDefBarre;limDroite(btex $"+latex(Lo[0])+"$ etex,"}+ if(k0==1){"1"}else{"0"}+ ");"; if(nz>2){ for(r:=1; r<=nz-2;r++){ krm:=evalf(Lo[r-1])< evalf(Lo[r]); krp:=evalf(Lo[r])> evalf(Lo[r+1]) ; lp:=lp+if(Z[r]==Z[r+1]){ "limGauche(btex$"+latex(Lo[r])+"$etex,"+ if(krm==1){"1);"} else{"0);"} }// fin if zr=zr+1 else{ if(Z[r]==Z[r-1]){ "nonDefBarre;limDroite(btex$"+latex(Lo[r])+"$etex,"+ if(krp==1){"1);"} else{"0);"} }//fin if zr=zr-1 else{ "valPos(btex $"+latex(Lo[r])+"$etex,"+ if(krp==1){"1);"}else{"0);"} }//fin else zr=zr-1 }//fin else zr=zr+1 }//fin for }//fin de if nz>2 lf:=if(member(Z[nz-1],F)==0){"valPos(btex $"+latex(Lo[nz-1])+"$ etex,"+ if(kz==1){"1);"}else{"0);"}} else{"limGauche(btex $"+latex(Lo[nz-1])+"$ etex,"+ if(kz==1){"1);nonDefBarre;"}else{"0);nonDefBarre;"}}; MetaLfc:= if(nz>2){"beginTableau("+nmr+")"+ l0+ li+ lp+ lf +" endTableau; ";}else{"beginTableau("+nmr+")"+ l0+ li+ lf +" endTableau; ";} //return(MetaLfc); sortie:=fopen("XCasmpfc.mp"); fprint(sortie,Unquoted,MetaLfc); fclose(sortie); }:; \end{VerbatimOut} %%$ %% %% traitement des fichiers produits par giac/xcas %% %% % pour l'échelle des tableaux taper \ech{facteur de réduction} \newcommand\echelle{1} \newcommand\ech[1]{\renewcommand\echelle{#1}} \newcommand\couleurtab{black} \newcommand\coultab[1]{\renewcommand\couleurtab{#1}} \newcommand{\dresse}[2]{% \ifthenelse{\boolean{xcas}}{% Avec l'option "XCas present" \executGiacmp{XCas#2.giac}% reconstituer le tableau % exporter le source mp % puis lancer metapost pour creer % l'image du tableau \immediate\write18{\cat XCasmpfc.mp >> \nomtravail_Tab.mp} \immediate\write18{\cat enteteMP.cfg >> \nomtravail_Tab#2.mp} \immediate\write18{\echod def beginTableau(expr c) =\echof >> \nomtravail_Tab#2.mp} \immediate\write18{\echod begingroup\echof >> \nomtravail_Tab#2.mp} \immediate\write18{\echod charcode:=c;\echof >> \nomtravail_Tab#2.mp} \immediate\write18{\echod clearxy; clearit; clearpen;\echof >> \nomtravail_Tab#2.mp} \immediate\write18{\echod pickup defaultpen;\echof >> \nomtravail_Tab#2.mp} \immediate\write18{\echod drawoptions(withcolor(#1));\echof >> \nomtravail_Tab#2.mp} \immediate\write18{\echod initTableau;\echof >> \nomtravail_Tab#2.mp} \immediate\write18{\echod enddef;\echof >> \nomtravail_Tab#2.mp} \immediate\write18{\cat XCasmpfc.mp >> \nomtravail_Tab#2.mp} \immediate\write18{\cat queue.mp >> \nomtravail_Tab#2.mp} \immediate\write18{mpost -jobname=\nomtravail_Tab \nomtravail_Tab#2.mp} \immediate\write18{\rem \nomtravail_Tab#2.mp} }% {% sinon, si le tableau est absent, alerte. \IfFileExists{\nomtravail_Tab.\theTVn}{}{% \PackageError{tablor}{Tableau absent non reconstituable.}{Pour compiler il faut, soit les fichiers de tableaux, soit le fichier \nomtravail_Tab.mp, soit disposer de XCas.}}} \begin{center} \includegraphics[scale=\echelle]{\nomtravail_Tab.\theTVn} \end{center} \stepcounter{TVn} } %% %% traitement des fichiers produits par giac/xcas avec possibilite %% de modifier le fichier metapost (environnement etoile)) %% \newcommand{\dressetoile}[2]{% \IfFileExists{\nomtravail_Tab.\theTVn}{% Test sur l'existence du tableau % Si oui, inclusion du fichier source de sauvegarde mp dans Tableaux \immediate\write18{\cat TSav-\theTVn.mp >> \nomtravail_Tab.mp}} % Si non, lancement des operations de fabrication {\executGiacmp{XCas#2.giac}% \immediate\write18{\editeur XCasmpfc.mp } % Modification avec l'editeur choisi \immediate\write18{\cat XCasmpfc.mp >> \nomtravail_Tab.mp} \immediate\write18{\cp XCasmpfc.mp TSav-\theTVn.mp} % Sauvegarde du % source mp sur le disque pour une % inclusion ulterieure dans Tableaux.mp. \immediate\write18{\cat enteteMP.cfg >> \nomtravail_Tab#2.mp} \immediate\write18{\echod def beginTableau(expr c) =\echof >> \nomtravail_Tab#2.mp} \immediate\write18{\echod begingroup\echof >> \nomtravail_Tab#2.mp} \immediate\write18{\echod charcode:=c;\echof >> \nomtravail_Tab#2.mp} \immediate\write18{\echod clearxy; clearit; clearpen;\echof >> \nomtravail_Tab#2.mp} \immediate\write18{\echod pickup defaultpen;\echof >> \nomtravail_Tab#2.mp} \immediate\write18{\echod drawoptions(withcolor(#1));\echof >> \nomtravail_Tab#2.mp} \immediate\write18{\echod initTableau;\echof >> \nomtravail_Tab#2.mp} \immediate\write18{\echod enddef;\echof >> \nomtravail_Tab#2.mp} \immediate\write18{\cat XCasmpfc.mp >> \nomtravail_Tab#2.mp}} % Inclusion du % source dans le % fichier % Tableaux \immediate\write18{\cat queue.mp >> \nomtravail_Tab#2.mp} \immediate\write18{mpost -jobname=\nomtravail_Tab \nomtravail_Tab#2.mp}% Reconstitution des tableaux % et creation du dernier. L'option % pallie l'absence de end en fin de % fichier \immediate\write18{\rem \nomtravail_Tab#2.mp} \begin{center} \includegraphics[scale=\echelle]{\nomtravail_Tab.\theTVn} \end{center} \ech{1} \setcounter{TVn}{\theTVnbis} % Restauration du compteur TVn } %% %% %%% %%% %%% les "giac" qui permettent d'executer la commande rentree dans le fichier tex %%% suivis des environnements qui permettront la saisie du code giac/xcas %%% Les versions etoilees permettent de modifier le code metapost produit initialement %%% \begin{VerbatimOut}{XCasa.giac} maple_mode(0); approx_mode:=0; read("config.cxx"); read("XcasTabSigna.cxx"); read("XCasa.user"); \end{VerbatimOut} \newenvironment{TSa} {\VerbatimEnvironment\begin{VerbatimOut}[commandchars=\\??]{XCasa.user}} {\end{VerbatimOut} \dresse{\couleurtab}{a} } \begin{VerbatimOut}{XCasQ.giac} maple_mode(0); approx_mode:=0; read("config.cxx"); read("XcasTabSignQ.cxx"); read("XCasQ.user"); \end{VerbatimOut} \newenvironment{TSq}% {\VerbatimEnvironment\begin{VerbatimOut}[commandchars=\\??]{XCasQ.user}} {\end{VerbatimOut}\dresse{\couleurtab}{Q}} \newenvironment{TSq*}[1]% {\setcounter{TVnbis}{\theTVn} \setcounter{TVn}{#1} \VerbatimEnvironment\begin{VerbatimOut}[commandchars=\\??]{XCasQ.user}} {\end{VerbatimOut}\dressetoile{\couleurtab}{Q}} \begin{VerbatimOut}{XCasL.giac} maple_mode(0); approx_mode:=0; read("config.cxx"); read("XcasTabSignL.cxx"); read("XCasL.user"); \end{VerbatimOut} \newenvironment{TS}% {\VerbatimEnvironment\begin{VerbatimOut}[commandchars=\\??]{XCasL.user}} {\end{VerbatimOut}\dresse{\couleurtab}{L}} \newenvironment{TS*}[1] {\setcounter{TVnbis}{\theTVn} \setcounter{TVn}{#1} \VerbatimEnvironment\begin{VerbatimOut}[commandchars=\\??]{XCasL.user}} {\end{VerbatimOut}\dressetoile{\couleurtab}{L}} \begin{VerbatimOut}{XCasTSc.giac} maple_mode(0); approx_mode:=0; read("config.cxx"); read("XcasTSc.cxx"); read("XCasTSc.user"); \end{VerbatimOut} \newenvironment{TSc*}[1]% {\setcounter{TVnbis}{\theTVn} \setcounter{TVn}{#1} \VerbatimEnvironment\begin{VerbatimOut}[commandchars=\\??]{XCasTSc.user}} {\end{VerbatimOut}\dressetoile{\couleurtab}{TSc}} \newenvironment{TSc}% {\VerbatimEnvironment\begin{VerbatimOut}[commandchars=\\??]{XCasTSc.user}} {\end{VerbatimOut}\dresse{\couleurtab}{TSc}} \begin{VerbatimOut}{XCasTV.giac} maple_mode(0); approx_mode:=0; read("config.cxx"); read("XcasTV.cxx"); read("XCasTV.user"); \end{VerbatimOut} \newenvironment{TV}{% \VerbatimEnvironment \begin{VerbatimOut}[commandchars=\\??]{XCasTV.user}}% {\end{VerbatimOut} \dresse{\couleurtab}{TV}} \newenvironment{TV*}[1]{% \setcounter{TVnbis}{\theTVn} \setcounter{TVn}{#1} \VerbatimEnvironment\begin{VerbatimOut}[commandchars=\\??]{XCasTV.user}}% {\end{VerbatimOut}\dressetoile{\couleurtab}{TV}} \begin{VerbatimOut}{XCasTVP.giac} maple_mode(0); approx_mode:=0; read("config.cxx"); read("XcasTVP.cxx"); read("XCasTVP.user"); \end{VerbatimOut} \newenvironment{TVP}% {\VerbatimEnvironment\begin{VerbatimOut}[commandchars=\\??]{XCasTVP.user}}% {\end{VerbatimOut} \dresse{\couleurtab}{TVP}} \newenvironment{TVP*}[1]{% \setcounter{TVnbis}{\theTVn} \setcounter{TVn}{#1} \VerbatimEnvironment\begin{VerbatimOut}[commandchars=\\??]{XCasTVP.user}}% {\end{VerbatimOut}\dressetoile{\couleurtab}{TVP}} \begin{VerbatimOut}{XCasTVZ.giac} maple_mode(0); approx_mode:=0; read("config.cxx"); read("XcasTVZ.cxx"); read("XCasTVZ.user"); \end{VerbatimOut} \newenvironment{TVZ}% {\VerbatimEnvironment\begin{VerbatimOut}[commandchars=\\??]{XCasTVZ.user}}% {\end{VerbatimOut} \dresse{\couleurtab}{TVZ}} \newenvironment{TVZ*}[1]{% \setcounter{TVnbis}{\theTVn} \setcounter{TVn}{#1} \VerbatimEnvironment\begin{VerbatimOut}[commandchars=\\??]{XCasTVZ.user}}% {\end{VerbatimOut}\dressetoile{\couleurtab}{TVZ}} \begin{VerbatimOut}{XCasTVapp.giac} maple_mode(0); approx_mode:=0; read("config.cxx"); read("XcasTVapp.cxx"); read("XCasTVapp.user"); \end{VerbatimOut} \newenvironment{TVapp}% {\VerbatimEnvironment\begin{VerbatimOut}[commandchars=\\??]{XCasTVapp.user}}% {\end{VerbatimOut} \dresse{\couleurtab}{TVapp}} \newenvironment{TVapp*}[1]{% \setcounter{TVnbis}{\theTVn} \setcounter{TVn}{#1} \VerbatimEnvironment\begin{VerbatimOut}[commandchars=\\??]{XCasTVapp.user}}% {\end{VerbatimOut}\dressetoile{\couleurtab}{TVapp}} \begin{VerbatimOut}{XCasTVI.giac} maple_mode(0); approx_mode:=0; read("config.cxx"); read("XcasTVI.cxx"); read("XCasTVI.user"); \end{VerbatimOut} \newenvironment{TVI}% {\VerbatimEnvironment\begin{VerbatimOut}[commandchars=\\??]{XCasTVI.user}}% {\end{VerbatimOut}\dresse{\couleurtab}{TVI}} \newenvironment{TVI*}[1]% {\setcounter{TVnbis}{\theTVn} \setcounter{TVn}{#1} \VerbatimEnvironment\begin{VerbatimOut}[commandchars=\\??]{XCasTVI.user}}% {\end{VerbatimOut}\dressetoile{\couleurtab}{TVI}} \begin{VerbatimOut}{XCasTVIex.giac} maple_mode(0); approx_mode:=0; read("config.cxx"); read("XcasTVIex.cxx"); read("XCasTVIex.user"); \end{VerbatimOut} \newenvironment{TVIex}% {\VerbatimEnvironment\begin{VerbatimOut}[commandchars=\\??]{XCasTVIex.user}}% {\end{VerbatimOut}\dresse{\couleurtab}{TVIex}} \newenvironment{TVIex*}[1]% {\setcounter{TVnbis}{\theTVn} \setcounter{TVn}{#1} \VerbatimEnvironment\begin{VerbatimOut}[commandchars=\\??]{XCasTVIex.user}}% {\end{VerbatimOut}\dressetoile{\couleurtab}{TVIex}} \begin{VerbatimOut}{XCasTVIapp.giac} maple_mode(0); approx_mode:=0; read("config.cxx"); read("XcasTVIapp.cxx"); read("XCasTVIapp.user"); \end{VerbatimOut} \newenvironment{TVIapp}% {\VerbatimEnvironment\begin{VerbatimOut}[commandchars=\\??]{XCasTVIapp.user}}% {\end{VerbatimOut}\dresse{\couleurtab}{TVIapp}} \newenvironment{TVIapp*}[1]% {\setcounter{TVnbis}{\theTVn} \setcounter{TVn}{#1} \VerbatimEnvironment\begin{VerbatimOut}[commandchars=\\??]{XCasTVIapp.user}}% {\end{VerbatimOut}\dressetoile{\couleurtab}{TVIapp}} \begin{VerbatimOut}{XCasTVPC.giac} maple_mode(0); approx_mode:=0; read("config.cxx"); read("XcasTVPC.cxx"); read("XCasTVPC.user"); \end{VerbatimOut} \newenvironment{TVPC}% {\VerbatimEnvironment\begin{VerbatimOut}[commandchars=\\??]{XCasTVPC.user}}% {\end{VerbatimOut} \dresse{\couleurtab}{TVPC}} \newenvironment{TVPC*}[1]{% \setcounter{TVnbis}{\theTVn} \setcounter{TVn}{#1} \VerbatimEnvironment\begin{VerbatimOut}[commandchars=\\??]{XCasTVPC.user}}% {\end{VerbatimOut}\dressetoile{\couleurtab}{TVPC}} \begin{VerbatimOut}{XCasTVS.giac} maple_mode(0); approx_mode:=0; read("config.cxx"); read("XcasTVS.cxx"); read("XCasTVS.user"); \end{VerbatimOut} \newenvironment{TVS}% {\VerbatimEnvironment\begin{VerbatimOut}[commandchars=\\??]{XCasTVS.user}}% {\end{VerbatimOut} \dresse{\couleurtab}{TVS}} \newenvironment{TVS*}[1]{% \setcounter{TVnbis}{\theTVn} \setcounter{TVn}{#1} \VerbatimEnvironment\begin{VerbatimOut}[commandchars=\\??]{XCasTVS.user}}% {\end{VerbatimOut}\dressetoile{\couleurtab}{TVS}} %% pour nettoyer les fichiers auxiliaires \AtEndDocument{\immediate\write18{\cat queue.mp >> \nomtravail_Tab.mp} } %% %% Zi end -> enjoy :)
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.