Khi các bạn đã cài TeXLive bản full thì chương trình tablor và tableauvariation.mp cũng đã cài đặt hoàn chỉnh. Để sử dụng được tablor các bạn còn phải cài xcas lên máy. Trong Ubuntu Linux việc này dễ dàng và không cần cấu hình gì thêm. Trong MS Winodws xem hướng dẫn ở cuối bài.
Để bảng biến thiên giống như chúng ta thường dùng, các bạn thực thi như sau:
1. Tạo một thư mục ví dụ tablor để ở bất cứ đâu. Soạn file TeX trong thư nục này.
2. Copy vào thư mục tablor nói trên hai file sau đây:
tableauVariation.mp
% This software is copyright (c) 2005 by Frédéric Mazoit
%
% This program is free software; you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation; either version 2 of the License, or
% any later version.
%
% This program is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU General Public License for more details.
deltaX=10pt;
deltaY=5pt;
deltaYvariations=20pt;
deltaXsignes=30pt;
Variables_=0;
Signes_=1;
Variations_=2;
Plus_=0;
Moins_=1;
Fleche_=2;
Valeur_=3;
ValeurBarree_=4;
Vide_=5;
Barre_=6;
NonDefBarre_=7;
NonDefRegion_=8;
VidePos_=9;
Trait_=10;
def mkLabel(expr x)=
if picture x: x
else: x infont defaultfont scaled defaultscale fi
enddef;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
def exists(expr i,j)=
known typeEl.[i][j]c
enddef;
def getH(expr x)= if known x: ypart(ulcorner x-llcorner x) else: 0 fi enddef;
def getL(expr x)= if known x: xpart(urcorner x-ulcorner x) else: 0 fi enddef;
vardef getHcase(expr i,j)=
max(
max(
deltaY,
if known el.[i][j]l: getH(el.[i][j]l) else: 0 fi),
max(
if known el.[i][j]c: getH(el.[i][j]c) else: 0 fi,
if known el.[i][j]r: getH(el.[i][j]r) else: 0 fi)
)
enddef;
vardef getMcase(expr i,j)=
save type;
if exists(i,j):
type:=typeEl.[i][j]c;
if (type=Plus_) or (type=Moins_) or (type=Fleche_) or (type=NonDefRegion_):
.5deltaXsignes+.5deltaX
elseif (type=NonDefBarre_): 2pt+.5deltaX
elseif (type=Barre_) or (type=Vide_): 0
else: .5deltaX+if known el.[i][j]c:.5getL(el.[i][j]c) else: 0 fi
fi
else: 0 fi
enddef;
vardef getGcase(expr i,j)=
getMcase(i,j)+ if known el.[i][j]l:getL(el.[i][j]l) else: 0 fi
enddef;
vardef getDcase(expr i,j)=
getMcase(i,j)+ if known el.[i][j]r:getL(el.[i][j]r) else: 0 fi
enddef;
vardef getGcol(expr col)=
save res,i;
res=0;
for i=1 upto nbLgn: res:=max(res,getGcase(i,col)); endfor;
res
enddef;
vardef getDcol(expr col)=
save res,i;
res=0;
for i=1 upto nbLgn: res:=max(res,getDcase(i,col)); endfor;
res
enddef;
vardef getHelsLgn(expr line)=
save res, j;
res:=11pt;
for j=0 upto nbCol: res:=max(res,getHcase(line,j)); endfor;
res
enddef;
vardef getHLgn(expr line)=
if typeLgn[line]=Variations_:
max(getHelsLgn(line)+deltaYvariations,40pt)+deltaY
else:
getHelsLgn(line)+deltaY
fi
enddef;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
def valSmp_(expr s,type,b)=
vd:=0;
col:=col+1;
if col>nbCol: nbCol:=col; fi;
typeEl.[nbLgn][col]c=type;
pos.[nbLgn][col]c=b;
el.[nbLgn][col]c=mkLabel(s);
enddef;
def sgnSmp_(expr s)=
vd:=0;
col:=col+1;
if col>nbCol: nbCol:=col; fi;
typeEl.[nbLgn][col]c=s;
enddef;
def chkCol(expr a)=
if (col mod 2)=a:
if ndf=1: sgn_(NonDefRegion_,1);
elseif (typeLgn[nbLgn]=Variations_) and (vd=0): fleche;
else: sgnSmp_(Vide_);
fi
fi
enddef;
def valeur_(expr s,type,b)=
chkCol(0);
valSmp_(s,type,b);
enddef;
def sgn_(expr s,a)=
chkCol(a);
sgnSmp_(s);
enddef;
def flecheBrisee(expr b)=
sgn_(Trait_,1);
vd:=0;
col:=col+1;
if col>nbCol: nbCol:=col; fi;
typeEl.[nbLgn][col]c=VidePos_;
pos.[nbLgn][col]c=b;
enddef;
def vide=vd:=1;sgn_(Vide_,0);enddef;
def val(expr s)=valeur_(s,Valeur_,.5); enddef;
def valBarre(expr s)=valeur_(s,ValeurBarree_,.5); enddef;
def valPos(expr s,b)=valeur_(s,Valeur_,b); enddef;
def valPosBarre(expr s,b)=valeur_(s,ValeurBarree_,b); enddef;
def barre=sgn_(Barre_,0); enddef;
def nonDefBarre=sgn_(NonDefBarre_,0); enddef;
def debutNonDef=ndf:=1;barre;enddef;
def finNonDef=barre;ndf:=0; enddef;
def bord=sgn_(Barre_,1); enddef;
def plus=sgn_(Plus_,1); enddef;
def moins=sgn_(Moins_,1); enddef;
def fleche=sgn_(Fleche_,1); enddef;
def limGauche(expr s,b)=
chkCol(0);
pos.[nbLgn][col+1]l=b;
el.[nbLgn][col+1]l=mkLabel(s);
enddef;
def limDroite(expr s,b)=
pos.[nbLgn][col]r=b;
el.[nbLgn][col]r=mkLabel(s);
enddef;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
def newLgn_(expr x,type)=
nbLgn:=nbLgn+1;
col:=0;
typeLgn[nbLgn]=type;
sgnSmp_(Vide_);
val(x);
bord;
enddef;
def newLigneVariables(expr x)=newLgn_(x,Variables_); enddef;
def newLigneSignes(expr x)=newLgn_(x,Signes_); enddef;
def newLigneVariations(expr x)=newLgn_(x,Variations_); enddef;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
def calculPos=
xCol[1]:=0;
for i=1 upto nbCol:
x[i]:=xCol[i]+getGcol(i);
xCol[i+1]:=x[i]+getDcol(i);
endfor
yLgn[1]:=0;
for i=1 upto nbLgn:
y[i]:=yLgn[i]-.5getHLgn(i);
yLgn[i+1]:=y[i]-0.5*getHLgn(i);
endfor;
enddef;
vardef calcP(expr i,j)=
if exists(i,j):
if (typeEl.[i][j]c=Valeur_) or (typeEl.[i][j]c=VidePos_):
(x[j],y[i]+(pos.[i][j]c-.5)*(getHLgn(i)-getHelsLgn(i)))
else: (x[j],y[i]) fi
else: (x[j],y[i]) fi
enddef;
vardef calcPl(expr i,j)=
(x[j]-2pt-.5(getL(el.[i][j]c)+getL(el.[i][j]l)),y[i]+(pos.[i][j]l-.5)*(getHLgn(i)-getHelsLgn(i)))
enddef;
vardef calcPr(expr i,j)=
(x[j]+2pt+.5(getL(el.[i][j]c)+getL(el.[i][j]r)),y[i]+(pos.[i][j]r-.5)*(getHLgn(i)-getHelsLgn(i)))
enddef;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
vardef drawFleche(expr i,j,k)=
save aa,bb;
aa=j-1;
bb=j+1;
forever:
exitif typeEl.[i][aa]c<>Vide_;
aa:=aa-1;
endfor
forever:
exitif typeEl.[i][bb]c<>Vide_;
bb:=bb+1;
endfor
if k=Fleche_: drawarrow else: draw fi(
if known el.[i][aa]r: calcPr(i,aa) else: calcP(i,aa) fi--
if known el.[i][bb]l: calcPl(i,bb) else: calcP(i,bb) fi)
if known el.[i][aa]r: cutbefore(bbox(thelabel(el.[i][aa]r,calcPr(i,aa)))) fi
if known el.[i][aa]c: cutbefore(bbox(thelabel(el.[i][aa]c,calcP(i,aa)))) fi
if known el.[i][bb]l: cutafter(bbox(thelabel(el.[i][bb]l,calcPl(i,bb)))) fi
if known el.[i][bb]c: cutafter(bbox(thelabel(el.[i][bb]c,calcP(i,bb)))) fi;
enddef;
vardef rempli(expr xa,xb,ya,yb)=
save i;
for i=1 step 5pt until xb-xa+ya-yb-1:
draw (xa,ya-i)--(xa+i,ya) cutbefore ((xa,yb)--(xb,yb)) cutafter ((xb,ya)--(xb,yb));
endfor
enddef;
def afficheSigne(expr s,i,j)=
label(s,.5[calcP(i,j-1),calcP(i,j+1)]);
enddef;
def afficheTableau=
calculPos;
for i=1 upto nbLgn:
for j=1 upto nbCol:
if exists(i,j):
t:=typeEl.[i][j]c;
if t=Plus_:
afficheSigne(btex $+$ etex,i,j);
elseif t=Moins_:
afficheSigne(btex $-$ etex,i,j);
elseif t=Fleche_:
drawFleche(i,j,Fleche_);
elseif t=Trait_:
drawFleche(i,j,Trait_);
elseif t=NonDefRegion_:
rempli(x[j-1],x[j+1],yLgn[i],yLgn[i+1]);
elseif t=Barre_:
draw (x[j],yLgn[i])--(x[j],yLgn[i+1]);
elseif t=ValeurBarree_:
label(el.[i][j]c,calcP(i,j));
draw (x[j],yLgn[i])--(x[j],yLgn[i+1]) dashed evenly;
elseif t=NonDefBarre_:
draw (x[j]-1pt,yLgn[i])--(x[j]-1pt,yLgn[i+1]);
draw (x[j]+1pt,yLgn[i])--(x[j]+1pt,yLgn[i+1]);
elseif t=Valeur_:
label(el.[i][j]c,calcP(i,j));
fi
if known el.[i][j]l: label(el.[i][j]l,calcPl(i,j)); fi;
if known el.[i][j]r: label(el.[i][j]r,calcPr(i,j)); fi;
fi
endfor
endfor
for i=2 upto nbLgn:
draw (xCol[1],yLgn[i])--(xCol[nbCol+1],yLgn[i]);
endfor;
% Bang bien thien cua VN khong co khung nen bo 4 dong sau day:
% draw(xCol[1],yLgn[1])--(xCol[nbCol+1],yLgn[1]);
% draw(xCol[1],yLgn[nbLgn+1])--(xCol[nbCol+1],yLgn[nbLgn+1]);
% draw(xCol[1],yLgn[1])--(xCol[1],yLgn[nbLgn+1]);
% draw(xCol[nbCol+1],yLgn[1])--(xCol[nbCol+1],yLgn[nbLgn+1]);
enddef;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
def beginTableau(expr c) =
begingroup
charcode:=c;
clearxy; clearit; clearpen;
pickup defaultpen;
drawoptions();
initTableau;
enddef;
def initTableau=
picture el.[][]l;
picture el.[][]c;
picture el.[][]r;
numeric typeLgn[];
numeric col;
numeric nbLgn;
numeric nbCol;
numeric pos.[][]l;
numeric pos.[][]c;
numeric pos.[][]r;
numeric typeEl.[][]c;
numeric yLgn[];
numeric xCol[];
ndf:=0;
vd:=0;
col=0;
nbLgn=0;
nbCol=0;
enddef;
def endTableau=
afficheTableau;
shipit;
endgroup
enddef;
tablor.cfg
\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 :)
3. Soạn một file TeX đặt tên là test.tex có nội dung, chẳng hạn như sau:
1 2 3 4 5 6 7 8 9 10 11 12
\documentclass{article}
\usepackage{graphicx}
\usepackage[utf8]{inputenc}
\usepackage[vietnam]{babel}
\usepackage[xcas]{tablor}
\begin{document}
\initablor
\begin{TV}
TV([-infinity,+infinity],[2],"g","x",(6x^2-2x+1)/(x-2),1,n,\tv)
\end{TV}
\nettoyer
\end{document}
4. Biên dịch file test.tex bằng pdflatex và view bằng acrobat reader.
Bảng biến thiên của hàm số: \(y=\dfrac{6x^2-2x+1}{x-2}\)
Tablor sử dụng Xcas làm tất cả mọi việc: lấy đạo hàm, tìm cực trị, tìm giá trị cực trị, tìm giới hạn và sử dung tableauVariation để lập bảng biến thiên .
các bạn cài giac từ file exe xcasinst.exe. Windows sẽ tự động cài giac vào C:\xcas. Các bạn phải cập nhật lại đường dẫn cho hệ thống như sau:
- Nhấp phải chuột vào My Computer, chọn Properties, Advanced Systems Settings, Environment Variables, trên dòng Path của User variables for user, các bạn chọn Edit, thêm dấu ; để ngăn cách biến cũ, sau đó thêm C:\xcas
- Chọn OK nhiều lần để ra ngoài, sau đó Reboot lại máy tính. Xcas sẽ được LaTeX nhận diện và sẽ tự động tiến hành tính toán.
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