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diff --git a/Master/texmf-dist/doc/latex/mfpic4ode/mfpic4ode.tex b/Master/texmf-dist/doc/latex/mfpic4ode/mfpic4ode.tex new file mode 100644 index 00000000000..a20ce6c6131 --- /dev/null +++ b/Master/texmf-dist/doc/latex/mfpic4ode/mfpic4ode.tex @@ -0,0 +1,192 @@ +%% +%% This is file `mfpic4ode.tex', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% mfpic4ode.dtx (with options: `tex') +%% +%% This is a generated file. +%% +%% Copyright (C) 2007 by Robert Marik <marik@mendelu.cz> +%% +%% This file may be distributed and/or modified under the conditions of +%% the LaTeX Project Public License, either version 1.2 of this license +%% or (at your option) any later version. The latest version of this +%% license is in: +%% +%% http://www.latex-project.org/lppl.txt +%% +%% and version 1.2 or later is part of all distributions of LaTeX version +%% 1999/12/01 or later. +%% +\catcode`\@=11 + +\newif\ifcolorODEarrow +%%%\colorODEarrowfalse +\colorODEarrowtrue + +%%% The line from one point to another +\def\ODEline#1#2{\lines{#1,#2}} + +%%% The variable ODErhs is used to store the function from the right +%%% hand side of ODE in the form y'=f(x,y). We use command +%%% ODEdefineequation to set up this variable. +\def\ODEdefineequation#1{\fdef{ODErhs}{x,y}{#1}} + +%%% Integral curve using Euler method. The step of this method is +%%% ODEstep and the number of steps is ODEstepcount. The points are +%%% stored in metapost variables x1,y1. +\def\trajectory#1#2{ + \mfsrc{x1:=#1;y1:=#2; + for i=1 upto ODEstepcount: + x2:=x1+ODEstep; + y2:=y1+ODEstep*ODErhs(x1,y1);} + \ODEline{z1}{z2} + \mfsrc{ + if ((y2>yneg) and (y2<ypos)): x1:=x2; y1:=y2 fi; + endfor + }} + +%%% Integral curve using Runge--Kutta method. +\def\trajectoryRK#1#2{ + \mfsrc{x1:=#1;y1:=#2; + for i=1 upto ODEstepcount: + k1:=ODErhs(x1,y1); + x3:=x1+(ODEstep/2); + y3:=y1+k1*(ODEstep/2); + k2:=ODErhs(x3,y3); + x2:=x1+ODEstep; + y2:=y1+ODEstep*k2;} + \ODEline{z1}{z2} + \mfsrc{ + if ((y2>yneg) and (y2<ypos)): x1:=x2; y1:=y2 fi; + endfor + }} +%%% Integral curve using fourth order Runge--Kutta method. +\def\trajectoryRKF#1#2{ + \mfsrc{x1:=#1;y1:=#2; + for i=1 upto ODEstepcount: + k1:=ODErhs(x1,y1); + x3:=x1+(ODEstep/2); + y3:=y1+k1*(ODEstep/2); + k2:=ODErhs(x3,y3); + y4:=y1+k2*(ODEstep/2); + k3:=ODErhs(x3,y4); + y5:=y1+k3*(ODEstep/2); + k4:=ODErhs(x3,y5); + kk:=(k1+2*k2+2*k3+k4)/6; + x2:=x1+ODEstep; + y2:=y1+ODEstep*kk;} + \ODEline{z1}{z2} + \mfsrc{ + if ((y2>yneg) and (y2<ypos)): x1:=x2; y1:=y2 fi; + endfor + }} +\def\ODEarrow#1#2{ + \mfsrc{x1:=#1; y1:=#2; + x3:=x1+(ODEarrowlength)/((xscale)++(ODErhs(#1,#2)*yscale)); + y3:=y1+(ODEarrowlength*ODErhs(#1,#2))/((xscale)++(ODErhs(#1,#2)*yscale)); + if y3>y1:ODEcolorarrow:=blue else: ODEcolorarrow:=red fi; + } + \ifcolorODEarrow + \drawcolor{ODEcolorarrow} \headcolor{ODEcolorarrow} + \fi + \draw\arrow\lines{z1,z3} +} + +\def\ODEarrows#1{\ODE@cycle@points#1;,;} +\def\trajectories#1{\ODE@cycle@IC#1;,;} +\def\ODE@last@point{} +\def\ODE@cycle@points#1,#2;{\def\temp{#1}\ifx\temp\ODE@last@point\let\next\relax + \else\ODEarrow{#1}{#2}\relax\let\next\ODE@cycle@points\fi\next} +\def\ODE@cycle@IC#1,#2;{\def\temp{#1}\ifx\temp\ODE@last@point\let\next\relax + \else + \trajectoryRKF{#1}{#2}\relax\let\next\ODE@cycle@IC\fi\next} +\mfsrc{path p,q;color ODEcolorarrow;} + +%%% Onedimensional autonomous systems y'=f(y) where '=d/dx +\def\ODEharrow#1{ + \mfsrc{x1:=#1; + if ODErhs(0,x1)>0: x3:=x1+ODEarrowlength else: x3:=x1-ODEarrowlength fi; + if ODErhs(0,x1)*ODErhs(0,x3)<0: x1:=-100;x3:=-100 fi; + if x3>x1:ODEcolorarrow:=blue else: ODEcolorarrow:=red fi; + } + \ifcolorODEarrow \drawcolor{ODEcolorarrow} + \headcolor{ODEcolorarrow} \fi + \pen{1.5pt} + \draw\arrow\lines{(x1,0),(x3,0)} +} + +\def\ODEvarrow#1{ + \mfsrc{x1:=#1; + if ODErhs(0,#1)>0: + x3:=x1+(ODEarrowlength/yscale) else: x3:=x1-(ODEarrowlength/yscale) fi; + if ODErhs(0,x1)*ODErhs(0,x3)<0: x1:=-100;x3:=-100 fi; + if x3>x1:ODEcolorarrow:=blue else: ODEcolorarrow:=red fi; + } + \ifcolorODEarrow \drawcolor{ODEcolorarrow} + \headcolor{ODEcolorarrow} \fi + \pen{1.5pt} + \draw\arrow\lines{(0,x1),(0,x3)} +} + +%%% Twodimensional autonomous systems x'=f(x,y), y'=g(x,y) where '=d/dt +\def\ASdefineequations#1#2{\fdef{ASf}{x,y}{#1}\fdef{ASg}{x,y}{#2}} + +\def\AStrajectory#1#2{ + \mfsrc{x1:=#1;y1:=#2; + for i=1 upto ODEstepcount: + x2:=x1+ODEstep*ASf(x1,y1); + y2:=y1+ODEstep*ASg(x1,y1);} + \ODEline{z1}{z2} + \mfsrc{ + if ((y2>yneg) and (y2<ypos)): x1:=x2; y1:=y2 fi; + endfor + }} +\def\ASarrow#1#2{ + \mfsrc{x1:=#1; y1:=#2; + x3:=x1+(ODEarrowlength*ASf(#1,#2))/((ASf(#1,#2)*xscale)++(ASg(#1,#2)*yscale )); + y3:=y1+(ODEarrowlength*ASg(#1,#2))/((ASf(#1,#2)*xscale)++(ASg(#1,#2)*yscale )); + if y3>y1:ODEcolorarrow:=blue else: ODEcolorarrow:=red fi; + } + \ifcolorODEarrow + \drawcolor{ODEcolorarrow} \headcolor{ODEcolorarrow} + \fi + \draw\arrow\lines{z1,z3} +} + +\def\ASarrows#1{\AS@cycle@points#1;,;} +\def\AS@cycle@points#1,#2;{\def\temp{#1}\ifx\temp\ODE@last@point\let\next\relax + \else\ASarrow{#1}{#2}\relax\let\next\AS@cycle@points\fi\next} +\def\AStrajectories#1{\AS@cycle@IC#1;,;} +\def\AS@cycle@IC#1,#2;{\def\temp{#1}\ifx\temp\ODE@last@point\let\next\relax + \else + \AStrajectoryRKF{#1}{#2}\relax\let\next\AS@cycle@IC\fi\next} +\def\AStrajectoryRKF#1#2{ + \mfsrc{x1:=#1;y1:=#2; + TIMEsteps:=abs(TIMEend/TIMEstep); + TIME:=0; + for i=1 upto TIMEsteps: + k1:=ASf(x1,y1); + l1:=ASg(x1,y1); + k2:=ASf(x1+(TIMEstep*k1/2),y1+(TIMEstep*l1/2)); + l2:=ASg(x1+(TIMEstep*k1/2),y1+(TIMEstep*l1/2)); + k3:=ASf(x1+(TIMEstep*k2/2),y1+(TIMEstep*l2/2)); + l3:=ASg(x1+(TIMEstep*k2/2),y1+(TIMEstep*l2/2)); + k4:=ASf(x1+(TIMEstep*k3),y1+(TIMEstep*l3)); + l4:=ASg(x1+(TIMEstep*k3),y1+(TIMEstep*l3)); + k5:=((k1)/6)+((k2)/3)+((k3)/3)+((k4)/6); + l5:=(l1/6)+(l2/3)+(l3/3)+(l4/6); + x2:=x1+(TIMEstep*k5); + y2:=y1+(TIMEstep*l5);} + \ODEline{z1}{z2} + \mfsrc{ + if ((y2>yneg) and (y2<ypos) and (x2<xpos) and (x2>xneg)): x1:=x2; y1:=y2 fi; + endfor + }} + +\catcode`\@12\relax +\endinput +%% +%% End of file `mfpic4ode.tex'. |