summaryrefslogtreecommitdiff
path: root/Master/texmf-dist/tex/generic/pgf/math/pgfmathode.code.tex
blob: d88378975580c7fe69a331fee0396596347fff62 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
% Copyright 2011 by Christophe Jorssen
%
% This file may be distributed and/or modified
%
% 1. under the LaTeX Project Public License and/or
% 2. under the GNU Public License.
%
% See the file doc/generic/pgf/licenses/LICENSE for more details.
\def\pgfmathode@stopadd{\pgfmathode@stopadd}

\def\pgfmathode@addindextofunc[#1]#2{%
  \begingroup
    \gdef\pgfmathode@func{}%
    \pgfmathode@addindextofunc@i[#1]#2\pgfmathode@stopadd}

\def\pgfmathode@addindextofunc@i[#1]#2#3{%
    \ifx#1#2
      \expandafter\gdef\expandafter\pgfmathode@func\expandafter{%
        \pgfmathode@func #2[\pgfmathode@timeindex]}%
    \else
      \expandafter\gdef\expandafter\pgfmathode@func\expandafter{%
        \pgfmathode@func #2}%
    \fi
    \ifx#3\pgfmathode@stopadd
      \expandafter\gdef\expandafter\pgfmathode@func\expandafter{%
        \expandafter{\pgfmathode@func}}%
      \let\next\endgroup
    \else
      \def\next{\pgfmathode@addindextofunc@i[#1]#3}%
    \fi
    \next}

%**********************************************************************
% Know limitation: fk cannot be a time function. Should be possible to
% use \Sol[0] as time in the set of equations.
%**********************************************************************

% Numerically solve a differential system with 1st order Runge-Kutta
% (Euler method) 
% dq1/dt = f1(t,q1,q2,...,qN)
% dq2/dt = f2(t,q1,q2,...,qN)
% ...
% dqN/dt = fN(t,q1,q2,...,qN)
% 
% #1: A pgf array where the solution will be stored.
% At the end:
% #1 -> {{t(0),q1(t(0)),q2(t(0)),...,qN(t(0))},
%        {t(1),q1(t(1)),q2(t(1)),...,qN(t(1))},
%        ...
%        {t(Nstep),q1(t(Nstep)),q2(t(Nstep)),...,qN(t(Nstep))}}
% #2: A pgf array with f1,f2,...,fN
% {f1}
% #3: A pgf array with initial conditions.
% {t(0),q1(t(0)),q2(t(0)),...,qN(t(0))}
% #4: t(Nstep)
% #5: Nstep (number of steps)
%
% * First order example: dq/dt = -q + 5 with IC=(t=0,q(0)=0)
% \pgfmathodeRKI{\Sol}{{-\Sol[1]+5}}{{0,0}}{3.5}{100}
% * Second order example: {dq1/dt=q2, dq2/dt=-q2/2-sin(q1)} (damped
%   pendulum) 
% \pgfmathodeRKI{\Sol}{{\Sol[2],-\Sol[2]/2-sin(deg(\Sol[1]))}}%
%   {{0,-1,5}}{15}{50}

\def\pgfmathode@generic@init@RK{%
  \begingroup
    % Sol array (#1) first element is given by the set of initial
    % conditions (#3). We need to enclose it in brace so that is a now
    % a "matrix" array.
    \gdef#1{{#3}}%
    % The set of rhs {f1(t,q1,q2,...,qN),...,fN(t,q1,q2,...,qN)} is
    % given in terms of #1[0],#1[1],...,#1[N]. We need to transform to
    % #1[\pgfmathode@timeindex][0],...,#1[\pgfmathode@timeindex][N], that is
    % add [\pgfmathode@timeindex] after every occurence of #1 in #2.
    % No need to brace #2 since it is already surounded by braces. The
    % result is stored in \pgfmathode@func
    \pgfmathode@addindextofunc[#1]#2%
    % Set time step. Actually the number of time steps is #5-1, so we
    % divide by (#5-1)+1 = #5.
    \pgfmathsetmacro\pgfmathode@Sol@t{#1[0][0]}%
    \pgfmathsetmacro\pgfmathode@Deltat{(#4-\pgfmathode@Sol@t)/#5}%
    % Compute the order of the system: it is equal to the number of IC
    % minus one (time).
    \pgfmathsetmacro\pgfmathode@dimfunc{dim(#3)-1}%
    \pgfmathsetmacro\pgfmathode@numstep{#5-1}}

\def\pgfmathodeRKI#1#2#3#4#5{%
    \pgfmathode@generic@init@RK
    % Start the time loop for Runge-Kutta 1st order
    \foreach \pgfmathode@timeindex in {0,...,\pgfmathode@numstep} {%
      \message{Step \pgfmathode@timeindex...}%
      % Start the function loop
      \foreach \pgfmathode@funcindex in {1,...,\pgfmathode@dimfunc} {%
        % The RKI algorithm
        % qk[t(i)] = Deltat*fk(t(i-1),q1(t(i-1)),...qN(t(i-1))) +
        %            qk[t(i-1)] 
        % Note: this part can be really slow due to array managment in
        % pgfmath. 
        % TODO: find a better way (at least inside this loop)
        \pgfmathparse{%
          \pgfmathode@Deltat*(\pgfmathode@func[\pgfmathode@funcindex-1])+
          #1[\pgfmathode@timeindex][\pgfmathode@funcindex]}%
        % Store the result in a helper macro
        \expandafter\xdef\csname pgfmathode@Sol@\pgfmathode@funcindex
          \endcsname{\pgfmathresult}%
      }
      % Store step i
      \pgfmathparse{\pgfmathode@Sol@t+\pgfmathode@Deltat}%
      \xdef\pgfmathode@Sol@t{\pgfmathresult}%
      \xdef\pgfmathode@temp{}%
      \foreach \pgfmathode@funcindex in {1,...,\pgfmathode@dimfunc} {%
        \ifx\pgfmathode@temp\pgfutil@empty
          \xdef\pgfmathode@temp{%
            \csname pgfmathode@Sol@\pgfmathode@funcindex\endcsname}%
        \else
          \xdef\pgfmathode@temp{%
            \pgfmathode@temp,
            \csname pgfmathode@Sol@\pgfmathode@funcindex\endcsname}%
        \fi}
      \xdef#1{%
        {\expandafter\pgfutil@firstofone#1,%
          {\pgfmathode@Sol@t,\pgfmathode@temp}}}%
    }
  \endgroup}

% RKIV: far from working yet!
\def\pgfmathodeRKIV#1#2#3#4#5{%
    \pgfmathode@generic@init@RK
    % Start the time loop for Runge-Kutta 4th order
    \foreach \pgfmathode@timeindex in {0,...,\pgfmathode@numstep} {%
      \message{Step \pgfmathode@timeindex...}%
      % Start the function loop
      \foreach \pgfmathode@funcindex in {1,...,\pgfmathode@dimfunc} {%
        % The RKIV algorithm
        % qk[t(i)] = qk[t(i-1)] + 
        %            (1/6) * (j1k + 2j2k + 2j3k + j4k)
        % where j1k = fk(t(i-1),q1(t(i-1)),...qN(t(i-1))) * Deltat
        %       j2k = fk(t(i-1)+.5*Deltat,
        %                q1(t(i-1))+.5*j11,...qN(t(i-1))+.5*j1N) * Deltat
        %       j3k = fk(t(i-1)+.5*Deltat,
        %                q1(t(i-1))+.5*j21,...qN(t(i-1))+.5*j2N) * Deltat
        %       j4k = fk(t(i-1)+Deltat,
        %                q1(t(i-1))+j31,...qN(t(i-1))+j3N) * Deltat
        %
        % Compute j1k
        \pgfmathparse{%
          \pgfmathode@Deltat*(\pgfmathode@func[\pgfmathode@funcindex-1])+
          #1[\pgfmathode@timeindex][\pgfmathode@funcindex]}%
        \expandafter\xdef\csname pgfmathode@RKIV@1@\pgfmathode@funcindex
          \endcsname{\pgfmathresult}%
        % Compute j2k
        \pgfmathparse{%
          \pgfmathode@Deltat*(\pgfmathode@func[\pgfmathode@funcindex-1])+
          #1[\pgfmathode@timeindex][\pgfmathode@funcindex]}%
        \expandafter\xdef\csname pgfmathode@RKIV@1@\pgfmathode@funcindex
          \endcsname{\pgfmathresult}%
      }
      % Store step i
      \pgfmathparse{\pgfmathode@Sol@t+\pgfmathode@Deltat}%
      \xdef\pgfmathode@Sol@t{\pgfmathresult}%
      \xdef\pgfmathode@temp{}%
      \foreach \pgfmathode@funcindex in {1,...,\pgfmathode@dimfunc} {%
        \ifx\pgfmathode@temp\pgfutil@empty
          \xdef\pgfmathode@temp{%
            \csname pgfmathode@Sol@\pgfmathode@funcindex\endcsname}%
        \else
          \xdef\pgfmathode@temp{%
            \pgfmathode@temp,
            \csname pgfmathode@Sol@\pgfmathode@funcindex\endcsname}%
        \fi}
      \xdef#1{%
        {\expandafter\pgfutil@firstofone#1,%
          {\pgfmathode@Sol@t,\pgfmathode@temp}}}%
    }
  \endgroup}


\endinput