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% PostScript prologue for pst-ode.tex.
% Version 0.5, 2013/11/11
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% Alexander Grahn (C) 2012, 2013
%
% This program can be redistributed and/or modified under the terms
% of the LaTeX Project Public License Distributed from CTAN archives
% in directory macros/latex/base/lppl.txt.
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/tx@odeDict 1 dict def
tx@odeDict begin
%some constants for step size calculation
/sfty 0.9 def /pgrow -0.2 def /pshrink -0.25 def
%helper functions
/addvect { % [1 2 3] [4 5 6] addvect => [5 7 9]
  aload pop xlength1 -1 roll {xlength1 -1 roll add} forall
  xlength array astore
} bind def
/subvect { % [1 2 3] [4 5 6] subvect => [-3 -3 -3]
  aload pop xlength1 -1 roll {xlength1 -1 roll sub} forall
  xlength array astore
} bind def
/mulvect { % [1 2 3] 4 mulvect => [4 8 12]
  /mul cvx 2 array astore cvx forall xlength array astore
} bind def
/edivvect { % [1 2 3] [4 5 6] edivvect => [0.25 0.4 0.5]
  aload pop xlength1 -1 roll {xlength1 -1 roll div} forall
  xlength array astore
} bind def
/eabsvect { % [-1 2 -3] eabsvect => [1 2 3]
  {abs} forall xlength array astore
} bind def
%/revstack { % (a) (b) (c) (d) 3 revstack => (a) (d) (c) (b)
%  -1 2 {dup 1 sub neg roll} for
%} bind def
/min { 2 copy gt { exch } if pop } bind def
/max { 2 copy lt { exch } if pop } bind def
%coefficient table (Butcher table) of RKF45
/a41 25 216 div def /a43 1408 2565 div def
/a44 2197 4104 div def /a45 1 5 div neg def
/a51 16 135 div def /a53 6656 12825 div def
/a54 28561 56430 div def /a55 9 50 div neg def
/a56 2 55 div def
/b21 1 4 div def /b31 3 32 div def /b32 9 32 div def
/b41 1932 2197 div def /b42 7200 2197 div neg def /b43 7296 2197 div def
/b51 439 216 div def /b52 8 neg def /b53 3680 513 div def
/b54 845 4104 div neg def /b61 8 27 div neg def /b62 2 def
/b63 3544 2565 div neg def /b64 1859 4104 div def /b65 11 40 div neg def
%Runge-Kutta-Fehlberg (RKF45) method
%performs one integration step over tentative step size ddt
%[state vector x(t)] RKF45 => [x(t)] [x(t+ddt) by RKF4] errmax
/RKF45 {
  /t tcur def
  dup ODESET ddt mulvect /k1 exch def
  /t tcur ddt 4 div add def
  dup k1 b21 mulvect addvect ODESET ddt mulvect /k2 exch def
  /t tcur ddt 3 mul 8 div add def
  dup k1 b31 mulvect addvect k2 b32 mulvect addvect ODESET ddt mulvect
    /k3 exch def
  /t tcur ddt 12 mul 13 div add def
  dup k1 b41 mulvect addvect k2 b42 mulvect addvect k3 b43 mulvect addvect
    ODESET ddt mulvect /k4 exch def
  /t tcur ddt add def
  dup k1 b51 mulvect addvect k2 b52 mulvect addvect k3 b53 mulvect addvect
    k4 b54 mulvect addvect ODESET ddt mulvect /k5 exch def
  /t tcur ddt 2 div add def
  dup k1 b61 mulvect addvect k2 b62 mulvect addvect k3 b63 mulvect addvect
    k4 b64 mulvect addvect k5 b65 mulvect addvect ODESET ddt mulvect
    /k6 exch def % => [x(t)]
  %fourth order solution (increment dx)
  dup dup k1 a41 mulvect k3 a43 mulvect addvect k4 a44 mulvect addvect
    k5 a45 mulvect addvect dup
    % => [x(t)] [x(t)] [x(t)] [dx by RKF4] [dx by RKF4]
  %fifth order solution (abs. error)
  k1 a51 mulvect k3 a53 mulvect addvect k4 a54 mulvect addvect
    k5 a55 mulvect addvect k6 a56 mulvect addvect subvect
    % => [x(t)] [x(t)] [x(t)] [dx by RKF4] [err]
  5 1 roll addvect 4 -2 roll % => [x(t)] [x(t+ddt) by RKF4] [err] [x(t)]
  %scaling vector for step size adjustment (Numerical Recipies)
  eabsvect k1 eabsvect addvect {1e-30 add} forall xlength array astore
  % => [x(t)] [x(t+ddt) by RKF4] [err] [xscale]
  %maximum rel. error
  /errmax 0 def
  edivvect {abs dup errmax gt {/errmax exch def}{pop} ifelse} forall errmax
  % => [x(t)] [x(t+ddt) by RKF4] errmax
} bind def
/ODEINT { % performs integration over output step [t,t+dt]
          % [ state vector x(t) ] ODEINT => [ state vector x(t+dt) ]
  %decrease overshooting step size
  tcur ddt add tout sub tcur ddt add tout sub dt add mul 0 gt
  {/ddt tout tcur sub def} if
  RKF45
  ode@tol div dup 1 gt {
    %failed step -> reduce step size
    exch pop pshrink exp 0.1 max sfty mul ddt mul /ddt exch def
    (-) odeprint tcur ddt add tcur eq {
      (error: step size underflow in ODEINT) print quit
    } if
    ODEINT %repeat step with new ddt
  }{
    %success
    3 -1 roll pop % remove previous state vector
    /tcur tcur ddt add def
    pgrow exp 5 min sfty mul ddt mul /ddt exch def
    %output step completed?
    tcur tout sub dt mul 0 ge {
      (o) odeprint /tcur tout def /tout tout dt add def
    }{
      (+) odeprint ODEINT %continue integration
    } ifelse
  }ifelse
} bind def
/writeresult { %write output vector to file
  {
    20 string cvs statefile exch writestring
    statefile ( ) writestring
  } forall
  statefile (\n) writestring
} bind def
/assembleresult { %assembles state vector for building table of results
  {
    dup (t) eq {
      pop tcur
    }{
      ode@laststate exch get
    } ifelse
  } forall
} bind def
end