rkf45.m

来自「Matlab numerical methods,examples of mat」· M 代码 · 共 88 行

function [T,Y] = rkf45(f,a,b,ya,m,tol)
%---------------------------------------------------------------------------
%RKF45   % Runge-Kutta-Fehlberg solution for y' = f(t,y) with y(a) = ya.
% Sample call
%   [T,Y] = rkf45('f',a,b,ya,m)
% Inputs
%   f    name of the function
%   a    left  endpoint of [a,b]
%   b    right endpoint of [a,b]
%   ya   initial value
%   m    initial guess for number of steps
% Return
%   T    solution: vector of abscissas
%   Y    solution: vector of ordinates
%
% NUMERICAL METHODS: MATLAB Programs, (c) John H. Mathews 1995
% To accompany the text:
% NUMERICAL METHODS for Mathematics, Science and Engineering, 2nd Ed, 1992
% Prentice Hall, Englewood Cliffs, New Jersey, 07632, U.S.A.
% Prentice Hall, Inc.; USA, Canada, Mexico ISBN 0-13-624990-6
% Prentice Hall, International Editions:   ISBN 0-13-625047-5
% This free software is compliments of the author.
% E-mail address:      in%"[e-mail]"
%
% Algorithm 9.5 (Runge-Kutta-Fehlberg Method RKF45).
% Section	9.5, Runge-Kutta Methods, Page 461
%---------------------------------------------------------------------------

a2 = 1/4; b2 = 1/4; a3 = 3/8; b3 = 3/32; c3 = 9/32; a4 = 12/13;
b4 = 1932/2197; c4 = -7200/2197; d4 = 7296/2197; a5 = 1;
b5 = 439/216; c5 = -8; d5 = 3680/513; e5 = -845/4104; a6 = 1/2;
b6 = -8/27; c6 = 2; d6 = -3544/2565; e6 = 1859/4104; f6 = -11/40;
r1 = 1/360; r3 = -128/4275; r4 = -2197/75240; r5 = 1/50;
r6 = 2/55; n1 = 25/216; n3 = 1408/2565; n4 = 2197/4104; n5 = -1/5;
big = 1e15;
h = (b-a)/m;
hmin = h/64;
hmax = 64*h;
max1 = 200;
Y(1) = ya;
T(1) = a;
j = 1;
tj = T(1);
br = b - 0.00001*abs(b);
while (T(j)<b),
  if ((T(j)+h)>br), h = b - T(j); end
  tj = T(j);
  yj = Y(j);
  y1 = yj;
  k1 = h*feval(f,tj,y1);
  y2 = yj+b2*k1;                         if big<abs(y2) break, end
  k2 = h*feval(f,tj+a2*h,y2);
  y3 = yj+b3*k1+c3*k2;                   if big<abs(y3) break, end
  k3 = h*feval(f,tj+a3*h,y3);
  y4 = yj+b4*k1+c4*k2+d4*k3;             if big<abs(y4) break, end
  k4 = h*feval(f,tj+a4*h,y4);
  y5 = yj+b5*k1+c5*k2+d5*k3+e5*k4;       if big<abs(y5) break, end
  k5 = h*feval(f,tj+a5*h,y5);
  y6 = yj+b6*k1+c6*k2+d6*k3+e6*k4+f6*k5; if big<abs(y6) break, end
  k6 = h*feval(f,tj+a6*h,y6);
  err = abs(r1*k1+r3*k3+r4*k4+r5*k5+r6*k6);
  ynew = yj+n1*k1+n3*k3+n4*k4+n5*k5;
  if ((err<tol)|(h<2*hmin)),
    Y(j+1) = ynew;
    if ((tj+h)>br),
      T(j+1) = b;
    else
      T(j+1) = tj + h;
    end
    j = j+1;
    tj = T(j);
  end
  if (err==0),
    s = 0;
  else
    s = 0.84*(tol*h/err)^(0.25);
  end
  if ((s<0.75)&(h>2*hmin)), h = h/2; end
  if ((s>1.50)&(2*h<hmax)), h = 2*h; end
  if ((big<abs(Y(j)))|(max1==j)), break, end
  mend = j;
  if (b>T(j)),
    m = j+1;
  else
    m = j;
  end
end