Code covered by the BSD License

# Numerical Methods Using MATLAB, 4e

### John Mathews (view profile)

23 Dec 2003 (Updated )

Companion software to accompany the book "Numerical Methods Using MATLAB"

j; end end R=[T' Y'];
```function R = rkf45(f,a,b,ya,m,tol)

% Input    - f  function
%             - a  left  endpoint of [a,b]
%             -b right endpoint of [a,b]
%             - ya initial value
%             -m initial guess for number of steps
% Output - T  solution: vector of abscissas
%             - Y  solution: vector of ordinates

%If f is an M-file function call R=rkf45(@f,a,b,ya,M).
%If f is an anonymous function call R=rkf45(f,a,b,ya,M).

%  NUMERICAL METHODS: Matlab Programs
% (c) 2004 by John H. Mathews and Kurtis D. Fink
%  Complementary Software to accompany the textbook:
%  NUMERICAL METHODS: Using Matlab, Fourth Edition
%  ISBN: 0-13-065248-2
%  Prentice-Hall Pub. Inc.
%  One Lake Street
%  Upper Saddle River, NJ 07458

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*f(tj,y1);
y2 = yj+b2*k1;                         if big<abs(y2) break, end
k2 = h*f(tj+a2*h,y2);
y3 = yj+b3*k1+c3*k2;                   if big<abs(y3) break, end
k3 = h*f(tj+a3*h,y3);
y4 = yj+b4*k1+c4*k2+d4*k3;             if big<abs(y4) break, end
k4 = h*f(tj+a4*h,y4);
y5 = yj+b5*k1+c5*k2+d5*k3+e5*k4;       if big<abs(y5) break, end
k5 = h*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*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
R=[T' Y'];```