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470 Chapter 11 Systems of Ordinary Differential Equations

procedure RK4 System1(n, h, t,(x i ), nsteps)
integer i, j, n; real h, t; real array (x i ) 1:n
allocate real array (y i ) 1:n ,(K i, j ) 1:n×1:4
output 0, t,(x i )
for j = 1 to nsteps do
call XP System(n, t,(x i ), (K i,1 ))
for i = 1 to n do
1
y i ← x i + hK i,1
2
end for
call XP System(n, t + h/2,(y i ), (K i,2 ))
for i = 1 to n do
1
y i ← x i + hK i,2
2
end for
call XP System(n, t + h/2,(y i ), (K i,3 ))
for i = 1 to n do
y i ← x i + hK i,3
end for
call XP System(n, t + h,(y) i ,(K i,4 ))
for i = 1 to n do
1
x i ← x i + h[K i,1 + 2K i,2 + 2K i,3 + K i,4 ]
6
end for
t ← t + h
output j, t,(x i )
end for
deallocate array (y i ), (K i, j )
end procedure RK4 System1

To illustrate the use of this procedure, we again use System (1) for our example. Of
course, it must be rewritten in the form of Equation (4). A suitable main program and a
procedure for computing the right-hand side of Equation (4) follow:

program Test RK4 System1
integer n ← 2, nsteps ← 100
real a ← 0, b ← 1
real h, t; real array (x i ) 1:n
t ← 0
(x i ) ← (1, 0)
h ← (b − a)/nsteps
call RK4 System1(n, h, t,(x i ), nsteps)
end program Test RK4 System1
procedure XP System(n, t,(x i ), ( f i ))
real array (x i ) 1:n ,( f i ) 1:n
integer n