GJC@MIT-MC 09/21/81 13:04:55
To: BROM at MIT-MC
CC: (FILE [SHARE2;NDIFFQ USAGE]) at MIT-MC
I changed the argument conventions for RUNGE_KUTTA systems
so that you can do things more efficiently.

If you call RUNGE_KUTTA(F,T,[X0,X1,X2]) THEN F WILL GET ARGUMENTS,

F(X_PRIME_VECTOR,T,X_VECTOR):= ....;

And F should do X_PRIME_VECTOR[0]: ...X_VECTOR[0] ...,
                X_PRIME_VECTOR[1]: ...................
                etc.

If you call RUNGE_KUTTA([F0,F1,F2],T,[X0,X1,X2]) then F will get arguments:

F0(T,X_VECTOR),
F1(T,X_VECTOR),
F2(T,X_VECTOR).

This kind of argument convention also generalizes properly to
second_order systems.


GJC@MIT-MC 06/27/81 12:49:21
To: BROM at MIT-MC
CC: BDB at MIT-MC, (FILE [SHARE2;NDIFFQ USAGE]) at MIT-MC
See the file SHARE2;NDIFFQ USAGE.

you can define 

FX(T,P):=(MODE_DECLARE(ARRAY(P),FLOAT),T+P[0]+P[1]...);
FY(t,P):=.....;
FZ(t,P):=....;

create arrays:

array([t,x,y,z],float,25);

time step:

init_float_array(t,0,1);

initial conditions:

x[0]:1.0;
y[0]:2.0;
z[0]:3.0;

run the problem:

runge_kutta([fx,fy,fz],t,[x,y,z]);

plot the result:

graph2(t,[x,y,z]);

You could also define
F(V,T,P):=(MODE_DECLARE(ARRAY(V,P),FLOAT),
           V[0]: ....,
           V[1]: ....,
           V[2]: ....);

And use the call Runge_Kutta(F,t,[X,Y,Z]);



GJC@MIT-MC 06/07/81 15:44-EDT  To: INFO-MACSYM
I moved the numerical solution to differential equation routines to 
SHARE so that you can get them by doing LOAD("NDIFFQ"); and then e.g.
Define_Variable(N,0.3,FLOAT);
Define_Variable(H,0.175,FLOAT);
F(X,E):=(Mode_Declare([X,E],FLOAT),N*EXP(X)/(E+X^(2*H)*EXP(H*X)));
Compile(F);
Array([X,E],FLOAT,35);
Init_Float_Array(X,1.0E-3,6.85); /* Fills X with the interval */
E[0]:5.0;                        /* Initial condition */
Runge_Kutta(F,X,E);              /* Solve it */
Graph2(X,E);                     /* Graph the solution */

p.s. Runge_Kutta(F,X,E,E_Prime) would be the call for a second-order 
equation.

p.p.s Runge_Kutta([f1,f2,...fn],x,[y1,y2,..yn]) would be the call for
n simultaneous first-order equations. The functions are called
2 arguments, the first being X, and the second being a vector of
values [y1,y2,...Yn], index from 0 to N-1.