#COPYRIGHT NOTICE:  Copyright (c) 1997-1998 by Sasha Cyganowski.  
#Permission is  
#granted to anyone to use, modify, or redistribute this software freely, 
#subject to the following restrictions:  1. The author accepts no
#responsibility for any consequences of this software and makes no guarantee
#that this software is free of defects.  2.  The origin of this software must  
#not be misrepresented, either by explicit claim or by omission. 
#3.  This notice and the copyright must be included with all copies or
#modified versions of this software.  4.  This software may not be included or
#redistributed as part of any package to be sold for profit unlesss the author
#has given explicit written permission to do so.    
# Sasha Cyganowski, School of Computing and Mathematics, Deakin University,
#Waurn Ponds, VIC, Australia.  Internet: sash@deakin.edu.au
# Telephone:  +61 3 5227 1382.


LJ:=proc(X,b,j)


local k;


sum('op(j,op(k,b))*diff(X,x[k])','k'=1..nops(b));


end:


L0:=proc(X,a,b)


local part1,part2,part3,k,j,l;





part1:=diff(X,t);


part2:=sum('a[k]*diff(X,x[k])','k'=1..nops(a));


part3:=1/2*sum('sum('sum('op(j,op(k,b))*op(j,op(l,b))*diff(X,x[k],x[l])','j'=1..nops(op(1,b)))','k'=1..nops(a))','l'=1..nops(a));


part1+part2+part3;


end:


WkTaylor2:=proc(a,b)


local u,i,n,soln,Y,Delta,W,j1,j2,V;


for i from 1 to nops(a) do


soln[i]:=Y.i[n+1]=Y.i[n]+a[i]*Delta[n]+1/2*L0(a[i],a,b)*Delta[n]^2+sum('(op(j,op(i,b))+1/2*Delta[n]*(L0(op(j,op(i,b)),a,b)+LJ(a[i],b,j)))*Delta*W.j[n]','j'=1..nops(op(1,b)))+1/2*sum('sum('LJ(op(j2,op(i,b)),b,j1)*(Delta*W.j1[n]*Delta*W.j2[n]+V[(j1,j2)])','j1'=1..nops(op(1,b)))','j2'=1..nops(op(1,b)));


for u from 1 to nops(a) do


soln[i]:=subs(x[u]=Y.u[n],soln[i]);


od;


print(soln[i]);


od;


end: