MaplePrimes - answers and comments on Question, Automatically reducing size of system of equations through substitution

eliminate

pagan — Thu, 16 Sep 2010 02:57:54 Z

You want to eliminate intermediate variables? One simple approach could involve calling the `eliminate` command. (How you would further analyze its output for more complicated examples is more interesting.)

restart:
sys:={y = z1 + z2^2,
      z1 = z2 + 3*exp(u),
      z2 = 2*u^3*z1}:
T:=eliminate(sys,[z1,z2]):
new:=normal(isolate(op(T[2]),y)):
expected:=y=3*exp(u)/(1-2*u^3) * (1 + 4*u^6*(3*exp(u)/(1-2*u^3))):
evalb(normal(rationalize(new))=normal(rationalize(expected)));
                              true

simplify

Oliver K — Thu, 16 Sep 2010 10:09:41 Z

You can also try ?simplify,siderels

Groebner basis

roman_pearce — Fri, 17 Sep 2010 08:38:29 Z

sys := {y = z1 + z2^2, z1 = z2 + 3*exp(u), z2 = 2*u3*z1}:

sys := subs(exp(u)=expu, map(lhs-rhs,sys)):

with(Groebner):   # make polynomials

vars := [SuggestVariableOrder(sys)];

elim := {z1,z2}:  # try to get rid of these

tord := lexdeg(selectremove(member, vars, elim));

gb := Groebner[Basis](sys, tord);

subs(expu=exp(u), gb);

This is not exactly "model reduction", as the Groebner basis will typically be larger, but it will eliminate variables. You can solve the first polynomial for the variable of lowest degree. In this case it is degree 1 in y, but you may not be so lucky in general.

Very nice! Somehow I was expecting a more

Paul V — Thu, 16 Sep 2010 09:41:05 Z

Very nice! Somehow I was expecting a more complicated answer, requiring the use of Groebner bases. I did not think to look for an "eliminate" function ... I was looking through the documentation for a "reduce" or "subs" function. This does what I want. Thank you.

Thanks, that's a useful function. In my case

Paul V — Thu, 16 Sep 2010 19:25:03 Z

Thanks, that's a useful function. In my case however, the side relations are actually in the system of equations themselves, so "eliminate" is probably a more general way to do the model reduction.

Thanks for the explanation. That's a really

Paul V — Fri, 17 Sep 2010 19:48:18 Z

Thanks for the explanation. That's a really nice introduction to getting a Groebner basis in Maple. It does result in more complicated expressions, but you're right it eliminates variables.

Also, I like the "map(lhs-rhs,sys)" trick: it's elegant and compact. I actually wrote a proc to do that...