MATLAB: Generate analytical solutions using MATLAB

analytical solutionsMATLABSymbolic Math Toolbox

I have a dynamic system of the form below and I am trying to find the general solution of f(n) and g(n) in terms of n and p, where n is an integer, from 1,2,…N and p is from 0 to 1.

 f(n) = (f(n-1))^2*p^2/[f(n-1)*p+g(n-1)*(1-p)] + ...
        (f(n-1))^2*(1-p)^2/[f(n-1)*(1-p)+g(n-1)*p]
 g(n) = f(n-1)*g(n-1)*p*(1-p)/[f(n-1)*p+g(n-1)*(1-p)] + ...
        f(n-1)*g(n-1)*p*(1-p)/[f(n-1)*(1-p)+g(n-1)*p]

where

 f(1) = p^2+(1-p)^2
 g(1) = 2*p*(1-p)

I am not able to solve it analytically myself. Would anyone have a clue on how I could do that in MATLAB?

Best Answer

What kind of answer were you hoping for? Looks like f would have to be something like the product of n-1 binomial terms raised to some integral power whose value is not immediately obvious to me.

f(10) calculated "raw" is over 100000 characters long, and simplifying it post-facto chewed right through the 2 Gb available on my system. Simplifying after each step would help a lot, at least if the polynomial representation is desired instead of a mass of nested fractions.

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MATLAB: OOP in Matlab – What to do when the output is a different class following method

Happens to work for me, in 2013a, but I think that is not the point, as that is probably coincidental for this particular example.

The problem with the _mult method you defined is that when you overload a function like _mult in MuPAD, you're actually responsible for the full multiplication, not just that of two arguments. I.e., in your last call, Bx::_mult is called with four arguments: 6, Bx(5,4), 3, Bx(4,3). It has to handle that, since MuPAD doesn't know whether you even want things like commutativity with numbers.

Here's what I think the structure you'll want could look like:

Bx::mult2 := proc(a : Bx, b : Bx)
  ... multiply two Bx objects ...
end_proc:
Bx::multBx := proc() // multiply arbitrarily many Bx
  local n, left, right;
begin
  case args(0)
  of 0 do return(1);
  of 1 do return(args(1));
  of 2 do return(Bx::mult2(args()));
  otherwise
    n := floor(args(0)/2); % split here
    left := Bx::multBx(args(1..n));
    right := Bs::multBx(args(n+1..args(0)));
    return(Bx::mult2(left, right));
  end_case;
end_proc:
Bx::_mult := proc()
  local Bxes, others, dummy;
begin
  [Bxes, others, dummy] := split([args()], testtype, Bx);
  assert(dummy = []);
  others := _mult(op(others));
  Bxes := Bx::multBx(op(Bxes));
  if others = 1 then
    return(Bxes);
  elif Bxes = 1 then % can this happen?
    return(others);
  else
    return(hold(_mult)(Bxes, others));
  end_if;
end_proc:

MATLAB: How to perform symbolic factorial

The MuPad factorial function name is fact() . This does not, however, appear to be exposed at the Matlab level, so you might need to use, (e.g.,)

f2=symsum((1/sym('factorial(n)'))*f^(n),n,0,10)

Best Answer

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MATLAB: OOP in Matlab – What to do when the output is a different class following method

MATLAB: How to perform symbolic factorial

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