MATLAB error - "??? Undefined function or variable 'x'."

Hi! I'm just getting started on MATLAB, and I need to solve the following:

((P1/T1)*(((Q/Cp)+T1*((x/P1)^((n-1)/n)))/x))==(((1-E*(((P2/x)^(1/n))-1))/(1-E*(((x/P1)^(1/n))-1)))*(VbA/VbB)))

where 'x' is the variable. I've assigned the values for all the constants and tried using fsolve(func,var), but I keep getting an error message as follows:

??? Undefined function or variable 'x'.

I haven't been able to find where the mistake lies, so help would be very much appreciated!

Answer 1

[Walter Roberson]

If you have the symbolic toolbox, you would use

syms x
S = solve(((P1/T1)*(((Q/Cp)+T1*((x/P1)^((n-1)/n)))/x)) - (((1-E*(((P2/x)^(1/n))-1))/(1-E*(((x/P1)^(1/n))-1)))*(VbA/VbB)), x)

Note: you had an extra ")" at the very end.

Unfortunately if n is a positive integer, this will end up involving the roots of a polynomial of order (n^2) (e.g., a degree 25 polynomial for n=5).

If n is positive and rational, the solution can end up involving the roots of a fairly high order polynomial whose order is not obvious to me. For example, n=5/9 involves a degree 61 polynomial

[Alejandra]

Hmm...I see what you mean...I'll see if I can get round that, since it should not be as complex (The equation is supposed to be for a simple Thermodynamics Problem, as part of my Engineering course so...)

Thanks a lot!!

[Walter Roberson]

Some values of n will lead to polynomials of even order. When that happens, there will always be an even number of real roots -- 0, 2, 4, 6, or as appropriate. If you used fsolve, you would end up with _one_ of the real roots, and it would not be controllable as to which one you got as fsolve() does not accept bounding constraints.

When polynomials of odd order are generated, there will be an odd number of real roots, including plausibly only a single real root -- but if there *are* multiple real roots, again fsolve() might give you any of them.

If you do have the symbolic toolbox, and you use solve(), and n is numeric and not symbolic, then you should in theory get out an expression involving roots of a polynomial. For any one value of n, you will be able to write that expression in terms of roots(), which would be able to calculate (approximations of) all of the roots given the numeric values of all of the parameters; you would then be able to filter the results as appropriate.

[Walter Roberson]

What sort of values do you expect for n ?

[Alejandra]

n=1.29

Hmm the program does not throw any result now...

[Walter Roberson]

Solving that with solve() might take quite a lot of time and memory. I was running n=314/100 (i.e., the first few digits of Pi) and after about 25 minutes Maple had gone through about 2 Gb of memory and I had to kill the program.

[Walter Roberson]

Could you give some sample values for the other parameters?

Answer 2

[Walter Roberson]

The fsolve() approach would be to use

fsolve(@(x) ((P1/T1)*(((Q/Cp)+T1*((x/P1)^((n-1)/n)))/x)) - (((1-E*(((P2/x)^(1/n))-1))/(1-E*(((x/P1)^(1/n))-1)))*(VbA/VbB)), x0)

where x0 is an initial guess. As I mentioned earlier, this will return a single root and you do not get any control over which one.

You could also use

fzero(@(x) ((P1/T1)*(((Q/Cp)+T1*((x/P1)^((n-1)/n)))/x)) - (((1-E*(((P2/x)^(1/n))-1))/(1-E*(((x/P1)^(1/n))-1)))*(VbA/VbB)), [x0,x1])

where x0 and x1 are starting values and you are sure that the expression will be of differing signs at the two points. This allows you to control the search interval but does require you have an idea of where the bounds are.

[Alejandra]

I still get no response from the program...