MATLAB solve function is taking a very long time

Hello,

I have a system of 17 equations and 23 unknowns and I'm using the MATLAB "solve" function to solve symbolically for 17 unknowns in terms of the other 6 unknowns. However, the "solve" function has been running for several hours and still has not gotten a solution.

I have used the MATLAB "solve" function in this way for several small systems of equations and it finds a solution very quickly (within minutes). Does anyone have any ideas as to why the "solve" function is taking so long in this instance? What is generally the upper limit of equations that the "solve" function can solve in a reasonable amount of time? (I know there is an upper limit to the number of equations that can be handled, but I'm very surprised that 17 equations is taking so long.) Does anyone have any ideas as to anything I can try (maybe options to use with the solve function, or some other function to use) in order to get a solution for my system of equations?

Thank you,

Kevin

Answer 1

[Walter Roberson]

The upper limit to the number of equations that solve() can resolve in a reasonable amount of time depends upon whether the equations are linear or not. If they are non-linear (including even just squared terms) then 4 can usually be done fairly quickly, 5 is about the practical limit, and 6 might take days.

The limits are slightly higher if you happen to have the Maple based symbolic toolbox and are using just squared terms, but even for that, 6 might take overnight and 7 is unlikely to finish after several days.

[Kevin Bachovchin]

If we call the 17 unknowns I'm solving for x1-x17 I'm solving in terms of my equations have a form like this

A*x1+B*x2+C*x3+...+D*x17 = 0

but the coefficients A,B,C,D are also MATLAB syms. So is this considered linear even though I have two syms multiplied by each other?

[Ahmed A. Selman]

The equation you wrote is linear, but with no possible solution.

[Kevin Bachovchin]

Why no possible solution? I have 17 equations, not just 1. I was just trying to give the form of my equations.

[Walter Roberson]

That sounds linear to me. I simply have not done enough linear work to have a good idea of what the effective limits are for it.

If you email me the equations (or post them here) I could give it a try.

[Kevin Bachovchin]

In the case which is taking a very long time, my system is non-linear. In my previous comment, I just wanted to clarify what was meant by linear. I've done some examples with linear equations too, and it's solved very quickly even for over 10 equations.

In the non-linear case, I did one example with 8 equations and it worked relatively quickly (a few minutes). I guess 17 equations is too computationally intensive though for the non-linear case. Are there any alternative functions that might worth trying or is it hopeless to try to solve the system of equations?

[Kevin Bachovchin]

Also, this is unrelated, but I thought maybe you might be able to help. I'm running into an issue with using a symbolic expression in an equation. I have one function that creates a symbolic expression, which is often lengthy. In another function used in conjunction with the differential equation solver ode45, I am loading that symbolic expression and would like to evaluate that symbolic expression based upon parameters defined above.

For example, in my first function I solve for the symbolic expression SymExpr = a*b*c. In my second function, I have a, b, and c defined as numeric values. When I just copy paste in the expression for x, it takes less than a minute for ode45 to run. However when I load the saved symbolic expression SymExpr and use x=subs(SymExpr) or x=eval(SymExpr), it ran for over an hour without finishing.

I know I could do this by just copying and pasting in the second function x = a*b*c, but my expressions from the first function can get very lengthy (in fact, so long that it can exceed the maximum line length and get truncated when printed to the screen), so I'm hoping to do this without having to copy/paste. It seems like if I can copy and paste and get the result quickly, that there should be some automated way to do this.

Any help you could give me on this would be greatly appreciated.

Answer 2

[Ahmed A. Selman]

Sometimes trying the (simplify) command successively after each solve command, helps making the syms math incredibly faster. It helps when the result of a mathematical operation in a certain operation is to be used in a following one. For example, it might take long time to find D if B was a complicated expression in:

... 
B=solve(function_Of_x_y_z);
D=solve(function_of_B);

adding the simplify usually helps reducing the runtime as:

...
B=solve(function_Of_x_y_z);
B=simplify(B);
D=solve(function_of_B);

even multiple simplifies can be useful as:

...
B=solve(function_Of_x_y_z);
B=simplify(simplify(B));
D=solve(function_of_B);

Note that simplify command by itself takes some extra time to finish, yet it helps since syms math operations depends on the length of the expressions. So

u = x^2 + 4*x + 4*cos(2*x)^2 + 4*sin(2*x)^2; % This is a lengthy expression 
u = simplify(u);% Results, equivalently, u = (x + 2)^2.

[Kevin Bachovchin]

I'm only using one "solve" command. For example

Solution = solve(Equations==0, XSolveCell{:})

where Equations is a 17x1 vector and XSolveCell is a cell array consisting of the 17 unknowns I want to solve for.

[Ahmed A. Selman]

17 unknowns with a SINGLE equation? How is that mathematically possible.. even if they are in a linear form. Besides, suppose I have the equation:

y(x)= a*x+c,

it might be mathematically equivalent to, e.g.,

y(x1,x2)= a*x1 + b*x2 + c,

if we let A={a,b}, X={x1,x2}, then Y={y(x1,x2)} leading to

Y=A*X +b  .. etc,

But, the function y(x) can be solved when

y(x)=0 -- single variable (x), with a single condition

but not

y(x1,x2)=0 -- two variables, with a single condition.

Simply, it is impossible to solve for x1 or x2 only (as x1=a or x2=b/c.. etc). The only possibility is to have a solution like :

x1= -(c+b*x2)/a -- still a function of x2.