So, I'm trying to establish some new stability criteria for my simulations, and this involves a lot of convoluted inequalities. I've worked through the math a few times by hand, and it's very laborous; so, I wanted to figure out a way to automate the process (as I'm trying to find the best integration scheme from a stability perspective). Is there anyway to solve inequalities symbolically in Matlab? Here's what I'm trying to solve. In the following expression, x refers to the gradient of a force function with respect to x, and t is the time step. In general, x < 0 and t > 0:
-(t*x + (2*t^3*x + t^2*x^2 - 2*t*x + 4*t + 1)^(1/2) + 1)/(x*t^2 - 2) < 1
Based on what I've looked at online, this seems to be possible in MuPAD, but using the following code does not give me any valid results:
solve(-(t*x + (2*t^3*x + t^2*x^2 - 2*t*x + 4*t + 1)^(1/2) + 1)/(x*t^2 - 2) < 1, t)
Any idea what I can do to make this work and automate the process?
First, since Wolfram Alpha gives you an answer (that I presume you've checked for correctness), I assume you want to use Matlab to solve other similar problems. However, this is a very non-trivial inequality due to the roots of the polynomials. I haven't been able to get Matlab/MuPAD to do anything with it. As I stated, regular Matlab can solve inequalities and systems of equalities in many cases, e.g., in R2013b
syms x real;
solve(x^3-1>1,x)
Even Mathematica 9 has trouble (the Reduce function can be used instead of it's Solve, but the output is not easy to use).
You can, however, solve for the real roots where x < 0 and t > 0 via
syms x t real;
assume(x<0);
assume(t>0);
f = -(t*x+sqrt(2*t^3*x+t^2*x^2-2*t*x+4*t+1)+1)/(x*t^2-2);
s = solve(f==1,t)
which returns:
s =
-(x + (x*(x - 2))^(1/2))/x
This simplifies to sqrt((x-2)/x)-1. Thus t > sqrt((x-2)/x)-1, one of the bounds provided by Wolfram Alpha. (The other more complicated bound is always less than zero and actually is the condition that ensures that t is real.)
But, do you even need to be solving this problem symbolically? Do you need explicit expressions for the various intervals in terms of all x? If not, this type of problem is probably better suited numeric approaches – either via root solving (e.g., fzero) or minimization (e.g., fmincon).