MATLAB: Symbolic expression bounds

formal verificationsymbolicSymbolic Math Toolboxuser input

I have a symbolic expression (the real expression can be anything the Symbolic Toolbox supports) but for example we can use the following:

expr = sym('a + b')

Also, I know that a and b are integers and that their "legal" values go from 1 to 7.

I would like to figure out what are the bounds of expr. In this case, the min value would be 2 and the max value would be 14.

Is there any way to do this type of analysis?

Thanks Joan

Best Answer

Unfortunately there is no theoretical way to do that for arbitrary expressions.

For the special case of linear operations, you can use the linear minimizer or maximizer: see http://www.mathworks.com/help/toolbox/mupad/linopt/index.html

More generally for expressions in one variable, you would differentiate the expression, find the zeros of that, differentiate the expression again, evaluate at the zeros to determine whether you are looking at a min, max, or saddle point. Solving for the zeros might be difficult or impossible though...

For expressions in multiple variables, the question of the range becomes equivalent to the question of finding the global extrema. It is known that there is no general algorithm for that, but there are some classes of expressions that it can be solved for.

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MATLAB: An empty sum result when using function solve

[x1,y1] = solve(dot(a,b)-1,dot(a,c)-2)

When you pass a string into solve(), the string is interpreted strictly in terms of MuPAD calls. MuPAD does not have a function named dot(). In MuPAD, the dot product routine is named linalg::scalarProduct()

Also keep in mind that when you pass a string to solve() and the string refers to a variable that has been defined at the MATLAB level, then unless you somehow transfer the variable to MuPAD then MuPAD is not going to know about the value of the variable.

Basically the only time you should pass strings to solve() is as variable names or when you know you are writing a MuPAD expression that the interface to the Symbolic Toolbox does not provide an easy way to write. For example in MATLAB up to about R2011b, MATLAB did not recognize the "==" operator on symbolic expressions in any useful way, so it was necessary to pass in strings to solve() that used the MuPAD equivalent, the "=" operator

solve('x^2+y^2=9', 'x=2*y+5')     %notice MuPAD syntax has been used!
solve('zeta(cos(x), 5) = besselJ(3,sin(x))')   %uses MuPAD functions!

but otherwise do not pass in strings.

MATLAB: Using relational operators with the symbolic math toolbox

Yes, there is a MuPad call named "is" that will test a condition. It is not strictly constrained to numeric relations: it can test properties such as data types, and when assumptions have been applied, it may be able to decide relationships between expressions that involve symbolic variables. Most of the time, though, expressions that involve symbolic variables without assumptions are not decidable, and is() will return false for those cases.

Coding the relational expression at the Matlab level can be a problem, so you usually end up having to express it as a string and using subs() to transfer the values from the Matlab engine to the symbolic engine.

Best Answer

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