Find the values of parameters so that the matrix is symmetric positive definite

Good evening , I have a matrix 20x20 in symbolic form. The matrix depends on 5 parameters (a,b,c,d,e). I would like to get the interval of each parameters that ensures that the matrix S is symmetric positive definite.

Should I first compute the symbolic expression of the eigenvalues and then solve a system of 20 inequalities to find the interval of the 5 parameters?

Or can I do it directly with some Mathematica command?

Thank you.

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As it is written like now it is more of an algorithmic rather than Mathematica related question. You can add more details, code you are working on or just migrate it to other SE site. –  Sektor Apr 20 '14 at 1:27
Reading en.wikipedia.org/wiki/… it seems to suffice that all 20 principal minors have determinants that are simultaneously positive for some combination of your 5 parameters. Simple enough: Reduce. Might take a few centuries, but the task can be considered solved in principle. But maybe your matrix has some 'nice' form? –  Wouter Apr 20 '14 at 16:37
Symmetric is easy; definite is also easy via the determinant. –  chris Apr 21 '14 at 13:55