Routh-Hurwitz stability calculator
This online Routh-Hurwitz calculator builds the Routh array for a characteristic polynomial and reports whether the system is stable, without finding the roots. Enter the polynomial as the denominator above.
How the Routh array works
The number of sign changes in the first column of the Routh array equals the number of roots in the right half-plane. Zero — with no row of zeros — means all roots are in the left half-plane and the system is stable. Special cases (a zero in the first column, or an entire row of zeros) are handled automatically.
Frequently asked questions
- What is the Routh-Hurwitz criterion?
- The Routh-Hurwitz criterion tells you how many roots of a polynomial lie in the right half-plane — and therefore whether a system is stable — without solving for the roots. It builds a table, the Routh array, from the polynomial's coefficients.
- How do you build a Routh array?
- Write the coefficients in two rows, then compute each following row from the two rows above it using the Routh formula. The number of sign changes in the first column equals the number of unstable, right-half-plane roots.
- What do a zero in the first column or a row of zeros mean?
- A single zero in the first column is handled with the ε (epsilon) method; a whole row of zeros signals symmetrically placed roots and is resolved using the derivative of the auxiliary polynomial. This calculator handles both special cases automatically.
- Is this Routh-Hurwitz calculator free?
- Yes, it is free and runs entirely in your browser.