A Routh array, also called a Routh table, is a triangular array constructed from the coefficients of a real polynomial. The first two rows contain alternating coefficients, and each subsequent row is obtained from the two preceding rows by determinant-like combinations normalized by the first entry of the preceding row.
The Routh-Hurwitz theorem states that, when no exceptional zero occurs in the construction, the number of sign changes in the first column equals the number of roots in the right half-plane. A polynomial therefore has all roots in the left half-plane precisely when the first-column entries are nonzero and have the same sign.
