Numerical test of stability of large sparse matrices and dynamical systems

The stability of an n-square complex matrix A is determined by the number â = max {Re (a1)}, where the a1 are the eigenvalues. A practical numerical technique is developed for computing the number â of any normal matrix (AA* = A*A). When applied to an arbitrary matrix, the technique yields a number not less than â, and hence, if the number is negative, the matrix is stable. The technique is particularly efficient when A is a large sparse matrix.