Symmetric Matrix Polynomial Equation: Interpolation Results

An interpolation approach is pursued to solve a bilateral matrix polynomial equation frequently arising in control and signal processing. It results in eecient and numerically reliable resolution methods. Abstract New numerical procedures are proposed to solve the symmetric matrix polynomial equation A T (?s)X(s) + X T (?s)A(s) = 2B(s) that is frequently encountered in control and signal processing. An interpolation approach is presented that takes fully advantage of symmetry properties and leads to an equivalent reduced-size linear system of equations. It results in a simple and general characterization of all solutions of expected column degrees. Several new theoretical results concerning stability theory and reduced Sylvester resultant matrices are also developed and used to conclude a priori on the existence of a solution. By means of numerical experiments, it is shown that our algorithms are more eecient than older methods and, namely, appear to be numerically reliable.