Deflating subspaces of T-palindromic pencils and algebraic T-Riccati equations

By exploiting the connection between solving algebraic ⊤-Riccati equations and computing certain deflating subspaces of ⊤-palindromic matrix pencils, we obtain theoretical and computational results on both problems. Theoretically, we introduce conditions to avoid the presence of modulus-one eigenvalues in a ⊤-palindromic matrix pencil and conditions for the existence of solutions of a ⊤-Riccati equation. Computationally, we improve the palindromic QZ algorithm with a new ordering procedure and introduce new algorithms for computing deflating subspaces of the ⊤-palindromic pencil, based on quadraticizations of the pencil or on an integral representation of the projector on the sought deflating subspace.