Transforming algebraic Riccati equations into unilateral quadratic matrix equations

The problem of reducing an algebraic Riccati equation XCX − AX − XD + B = 0 to a unilateral quadratic matrix equation (UQME) of the kind PX ² + QX + R = 0 is analyzed. New transformations are introduced which enable one to prove some theoretical and computational properties. In particular we show that the structure preserving doubling algorithm (SDA) of Anderson (Int J Control 28(2):295–306, 1978) is in fact the cyclic reduction algorithm of Hockney (J Assoc Comput Mach 12:95–113, 1965) and Buzbee et al. (SIAM J Numer Anal 7:627–656, 1970), applied to a suitable UQME. A new algorithm obtained by complementing our transformations with the shrink-and-shift technique of Ramaswami is presented. The new algorithm is accurate and much faster than SDA when applied to some examples concerning fluid queue models.