Page 19 - Textos de Matemática Vol. 36

1.2. Boundary value problems 9
and the Riccati difference equation
W (k + 1) = −M21 − M22W (k)(I − M12W (k))−1M11, W (0) = W0. (DRDE)
Notice that DARE and DRDE can also be written as
W = −M21 − M22(I − W M12)−1W M11, (DARE)
W (k + 1) = −M21 − M22(I − W M12(k))−1W (k)M11, W (0) = W0. (DRDE)
Sometimes one considers instead of DRDE the reverse Riccati difference equa- tion
W (k) = −M21 − M22W (k + 1)(I − M12W (k + 1))−1M11, (RDRDE)
with W(0) = W0 for −k ∈ N. Non-symmetric Riccati equations of this form appear in polynomial factorization problems and have been investigated in [ClAn76], where the corresponding continuous-time Riccati equations have also been studied.