Solve discrete-time Lyapunov equations
X = dlyap(A,Q)
X = dlyap(A,B,C)
X = dlyap(A,Q,,E)
X = dlyap(A,Q) solves the discrete-time
Lyapunov equation AXAT − X + Q =
where A and Q are n-by-n matrices.
The solution X is symmetric when Q is symmetric, and positive definite when Q is positive definite and A has all its eigenvalues inside the unit disk.
X = dlyap(A,B,C) solves the Sylvester equation AXB – X + C =
where A, B, and C must have compatible dimensions but need not be square.
X = dlyap(A,Q,,E) solves the generalized
discrete-time Lyapunov equation AXAT – EXET + Q =
where Q is a symmetric matrix. The empty
, are mandatory. If you place
any values inside them, the function will error out.
The discrete-time Lyapunov equation has a (unique) solution if the eigenvalues α1, α2, …, αN of A satisfy αiαj ≠ 1 for all (i, j).
If this condition is violated,
the error message
Solution does not exist or is not unique.
dlyap uses SLICOT routines SB03MD and SG03AD
for Lyapunov equations and SB04QD (SLICOT) for Sylvester equations.
 Barraud, A.Y., “A numerical algorithm to solve A XA - X = Q,” IEEE® Trans. Auto. Contr., AC-22, pp. 883-885, 1977.
 Bartels, R.H. and G.W. Stewart, "Solution of the Matrix Equation AX + XB = C," Comm. of the ACM, Vol. 15, No. 9, 1972.
 Hammarling, S.J., “Numerical solution of the stable, non-negative definite Lyapunov equation,” IMA J. Num. Anal., Vol. 2, pp. 303-325, 1982.
 Higham, N.J., ”FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation,” A.C.M. Trans. Math. Soft., Vol. 14, No. 4, pp. 381-396, 1988.
 Penzl, T., ”Numerical solution of generalized Lyapunov equations,” Advances in Comp. Math., Vol. 8, pp. 33-48, 1998.
 Golub, G.H., Nash, S. and Van Loan, C.F. “A Hessenberg-Schur method for the problem AX + XB = C,” IEEE Trans. Auto. Contr., AC-24, pp. 909-913, 1979.
 Sima, V. C, “Algorithms for Linear-quadratic Optimization,” Marcel Dekker, Inc., New York, 1996.