This is machine translation

Translated by Microsoft
Mouseover text to see original. Click the button below to return to the English version of the page.

Note: This page has been translated by MathWorks. Click here to see
To view all translated materials including this page, select Country from the country navigator on the bottom of this page.


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 AXATX + Q = 0,

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 AXBX + C = 0,

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 AXATEXET + Q = 0,

where Q is a symmetric matrix. The empty square brackets, [], 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, dlyap produces 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.


[1] Barraud, A.Y., “A numerical algorithm to solve A XA - X = Q,” IEEE® Trans. Auto. Contr., AC-22, pp. 883-885, 1977.

[2] Bartels, R.H. and G.W. Stewart, "Solution of the Matrix Equation AX + XB = C," Comm. of the ACM, Vol. 15, No. 9, 1972.

[3] Hammarling, S.J., “Numerical solution of the stable, non-negative definite Lyapunov equation,” IMA J. Num. Anal., Vol. 2, pp. 303-325, 1982.

[4] 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.

[5] Penzl, T., ”Numerical solution of generalized Lyapunov equations,” Advances in Comp. Math., Vol. 8, pp. 33-48, 1998.

[6] 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.

[7] Sima, V. C, “Algorithms for Linear-quadratic Optimization,” Marcel Dekker, Inc., New York, 1996.

See Also


Introduced before R2006a