# dlyap

Solve discrete-time Lyapunov equations

## Syntax

`X = dlyap(A,Q)X = dlyap(A,B,C)X = dlyap(A,Q,[],E)`

## Description

`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.

## Diagnostics

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. ```

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### Algorithms

`dlyap` uses SLICOT routines SB03MD and SG03AD for Lyapunov equations and SB04QD (SLICOT) for Sylvester equations.

## References

[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.