# lyap

Solves continuous Lyapunov or Sylvester equations.

## Syntax

X = lyap(A, B) % Lyapunov Equation

X = lyap(A, B, C) % Sylvester Equation

X = lyap(A, B, [], E) % Generalized Lyapunov Equation

## Inputs

A
Real square matrix.
B
Real matrix.
C
Real matrix.
E
Real matrix.

## Outputs

X
Returns the solution to the continuous Lyapunov Equation. X is a real matrix.

## Examples

Solve a Lyapunov equation:

A = [10  2;
-3 -40];

B = [3  10;
10  1];

X1 = lyap(A, B)
X1 = [Matrix] 2 x 2
-0.22090   0.35448
0.35448  -0.01409
Solve a Sylvester equation:

A = [5];
B = [40 3;
4 30];
C = [2 1];

X2 = lyap(A, B, C)
X2 = [Matrix] 1 x 2
-0.04223  -0.02495
Solve a Generalized Continuous Lyapunov equation:

A = [30  1  1;
1  30  0;
1  0  20];

E = [1  3  10;
3  20  0;
0  1  1];

B = [6.40   73.0   28.0;
73.0   7.0   25.0;
28.0   25.0   1.8];

X3 = lyap (A, B, [], E)
X3 = [Matrix] 3 x 3
4.24365  -0.72106  -0.21641
-0.72106   0.10514  -0.02917
-0.21641  -0.02917   0.03104

## Comments

X = lyap(A, B) solves the continuous Lyapunov equation AX + XA' = -B.

X = lyap(A, B, C) solves the Sylvester equation AX + XB = -C.

X = lyap(A, B, [], E) solves the generalized continuous Lyapunov equation AXE' + EXA' = - B.

Based on the SLICOT library functions SB03MD, SB04MD, and SG03AD.