## 2 Matrix Sign Function-Based Solvers
This growth can be controlled by computing a RRQR (rank-revealing QR) factorization, which introduces a relatively low overhead. This approach reports important gains when the number of iterations needed for convergence is large, or when the number of columns of \(B\) is large. The RRQR decomposition can be obtained by means of the traditional QR factorization with column pivoting [11], plus a reliable rank estimator. Note that \(Q_{s}\) is not accumulated, as it is not needed in subsequent computations. This reduces the cost of computing the RRQR significantly. ### Solution of Algebraic Riccati Equations

The sign function method can also be employed to compute the stabilizing solution of an algebraic Riccati equation (ARE) of the form:

\[F^{T}X+XF-XGX+Q\ =\ 0 \tag{2}\]

where \(F\), \(G\), \(Q\in\mathbb{R}^{n\times n}\) and the solution, \(X\in\mathbb{R}^{n\times n}\), is symmetric and satisfies that \(F-GX\) is c-stable. In particular, the solution to the ARE can be obtained from the c-stable invariant subspace of the Hamiltonian matrix [12]:

\[H=\left[ \begin{array}{cc}F&-G\\ -Q&-F^{T}\end{array} \right] \tag{3}\]

which can be extracted by first obtaining the matrix sign function of \(H\):

\[\text{sign}(H)=Y=\left[ \begin{array}{cc}Y_{11}&Y_{12}\\ Y_{21}&Y_{22}\end{array} \right] \tag{4}\]

and then solving the over-determined system:

\[\left[ \begin{array}{c}Y_{11}\\ Y_{12}+I_{n}\end{array} \right]X=\left[ \begin{array}{c}I_{n}-Y_{21}\\ -Y_{11}\end{array} \right] \tag{5}\]

Algorithm 3--GERSG below summarizes the above steps to solve the ARE in Equation (2) with this method. ```
\(H_{0}\leftarrow\left[ \begin{array}{cc}F&-G\\ -Q&-F^{T}\end{array} \right]\)  Apply Algorithm 1--GESINE to compute \(Y=\left[ \begin{array}{cc}Y_{11}&Y_{12}\\ Y_{21}&Y_{22}\end{array} \right]\leftarrow\text{sign}(H_{0})\)  Solve \(\left[ \begin{array}{c}Y_{11}\\ Y_{12}+I_{n}\end{array} \right]X=\left[ \begin{array}{c}I_{n}-Y_{21}\\ -Y_{11}\end{array} \right]\) for \(X\)
```
**Algorithm 3**GERGSG**:

### Solution of Differential Riccati Equations

The matrix sign function can also be applied in combination with the Rosenbrock method for the solution of an autonomous symmetric differential Riccati equation (DRE) [13] of the form:

\[ \begin{array}{rcl}\dot{X}(t)&=&Q(t)+X(t)A(t)+A(t)^{T}X(t)-X(t)S(t)X(t)\equiv F (t,X(t))\\ X(t_{0})&=&X_{0}\end{array}  \tag{6}\]where \(t\in[t_{0},t_{f}]\), \(A(t)\in\mathbb{R}^{n\times n}\), \(Q(t)=Q(t)^{T}\in\mathbb{R}^{n\times n}\), \(S(t)=S(t)^{T}\in\mathbb{R}^{n\times n}\) and \(X(t)=X(t)^{T}\in\mathbb{R}^{n\times n}\). Here, we assume that the coefficient matrices are piecewise continuous locally bounded matrix-valued functions, which ensures the existence and uniqueness of the solution to Equation (6); see, e.g., Theorem 4.1.6 in [1]. The application of the Rosenbrock method of order one to an autonomous symmetric DRE of the form Equation (6) yields:

\[\tilde{A}_{k}^{T}X_{k+1}+X_{k+1}\tilde{A}_{k}=-Q-X_{k}SX_{k}-\frac{1}{h}X_{k} \tag{7}\]

where \(X_{k}\approx X(t_{k})\) and \(\tilde{A}_{k}=A-SX_{k}-\frac{1}{2h}I\); see [14,15] for details. In addition, we assume: \(Q=C^{T}C\), \(C\in\mathbb{R}^{p\times n}\), \(S=BB^{T}\), \(B\in\mathbb{R}^{n\times m}\), \(X_{k}=Z_{k}Z_{k}^{T}\), \(Z_{k}\in\mathbb{R}^{n\times z_{k}}\), with \(p\), \(m\), \(z_{k}\ll n\). If we denote \(N_{k}=[\,C^{T}\ Z_{k}(Z_{k}^{T}B)\ \sqrt{h^{-1}Z_{k}}\,]\), then the Lyapunov Equation (7) results in:

\[\tilde{A}_{k}^{T}X_{k+1}+X_{k+1}\tilde{A}_{k}=-N_{k}N_{k}^{T} \tag{8}\]

where \(\tilde{A}_{k}=A-B(Z_{k}(Z_{k}^{T}B))^{T}-\frac{1}{2h}I\). The procedure that is obtained from this elaboration is presented in Algorithm 4--ROS1. Observing that \(\text{rank}(N_{k})\leq p+m+z_{k}\ll n\), we can use the sign function method to solve Equation (8), as illustrated in Algorithm 2--GECINC (see Section 2.1).