做這篇論文的目的是想了解近十年來學者們對於解 Algebraic Riccati Equations 的數值方法有哪些。原因是由控制系統中的問題可以轉換成 Algebraic Riccati Equations 求解，因此才會衍生出整理" 解 Algebraic Riccati Equations 的方法" 這個念頭。
然而方法有很多種，不可能一一盡知。所以在此篇論文中只列出四個比較重要的方法: Newton's Method、 Schur Method、 Matrix Sign Function Method 和 SDA Method。
解 Algebraic Riccati Equations(不管是CARE還是DARE) 其實可以將其看成函數解其根，只是去解矩陣多項式的根，因此很直接的會想到用牛頓法解 Algebraic Riccati Equations。但是在矩陣中很難定義矩陣的微分(雖然有定義)，因此會用迭代的觀念導出求根的迭代式，而迭代也不一定要往同一個方向迭代，因此有了往不同方向迭代的方法 -- Line Search。若不從解根的這個方向看的話， Algebraic Riccati Equations 可以轉換成一個 Hamiltonian matrix，而解 Algebraic Riccati Equations 就相當於去求 Hamiltonian matrix 的不變子空間，因此列舉了 Schur Method 和 Matrix Sign Function Method 這兩種方法來說明。至於 SDA Method 是利用保持矩陣結構的不變性去找到三個遞迴式，而且其中一項的收斂值是 Algebraic Riccati Equations 的解，但是其收斂性要在某些條件下才會成立，這些都在 SDA Method 那一部份會提到。

In the last decades, a number of numerical methods for solving algebraic Riccati equations are proposed. According to that the kernel of the optimal control problems is used to be the stabilizing solutions of corresponding AREs, the thesis hence aims to survey several popular numerical methods for solving AREs, for
instance, Newton's method, matrix sign function method, Schur method and SDA method.

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