現今科學發展，大型運算成為重要的研究課題。Algebraic Riccati equations 是從二次優化控制問題中而來。在這篇論文中，我們探討大型稀疏 algebraic Riccati equations 低秩逼近解與控制系統的關係，利用控制系統中的可控性及可觀測性，得到可以利用低秩解去逼近大型稀疏 algebriac Riccati equations；在求解 algebriac Riccati equations，我們利用牛頓法(Newton's method)進行迭代，一般而言牛頓法為二次收斂，但在每一次迭代時需要求解 Lyapunov equation，使得收斂速度大幅降低，在大型稀疏矩陣下求解 Lyapunov equation，有 Cholesky Factor Alternating Direction Implicit 迭代法可以保持低秩解結構，進而減少其運算量；更進一步利用兩個策略：Guess initial、Relaxed CFADI，減少內部迭代次數，加速整體牛頓法的迭代速度，最後以數值實驗觀察其迭代速度的變化。

In recent years, large-scale computing has become an important research topic. Algebraic Riccati equations is a control problem comes from the quadratic optimization. In this paper, we study the relationship between the large-scale sparse algebraic Riccati equations low-rank approximate solutions and the control systems, the control system controllability and observability that can be used to obtain low-rank approximate solution of large sparse algebriac Riccati equations; we use Newton's method solving the algebriac Riccati equations, the convergence rate of Newton's method is quadratic, but each iteration requires solving the Lyapunov equation, so that the convergence rate significantly lower, for solving the Lyapunov equations Cholesky Factor Alternating Direction Implicit iterative method can be kept low-rank structure of the solution, thereby reducing its computation; further use of two strategies: Guess initial value, Relaxed CFADI, reducing the total number of inner iteration to accelerate the convergence rate of Newton's method, and finally provide some numerical results.

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