迭代法是一種重複改善線性系統 Ax=b 的近似解，最後會逐漸趨近真解的方法。在本篇論文中將回顧一些迭代法包括 IGCG、CGS、BICGSTAB等方法並探討是否會發生中斷的現象。再從中挑選幾種類型的 Krylov 子空間法來求解一些利用有限差分法離散化偏微分方程式所化為的大型線性稀疏系統，並比較其迭代的時間與收斂的速度。

Each iterative method uses successive procedure to produce a more accurate solution to a linear system at each iteration. In this paper, we review some famous iterative methods such as IGCG, CGS, and BICGSTAB and discuss at what situation the breakdown will occur. We also choose two partial differential equations with boundary value conditions as our test problems and discretisize them to linear systems by the central difference. Furthermore, we choose some types of Krylov subspace methods to solve these test problems, and compare the computing time and the rate of convergence of those Krylov subspace methods.

Chapter 1 Introduction 1
1-1 The Background 1
1-2 The Purpose 3
Chapter 2 Constructing Krylov Subspace methods 4
2-1 The Basic Properties of the Krylov Subspace
and Auxiliary Matrix 4
2-2 Stopping Criteria 9
Chapter 3 Krylov Subspace Methods 15
CG METHOD 15
CGNE AND CGNR METHODS\hfill 16
GMRES METHOD 16
IGCG METHODS 19
BICG METHOD 20
CGS METHOD 21
BICGSTAB METHOD 23
Chapter 4 Numerical Experiments 27
Chapter 5 Conclusion 32
Reference 33

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