本篇主要在探討不完全LU預條件化應用在兩個迭代方法GMRES方法和DQGMRES方法上,目的是用來解決
大型稀疏線性系統Ax=b, 其中A是n維的非奇異方陣, b是n維的行向量, x是要求解的未知n維行向量, 我們利用布林矩陣產生pattern來決定如何進行ILU分解,
最後我們選擇不同問題來做數值實驗來驗證這些方法, 並比較不同方法的迭代數和CPU時間.

In this dissertation presents a method
that applying ILU preconditioner to the well-known
iterative methods, GMRES and DQGMRES for solving a large linear system Ax = b,where $A$ is a sparse nonsigular square matrix of order $n imes n$,the given $b$ and the unknown $x$ are $n$-dimensional column vectors.
The forementioned approach is to generate a pattern from Boolean
matrices in practical. We compare the iteration number and
CPU time on several problems for various numerical methods
in order to verify our methods.

1.Introduction 1
2.ILU decomposition 2
2.1 ILU decomposition with M-matrix 2
2.2 A matrix pattern vs. Boolean matrices 10
3.Storagement of sparse matrices 13
4.GMRES and DQGMRES method 16
4.1 GMRES method 16
4.2 DQGMRES method 21
5.Right and left preconditioner 25
6.Numerical experiments 28
7.Conclusions 29
8.Reference 30

J.-Y. Chen, D. R. Kincaid and D. M. Young: Generalizations and modifications of the GMRES iterative mothod.
Num. Algorithms 21(1999), 119-146.

G. H. Golub and C. F. Van Loan: Matrix Computations,
it The Johns Hopkins University Press.(1996).

P. Lancaster and M. Tismenetsky: The Theory of Matrioes Seoond Edition with Applictions,
Academic Press.(1985).

R. C. Mittal and A. H. Al-Kurdi: LU-decomposition and numerical structure for solving large sparse nonsymmetric
linear systems,
Computers Math. Applic. 43 (1/2)(2001), 131-155.

R. C. Mittal and A. H. Al-Kurdi: An efficient method for constructing an ILU preconditioner for solving large
sparse nonsymmetric linear systems by the GMRES method,
Computers Math. Applic. 45(2003), 1757-1772.

Y. Saad and M. H. Schultz, GMRES: A generalized residual algorithm for solving nonsymmetric linear systems,
SIAM J. Sci. Stat. Comp. 7(1986), 856-869.

Y. Saad and K. Wu. DQGMRES: A direct quasi-minimal residual algorithm based on incomplete orthogonalization,
Numerical Linear Algebra with Applications. 3 (1996), 329-343.

Y. Saad: Iterative Methods for Sparse Linear Systems,
Yousef Saad (2000).