當電腦不停進步，過去許多無法計算的難題都有了解答的機會，計算線性系統的解一直都是數學家與電腦學家的目標。從一九五零年代左右發表了許許多多的論文在討論這個問題。當所要解的線性系統越來越大，所需要的電腦效能也越來越大。以至於求出大型線性系統的解成為一個困難的問題。到了現在，這個問題也漸漸露出曙光。
　　在本論文中，將探討幾個專門用來求得線性系統解的演算法，以及它們提出的背景與想法，並且加以實作。

As computers keeping advancing, many difficult problems which were unable to compute formerly now have the chance to get answered. It is always the goal of mathematicians and computer scientists to compute and get the answers of the linear systems. Since 1950s, there have been a lot of published papers discussing the issue. As the linear systems larger and larger, the computer efficiency required is higher and higher, so that it is very difficult to get the answers of large linear systems. Now, the problems are showing aurora.
In this dissertation, several mathematical calculations to compute the linear systems will be discussed, as well as their background and theory. Moreover, they will also be practiced.

Chapter 1 序論 ……………………………………………… 5
1.1 研究動機 ……………………………………………… 5
1.2 研究目標 ……………………………………………… 8
Chapter 2 各種演算法的簡介 ……………………………… 10
Chapter 3各類演算法及pseudocode ……………………… 12
3.1 Conjugate Gradient method ……………………… 12
3.2 Bi-Conjugate Gradient method …………………… 14
3.3 Squaring the Conjugate Gradient method ……… 16
Chapter 4 實作的過程 ……………………………………… 18
4.1 CGS algorithm …………………………………………19
4.2 驗證解的正確性 ……………………………………… 19
Chapter 5 實作數據 ………………………………………… 21
5.1 在實數系上的實驗 …………………………………… 22
5.2 在GF(2)上的實驗 ………………………………………23
Chapter 6 結論與討論 ……………………………………… 28
參考文獻 ……………………………………………………… 30

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[2].R.Fletcher. Conjugate Gradient Methods for Indefinite Systems, Lecture Notes in Mathematics 506, Springer-Verlag, Berlin, Heidelberg, New York, 1976, pp73-89
[3].P. Sonneveld, CGS, a fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 10(1989), pp 36-52.
[4].－,Solution of systems of linear equations by minimized iterations, J. Res. Nat. Bur. Stand., 49(1952), pp. 33-53.
[5].C.Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Nat. Bur. Stand., 45(1950), pp. 255-282
[6] Positive Definite Matrix, mathworld,
http://mathworld.wolfram.com/PositiveDefiniteMatrix.html