中文摘要
TORT (三維 Oak Ridge discrete ordinates 中子/光子遷移程式)是一個用於核反應器和核廢料儲存時的輻射屏蔽計算程式。TORT 程式主要使用不連續的縱座標法(discrete ordinates method)也稱做Sn 方法， 把遷移方程轉換成一個大型的線性系統。 典型的數值方法是採用SOR (Successive Over-Relaxation)解線性系統，而TORT 程式也同樣採用這種數值法解其中的線性系統。近來，已有其他的數值方法成功地應用在安裝於32位元個人電腦的TORT程式中，諸如：預先條件化的共扼梯度平方法 (conjugate gradient squared, CGS)和預先條件化的雙共扼梯度穩定法(Biconjugate Gradient Stabilized, BICGSTAB)，其中並搭配兩種不同的預先條件，分別是ILU(0) (point-wise incomplete LU factorization) 和MILU (modified point-wise incomplete LU factorization)兩類。 在我們的這個研究中，我們分別建立兩個採用預先條件化的TFQMR (transpose-free quasi-minimal residual) 和QMRCGSTAB (Quasi-minimal Residual Biconjugate Gradient Stabilized) 副程式去連接TORT程式。 在早先的研究中，上述兩個方法在32位元的個人電腦中計算時有難以收斂的現象。由於64位元的中央處理器(central processing unit , CPU)較32位元的處理器有較小的計算進位誤差(round off error)，因此引發我們把數值問題安裝於64位元的個人電腦上做測試。我們安裝TORT程式於 AMD ATHLON 64x2 雙核心處理器的個人電腦，使用的操作介面是Mandriva Linux, 10.0版。 我們發現預先條件化的TFQMR和QMRCGSTAB 搭配使用ILU(0) 預先條件法的收斂情形是較搭配使用MILU預先條件法來得好。並且，預先條件化的TFQMR和QMRCGSTAB 搭配使用ILU(0) 預先條件法於64位元的個人電腦上的收斂情形較在32位元的個人電腦上來得快。除此之外，無論是安裝於64或32位元的個人電腦上，其數值計算結果也與由 SOR， CGS 或BICGSTAB 連結TORT 程式計算結果相同。

Abstract
TORT (Three Dimensional Oak Ridge Discrete Ordinates Neutron/photon Transport Code) is a program for the calculation of the radiation shielding in nuclear power plant and nuclear spent fuel storage. The simulation of TORT relies on the discrete ordinates method known as the Sn method which shifts transport equation to a large linear system. The Numerical method typically used to solve linear systems is SOR (Successive Over-Relaxation) and it had been applied to TORT. Recently, other methods are proposed successfully for TORT, such as the preconditioned CGS (conjugate gradient squared) and the preconditioned BICGSTAB (Biconjugate Gradient Stabilized) with the point-wise incomplete LU factorization (ILU(0)) and modified point-wise incomplete LU factorization (MILU) preconditioners to be used in 32-bit personal computer. In this study, we developed subroutines based on the preconditioned TFQMR (transpose-free quasi-minimal residual) and QMRCGSTAB (Quasi-minimal Residual Biconjugate Gradient Stabilized) to be connected to TORT. In previous studies, the two numerical methods are difficult to converge in 32-bit personal computer. This leads to testing the numerical methods in 64-bit central processing unit (CPU) since more bits in CPU always yields less round off error. Here we test TORT program in AMD ATHLON 64x2 dual-core processor with operational system Mandriva Linux, the 2007 edition. We find that the convergence behaviors of the preconditioned TFQMR and QMRCGSTAB methods with preconditioner ILU(0) are more efficient than those with the preconditioner MILU. Also, the convergences of the preconditioned TFQMR and the preconditioned QMRCGSTAB in the 64-bit personal computer are faster than those in 32-bit personal computer. Moreover, the numerical results are the same as the results obtained from using SOR, CGS or BICGSTAB in TORT with both 32-bit and 64-bit personal computers.

References
[1] Rhoades, W. A. and Simpson, D. B. (1995) TORT version 3. Three‑Dimensional Discrete Ordinates Neutron /Photon Transport Code System, Oak Ridge National Laboratory Report.
[2] James J. Duderstadt, William R. Martin, Transport theory, New York: Wiley, 1979, p. 225~236, p. 420~458.
[3] Saad, Y. Iterative Method for Sparse Linear Systems, 2nd edition, SIAM, Philadelphia, 2003, p. 144~168, p. 205~226, p. 245~287.
[4] Pobert D. Richtmyer, K. W. orton, Difference methods for initial-value problems 2nd ed., John Wiley & Sons. Inc., 1967.
[5] James J. Duderstadt, Louis J. Hamilton, Nuclear Reactor analysis, John Wiley & Sons, Inc.,1976, p. 103~122, p. 179~195.
[6] G. Arfken and H. Weber, Mathematical Methods for Physicists, 5th Edition, Academic Press., 2001.
[7] Chen G. S. and R. D. SHEU, Application of two preconditioned generalized conjugate methods to three-dimensional neutron and photon transport equations, Progress in Nuclear Energy. Vol.45, No.1, pp.11-23, 2004.
[8] Richard L. Burden, J. Douglas Faires, Numerical analysis , 7nd edition, Thomson, 2003, p. 241~284, p. 325~334, p. 406~411.
[9] Roland W. Freund, SIAM J.SCI COMPUT, Vol. 14, No. 2, pp.470-482, March 1993.
[10] T. F. Chan, E. Gallopoulos, V. Simoncini, T. Szeto, and C. H. Tong, SIAM J.SCI COMPUT, Vol. 15, No. 2, pp.338-347, March 1994.