NCX鈉鈣離子交換通道，是迅速將細胞內的鈣離子與外部鈉離子交換的重要機制。本研究為優化劉晉良[7]利用Poisson-Fermi方程探討NCX離子通道的3DPNPF C++數值程式。
當我們處理稀疏矩陣時，通常都會進行壓縮，本研究考慮兩種壓縮格式：對角化壓縮及座標化壓縮，比較這兩種格式的計算速度。
再來，我們通常解線性系統的數值方法為的是Successive OverRelaxation
(SOR) 演算法[4]，我們也測試了Jacobi Davidson (JD) 演算法[3]以及BiConjugate Gradient (BiCG) 演算法[4]，在兩種壓縮格式下比較計算速度。
藉由本次比較，我們可以知道這三種演算法在這兩種壓縮格式下的計算速度，嘗試優化3DPNPF C++程式，達到節省時間，優化效能之外，也能作為未來平行化的參考。

The sodium-calcium ion exchange (NCX) channel has an important mechanism that can changes the calcium ions inside the cell with sodium ions outside rapidly. Liu [7] has developed a C++ program (named 3DPNPF) based on the proposed Poisson-Fermi model to investigate the NCX channel. The all resulting linear systems have solved by Successive OverRelaxation (SOR) algorithm. This thesis aims at test the performances of several linear system solvers for 3DPNPF, including SOR, Jacobi-Davidson (JD) and BiConjugate Gradient (BiCG) methods. We also consider two compression formats, diagonal structured and coordinate format, for the large and sparse matrices, while the 3DPNPF only consider the former one. The numerical results reveal that the SOR has better performance for most cases. However, the computational times of JD or BiCG methods are just 2-4 times slower than SOR. This shows that these two methods can be parallelized.

1. 緒論 1
2. 壓縮矩陣儲存方式 2
2.1 對角化壓縮 2
2.2 座標化壓縮 2
2.3 範例介紹 3
3. 數值方法 4
3.1 Jacobi-Davidson 演算法 4
3.2 Successive-OverRelaxation 演算法 6
3.3 BiConjugate Gradient 演算法 8
4. 數值成果 11
4.1 NCX通道及Poisson Fermi方程 11
4.2 應用 12
5. 參考文獻 16
附錄：程式碼 17
I. 對角化壓縮 17
II. 座標化壓縮 34

[1] J. Liao, H. Li, W. Zeng, D. B. Sauer, R. Belmares, Y. Jiang, Structural insight into the ion-exchange mechanism of the sodium/calcium exchanger, Science 335, 686 (2012).
[2] Yousef Saad. (2000). Iterative methods for sparse linear systems (3rd).
[3] J.-H. Chen and J. -L. Liu, Jacobi-Davidson method for solving linear system.
[4] R. Barrett, M. Berry, T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. Van der Vorst. (1994). Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods(2nd edition). SIAM, Philadelphia .
[5] J.-L. Liu, Numerical methods for the Poisson-Fermi equation in electrolytes, J. Comp. Phys. 247, 88-99 (2013).
[6] J.-L. Liu and B. Eisenberg, Poisson-Nernst-Planck-Fermi theory for modeling biological ion channels, J. Chem. Phys. 141, 22D532 (2014).
[7] J.-L. Liu and B. Eisenberg, Numerical methods for a Poisson-Nernst- Planck-Fermi model of biological ion channels, Phys. Rev. E 92, 012711 (2015).