The method to solve linear system
by 小浣熊
Huang-wen Hsiao
-------------------------------------------------------------------------
Introduction:
The first four methods are the methods we usually used. A series
of theorems show that the Crout reduction has the better performance.
The last five methods are in the recursive form. Therefore, we can
implement these algorithm more easily. However, we must have the attention
about the condition number of the matrix A which will do much influence
to the linear system.
1. Direct method
Ax = b => x = A^(-1) b
2. Gauss Elimination
3. Gauss-Jordan method
4. Crout Reduction ( Cholesky Reduction )
A = L * U
b = Ax = ( L * U )x = L * Ux
=> Ux = L^(-1) b
=> x = U^(-1) L^(-1) b
5. Error recursion
let e = x - x' , r = b - Ax'
Ae = b - Ax' = r
x_(n+1) = x_n + e_n
6. Jacobi method
A = L + D + U
b = Ax = ( L + D + U )x = Dx + ( L + U )x
=> x = D^(-1) b - D^(-1) ( L + U )x
x_(n+1) = D^(-1) b - D^(-1) ( L + U )x_n
7. Gauss seidel method
To be continue.....
8. Relaxation method
To be continue....
9. Overrelaxation method
To be continue....
Reference:
http://www.esc.auckland.ac.nz/Academic/DES/675.331/course_note/ch2.pdf
Advance Reference:
http://www.netlab.org/linalg/html_templates/node20.html
--
※ 發信站: 批踢踢實業坊(ptt.csie.ntu.edu.tw)
◆ From: linux3
※ 編輯: astronomer 來自: linux3 (12/28 22:27)