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1.4.Nhân tử hóa ma trận.pptx
1.
4. Nhân tử
hóa ma trận Một phép nhân tử hóa ma trận A là một phương trình biểu diễn A như một tích của hai hay nhiều ma trận © 2012 Pearson Education, Ltd. Slide 2.2- 1
2.
Nhân tử hóa
LU Giả sử A là ma trận cấp mxn mà có thể rút gọn hàng về dạng bậc thang mà không được phép đổi chỗ các hàng cho nhau. Khi đó A có thể viết được dưới dạng A = L.U, với L là ma trận tam giác dưới cấp mxm với các phần tử chéo là 1 và U là một dạng bậc thang của ma trận A có cấp mxn. Phép phân tích như vậy được gọi là phép nhân tử hóa LU của ma trận A. Ma trận L là khả nghịch và được gọi là ma trận tam giác dưới đơn vị. © 2012 Pearson Education, Ltd. Slide 2.2- 2
3.
Nhân tử hóa
LU Dạng nhân tử hóa LU của ma trận A © 2016 Pearson Education, Ltd. Slide 2.5- 3 2 4 1 5 2 4 5 3 8 1 2 5 4 1 8 6 0 7 3 1 A Ví dụ: Phân tích LU ma trận sau:
4.
© 2012 Pearson
Education, Ltd. Slide 2.2- 4 Nhân tử hóa LU
5.
Nhân tử hóa
LU © 2012 Pearson Education, Ltd. Slide 2.2- 5 1 0 0 0 2 4 1 5 2 2 1 0 0 0 3 1 2 3 . 1 3 1 0 0 0 0 2 1 3 4 2 1 0 0 0 0 5 A LU Vậy
6.
Ví dụ 2:
Phân tích LU của ma trận sau: © 2012 Pearson Education, Ltd. Slide 2.2- 6 2 4 2 3 6 9 5 8 2 7 3 9 4 2 2 1 6 3 3 4 A Nhân tử hóa LU
7.
© 2012 Pearson
Education, Ltd. Slide 2.2- 7 2 2 1 3 3 1 3 3 2 4 4 1 4 4 2 5 5 1 5 5 2 3 2 2 3 3 2 4 2 3 2 4 2 3 6 9 5 8 0 3 1 1 2 7 3 9 0 3 1 6 4 2 2 1 0 6 2 7 6 3 3 4 0 9 3 13 2 4 2 3 0 3 1 1 0 0 0 5 0 0 0 5 0 0 0 10 h h h h h h h h h h h h h h h h h h h h h A 4 4 3 5 5 3 2 2 4 2 3 0 3 1 1 0 0 0 5 0 0 0 0 0 0 0 0 h h h h h h U
8.
© 2012 Pearson
Education, Ltd. Slide 2.2- 8 2 6 3 2 3 5 4 6 5 6 9 10 1 3 1 1 1 1 2 2 1 3 3 2 1 0 0 0 0 3 1 0 0 0 1 1 1 0 0 2 2 1 1 0 3 3 2 0 1 L 1 0 0 0 0 2 4 2 3 3 1 0 0 0 0 3 1 1 . 1 1 1 0 0 0 0 0 5 2 2 1 1 0 0 0 0 0 3 3 2 0 1 0 0 0 0 A LU Vậy:
9.
Nhân tử hóa
LU Vấn đề là tại sao chúng ta đi phân tích nhân tử hóa LU của ma trận • Khi A = LU, phương trình AX = B có thể viết thành L(UX) = B. • Đặt Y= UX, chúng ta có thể tìm X qua việc giải cặp phương trình • Đầu tiên giải LY = B để tìm Y, sau đó giải UX = Y để tìm X . • Mỗi phương trình đều giải được một cách dễ dàng vì L và U đều là ma trận tam giác. © 2016 Pearson Education, Ltd. Slide 2.5- 9 LY = B UX = Y
10.
Nhân tử hóa
LU VD 1: Giả sử ma trận A được phân tích LU như sau Sử dụng phép phân tích LU này của A để giải phương trình AX = B, với B = −9 5 7 11 © 2016 Pearson Education, Ltd. Slide 2.5- 10
11.
© 2012 Pearson
Education, Ltd. Slide 2.2- 11 2 2 1 3 3 1 4 4 1 3 3 1 4 4 3 4 4 1 : : 2 : 3 : 5 : 3 : 8 1 0 0 0 9 1 0 0 0 9 1 1 0 0 5 0 1 0 0 4 2 5 1 0 7 0 5 1 0 25 3 8 3 1 11 0 8 3 1 16 1 0 0 0 9 1 0 0 0 1 0 0 4 0 0 1 0 5 0 0 3 1 16 h h h h h h h h h h h h h h h h h h L B 0 9 0 1 0 0 4 0 0 1 0 5 0 0 0 1 1 Ta giải LY = B 9 4 5 1 Y
12.
© 2012 Pearson
Education, Ltd. Slide 2.2- 12 3 7 2 2 9 0 2 1 2 4 0 0 1 1 5 0 0 0 1 1 U Y Ta giải UX = Y 3 4 6 1 X
13.
© 2012 Pearson
Education, Ltd. Slide 2.2- 13 1 0 0 1 2 1 2 1 2 1 0 0 1 0 1 2 1 3 1 0 0 0 2 1 A é ùé ù - ê úê ú ê úê ú = - ê úê ú ê úê ú - ê úê ú ë ûë û b T 1 1 2 Ví dụ: Cho ma trận A có phân tích LU như sau: . Hãy giải phương trình Ax = b trong đó Ta giải Ly = b 1 0 0 1 1 2 1 0 1 1 1 3 1 2 2 L b y é ù é ù - - ê ú ê ú ê ú ê ú é ù= - Þ = ê ú ê ú ê ú ë û ê ú ê ú - - ê ú ê ú ë û ë û
14.
© 2012 Pearson
Education, Ltd. Slide 2.2- 14 1 2 3 4 5 3 3 3 2 1 2 1 2 1 1 2 0 1 0 1 2 1 , 0 0 0 2 1 2 1 1 2 b a x a x U y x x a b R b x a x a Ta giải Ux = y
15.
© 2012 Pearson
Education, Ltd. Slide 2.2- 15
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