Systems of Linear Algebraic Equations - Numerical Methods in Engineering with MATLAB - page 53

3. Utilize the results of LU decomposition

⎡
⎣

⎤
⎦

⎡
⎣

⎤
⎦

1 00 2 − 3 − 1

A = LU = 3 / 210 0 13 / 2 − 7 / 2

1 / 211 / 13 1 0032 / 13

1

12 where b T

to solve Ax = b , = − .

4. Use Gauss elimination to solve the equations Ax = b , where

⎡
⎣

⎤
⎦

⎡
⎣

⎤
⎦

2 − 3 − 1 3

A = 32 − 5 b = − 9

2 4 − 1 − 5

5. Solve the equations AX = B by Gauss elimination, where

⎡
⎣

⎤
⎦

⎡
⎣

⎤
⎦

2 0 1 0

0 120

− 1 0

00
0 1
00

A = B =

− 12 0 1

00 1 − 2

6. Solve the equations Ax = b by Gauss elimination, where

⎡
⎣

⎤
⎦

⎡
⎣

⎤
⎦

00 212
0 1

1
1 0 2 − 1

A = b =

12 0 − 2 0 − 4

00 0 − 11 − 2

− − −

0 1 11 1 1

Hint: reorder the equations before solving.
7. Find L and U so that

⎡
⎣

⎤
⎦

4 − 1 0

A = LU = − 1 4 − 1

0 − 1 4

using (a) Doolittle's decomposition; (b) Choleski's decomposition.

8. Use Doolittle's decompositionmethod to solve Ax = b , where

⎡
⎣

⎤
⎦

⎡
⎣

⎤
⎦

− 36 − 4 − 3

65

A = 9 − 8 4 b =

− 12 24 − 26 − 42

9. Solve the equations Ax = b by Doolittle's decompositionmethod, where

⎡
⎣

⎤
⎦

⎡
⎣

⎤
⎦

2 . 34 − 4 . 10 1 . 78 0 . 02

A = − 1 . 98 3 . 47 − 2 . 22 b = − 0 . 73

2 . 36 − 15 . 17 6 . 18 − 6 . 63