Use each of these three algorithms with three-digit rounding arithmetic to

solve the linear systems below.

a) Gaussian elimination with backward substitution (alg061.mpl )

b) Gaussian elimination with partial pivoting (alg062.mpl )

c) Gaussian elimination with scaled partial pivoting (alg063.mpl )

i) 0. 03 x1  + 58. 9 x2  = 59. 2

5. 31 x1 −  6. 10 x2  = 47. 0

ii) 3. 3330 x1  + 15920 x2  + 10. 333 x3  = 7953

2. 2220 x1  + 16. 710 x2  + 9. 6120 x3  = 0. 965

− 1. 5611 x1  + 5. 1792 x2 −  1. 6855 x3  = 2. 714

iii) 1. 19 x1  + 2. 11 x2 −  100 x3  + x4  = 1. 12

14. 2 x1 −  0. 122 x2  + 12. 2 x3 − x4  = 3. 44

100 x2 −  99. 9 x3  + x4  = 2. 15

15. 3 x1  + 0. 110 x2 −  13. 1 x3 − x4  = 4. 16

2.  Consider the following linear system.

x1 − x2  + x3  = − 2

−x1  + 2 x2 − x3  = 3

x1  + x2  + x3  = 2

a) Find all values of   for which this system has no solution.

b) Find all values of   for which this system has infinitely many solutions.

c) Assuming a unique solution exists for a given , find the solution.

3.  Give LU  factorizations (with L1,1  = · · ·  = Ln,n  = 1) of the matrices below.

i)

@

 2 − 1 1

3 3 9

3 3 5

1

A  ii)

@

 1. 012 − 2. 132 3. 104

− 2. 132 4. 096 − 7. 013

3. 104 − 7. 013 0. 014

1

A

 iii)

BB@

 2 0 0 0

1 3

2  0 0

0 − 3 1

2  0

2 − 2 1 1

1

CCA

 iv)

BB@

 2. 1756 4. 0231 − 2. 1732 5. 1967

− 4. 0231 6. 0000 0 1. 1973

− 1. 0000 − 5. 2107 1. 1111 0

6. 0235 7. 0000 0 − 4. 1561

1

CCA

4.  For i  = 1,  2,  3,  4, define Ai  as follows.

A1  =

@

 2 − 1 0

− 1 2 − 1

0 − 1 2

1

A A2  =

BB@

 4 1 1 1

1 3 − 1 1

1 − 1 2 0

1 1 0 2

1

CCA

A3  =

BB@

 4 1 − 1 0

1 3 − 1 0

− 1 − 1 5 2

0 0 2 4

1

CCA

A4  =

BB@

 6 2 1 − 1

2 4 1 0

1 1 4 − 1

− 1 0 − 1 3

1

CCA

 i) For i  = 1,  2,  3,  4 find a factorization of the form Ai  = LDLt .

ii) For i  = 1,  2,  3,  4 find a factorization of the form Ai  = LLt .

iii) Let Y1  = (3, − 3,  1)t, Y2  = (0. 65,  0. 05,  0. 00,  0. 50)t,

Y3  = (7,  8, − 4,  6)t, Y4  = (0,  7, − 1, − 2)t.

 Use the factorizations from part (i) to solve the linear system AiX  = Yi ,

for i  = 1,  2,  3,  4.

5.  Obtain factorizations of the form A  = PtLU  for the following matrices.

i)

@

 1 2 − 1

1 2 3

2 − 1 4

1

A  ii)

BB@

 1 − 2 3 0

3 − 6 9 3

2 1 4 1

1 − 2 2 − 2

1

CCA

 iii)

BB@

 1 − 2 3 0

1 − 2 3 1

1 − 2 2 − 2

2 1 3 − 1

1

CCA

4