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# \$100.00Numerical Methods

• From Computer-Science: General-CS
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• Due on Mar. 15, 2012
• Asked on Mar 12, 2012 at 9:30:22PM

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Q:

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 x + 58. 9 x = 59. 2

5. 31 x1 −  6. 10 x = 47. 0

ii) 3. 3330 x + 15920 x + 10. 333 x = 7953

2. 2220 x + 16. 710 x + 9. 6120 x = 0. 965

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

iii) 1. 19 x + 2. 11 x2 −  100 x + x = 1. 12

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

100 x2 −  99. 9 x + x = 2. 15

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

2.  Consider the following linear system.

x1 x + x = − 2

x + 2 x2 x = 3

x + x + x = 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, = · · ·  = 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

0 0

0 − 3 1

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 = 1 2 3 4, define A as follows.

A =

@

2 − 1 0

− 1 2 − 1

0 − 1 2

1

A A =

BB@

4 1 1 1

1 3 − 1 1

1 − 1 2 0

1 1 0 2

1

CCA

A =

BB@

4 1 − 1 0

1 3 − 1 0

− 1 − 1 5 2

0 0 2 4

1

CCA

A =

BB@

6 2 1 − 1

2 4 1 0

1 1 4 − 1

− 1 0 − 1 3

1

CCA

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

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

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

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

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

for = 1 2 3 4.

5.  Obtain factorizations of the form = 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

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