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solve the questions in the file. question 3 needs to be solved in matlab. rest of the questions are all handwritten problems.

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Midterm (Math 5532)

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1. (25 pts) Let

A=

2

4

8

6

1

3

7

7

1

3

9

9

0

1

5

8

and b =

4

11

29

30

Using the exact arithmetic, do the following:

(a) Compute P LU -factorization of A, that is, P A = LU , using Gaussian

elimination with partial pivoting.

(b) Compute c = P b.

(c) Solve Ld = c for d.

(d) Solve U x = d for x.

2. (25 pts) Let

A=

1

0.5 0.333 0.25

0.5 0.333 0.25

0.2

0.333 0.25

0.2 0.167

0.25

0.2 0.167 0.143

and x =

1

1

1

1

Using 3 significant digit rounding arithmetic, do the following:

(a) Compute b = Ax.

(b) Compute LU -factorization of A that is, A = LU .

(c) Solve Lc = b for c.

(d) Solve U x̃ = c for x̃.

(e) Compute max1≤i≤4 |xi − x̃i |.

(f) Could Gaussian elimination with partial pivoting have helped the situation? Justify your answer.

1

3. (25 pts) (MATLAB) Let

A=

1

1

0

3

2

1 −1

1

3 −1 −1

2

−1

2

6 −1

and g =

4

−3

1

4

(a) Compute P LU factorization of A, that is, P A = LU , using Gaussian

elimination with partial pivoting.

(b) Make necessary MATLAB functions to solve AT y = g using the P LU factorization of A.

(c) Print out the solution vector y.

4. (25 pts) Let

A=

d1 e1 0 0

c2 d2 e2 0

0 c3 d3 e3

0 0 c4 d4

Suppose that none of c2 , c3 , and c4 is zero and that none of e1 , e2 , and e3 is

zero. Further assume that

|d1 | > |e1 |,

|d2 | ≥ |c2 | + |e2 |,

|d3 | ≥ |c3 | + |e3 |,

Prove that A is invertible.

2

|d4 | ≥ |c4 |.

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Tags:

equations

numerical methods

exact arithmetic

MATLAB functions

Gaussion eliminatio method

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