# MATH 5532 UMKC MATLAB Functions Advanced Numerical Methods Questions

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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)
Show your work in detail for full credit.
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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