APM3711 Assignment 3 Complete Solutions UNISA 2026 Numerical Analysis II

APM3711
ASSIGNMENT 3
Numerical Analysis II

FULL
SOLUTIONS
COMPLETE SOLUTIONS

MEMORANDUM
UNISA 2026
Page 1 of 29

,SOLUTIONS:
Question 1


We are given the matrix A with the initial vector

4 1 1 1
(0)
𝐴 = [1 3 0] , 𝐱 = [1]
1 0 2 0




(a) First Two Iterations

Iteration 1:

Compute 𝐲 (1) = 𝐴𝐱 (0) :


Page 2 of 29

, 4 1 1 1
𝐲 (1) = [1 3 0] [1]
1 0 2 0



Calculate each component:

𝑦1 = 4(1) + 1(1) + 1(0) = 4 + 1 + 0 = 5

𝑦2 = 1(1) + 3(1) + 0(0) = 1 + 3 + 0 = 4

𝑦3 = 1(1) + 0(1) + 2(0) = 1 + 0 + 0 = 1

So:

5
𝐲 (1) = [4]
1



The dominant component (largest absolute value) is 5. Normalize 𝐲 (1) by dividing by this
component:

1 5 1
(1)
𝐱 = [4] = [0.8000]
5
1 0.2000



The eigenvalue approximation is the dominant component:

𝜆(1) = 5




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