Clearly showing your calculations, work out by hand the lower triangular matrix L in Cholesky\'s decomposition of the matrix,

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

Question 1

a) Clearly showing your calculations, work out by hand the lower triangular matrix L in Cholesky's decomposition of the matrix,

b) Next, write your own R function to produce the lower triangular matrix L in Cholesky's decomposition of an arbitrary n x n positive definite, symmetric matrix and use your function to check your answer in part (a).

You should not use ANY existing R function for Cholesky's decomposition (e.g. chol()) in your function.

[8 marks]

Question 2

a) Clearly showing your calculations, work out by hand the singular values and the matrices

1 2

1 0

in the singular value decomposition of A

b) Write R code to check your answer in part (a) and comment on any differences between your answer in part (a) and the R output.

Question 3

Consider a system with differential equation given by

d3y dy dy dt3 dt2 dt

-y=et.

[6 marks]

a) Determine the transient and steady state solutions for y, and hence find the particular solution if y = 0, y = 1 and -- at t=0.

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