3a Use Gaussian Elimination to find all solutions to the system of equations

\begin{aligned}x +  ay & = 0 \\ x + (a + 2)y + az & = 5 \\ 2x +  ay + az & = 3 − a \end{aligned}

in terms of the parameter a. Pay particular attention to any special values of a which may arise.

3b Find the inverse of the matrix

A = ( \begin{matrix}1 & −1 & 0 \\ 1 & 1 & −1 \\ 2 & −1 & −1 \end{matrix} )

Use this inverse to solve A\underline{x} = (0, 5, 4)^{T}, and compare the solution with part (3a) when a = −1.

Parts a and b carry respectively 60% and 40% of the marks 4a Define the eigenvectors and eigenvalues of a matrix A. What can be said

about the eigenvalues and eigenvectors of a real, symmetric matrix?

4b Show directly from the definition that (1, 1, 1)^{T} and (1, 0, −1)^{T} are eigenvectors of the matrix

A = ( \begin{matrix}3 & 4 & 1 \\ 4 & 0 & 4 \\ 1 & 4 & 3 \end{matrix} )

and give the corresponding eigenvalues.

4c Use the results of (4a) to find the third eigenvector. What is its eigenvalue?

Verify the relation between eigenvalues and determinant for A.

4d For A as above, describe the behaviour as n → ∞ of the vector \underline{x}_{n} defined

by

\underline{x}_{n+1} = A\underline{x}_{n} with \underline{x}_{0} = (2, 0, −2) + \underline{ε}_{0} ,

where \underline{ε}_{0} denotes the small (|\underline{ε}_{0}| ≃ 10^{−16}) rounding error on a machine of finite precision.

Parts a, b, c, d carry respectively 15%, 15%, 45%, 25% of the marks.

\begin{matrix}c \\ ◯ \end{matrix} University of London 1997  Paper 1.9 Page 4  TURN OVER

5a If X = f(X) for a differentiable function f(x), prove that a sufficient condition for the iterative scheme

x_{n+1} = f(x_{n})

to converge to X for a sufficiently close initial estimate x_{0}, is that

|f′(X)| < 1 .

5b Show that the equation x = e^{−x} is equivalent to x = − ln x, and draw rough graphs to show that this equation has a unique root X with 0 < X < 1.

5c Discuss whether the two sequences {x_{n}} and {y_{n}} defined by

x_{n+1} = e^{−x_{n}} and y_{n+1} = − ln y_{n} ,

with x_{0} = y_{0} = 0.5, are likely to converge to X.

5d Write down a Newton scheme for this problem.

Parts a, b, c, d carry respectively 40%, 15%, 30%, 15% of the marks.

6a The price, P, of hiring a computer depends on c, the amount of CPU time, and s, the amount of storage required, according to the formula

P (c, s) = cs(c + 2s)

when P, c and s are measured in suitable units.

If P, c and s depend on a parameter N, obtain a relation between \frac{dP}{dN}, \frac{dc}{dN} and \frac{ds}{dN}.

6b Currently, a program requires c = 1 and s = 1. It would be possible to rewrite the code to use slightly less storage but more CPU. Obtain a relation between the small changes δc and δs to determine when such a change would be worthwhile.

6c The major part of the program involves the storing and inverting of N × N matrices. If c = 1 and s = 1 for N = 100, deduce formulae for c and s in terms of N.

Show that currently \frac{dP}{dN} = 0.22.

6d Find a stationary point of the function P (c, s), and evaluate the matrix of second derivatives at that point.

Parts a, b, c, d carry respectively 15%, 25%, 35%, 25% of the marks.

\begin{matrix}c \\ ◯ \end{matrix} University of London 1997 Paper 1.9 Page 6 END OF PAPER