3a Show that if \underline{x} = (x, y, z)^{T} is any 3-vector and A is the matrix

A = ( \begin{matrix}1 & 2 & 2 \\ 2 & 1 & −2 \\ 2 & −2 & 1 \end{matrix} ) then |A\underline{x}|^{2} = 9|\underline{x}|^{2} .

b Deduce that all the eigenvalues of A have modulus 3, and by considering properties of the matrix A, such as its trace, T r(A), infer that the eigenvalues of A are 3, 3 and −3.

Demonstrate that (1, 1, 0)^{T} and (−1, 1, −2)^{T} are eigenvectors of A. Explain why the third eigenvector is perpendicular to these two, find it, and verify that it has the expected eigenvalue.

c Viewing A as a transformation matrix of 3-vectors, identify its action in terms of scalings, projections, reflections and rotations.

Parts a, b and c carry respectively 28%, 52% and 20% of the marks 4a Show that

(1 + i)^{n} − (1 − i)^{n} = 2^{(\frac{n+2}{2})} i sin (\frac{1}{4}nπ) .

b The numbers {u_{n}} for n = 0, 1, 2 … are defined by the recurrence relation

u_{n+1} − 2u_{n} + 2u_{n−1} = n with u_{0} = 1, u_{1} = 1 .

Obtain an explicit formula for u_{n}, and describe qualitatively the behaviour of u_{n} as n → ∞.

Parts a and b carry respectively 24% and 76% of the marks

\begin{matrix}c \\ ◯ \end{matrix} University of London 1996  Paper 1.9 Page 4  TURN OVER

5a An iterative scheme is defined for a given smooth function f(x) by

x_{n+1} = (1 − k)x_{n} + kf(x_{n})  where k > 0  is constant.

If the sequence {x_{n}} converges to a limit X, infer an equation satisfied by X.

Defining ε_{n} = x_{n} − X for every n, show that

ε_{n+1} = ε_{n}(1 + k[f′(X) − 1]) + O(ε_{n}^{2}) .

b Write down a necessary condition for the scheme to converge.

What value of k gives fastest convergence? Compare the scheme for this value with Newton’s method, and explain why Newton’s method is more practical.

c If ε_{n} → 0 as n → ∞, discuss whether ∑_{n=1}^{∞} x_{n} is convergent, stating which convergence test you use.

Parts a, b and c carry respectively 44%, 36% and 20% of the marks 6a A tri-diagonal matrix is one all of whose elements are zero apart from

the leading diagonal and the diagonals on either side. That is, its (i, j)th element is zero unless i = j, i = j − 1 or i = j + 1.

Show that (6N − 4) floating point operations (counting multiplications and reciprocals only) are necessary to solve A\underline{x} = \underline{b}, where A is an N × N tri-diagonal matrix.

b Write down 4 equations satisfied by the stationary point of the function

f(w, x, y, z) = w^{2} + x^{2} + y^{2} + z^{2} + wx + xy + yz − w + 2x + 2y − z

and solve them using Gaussian elimination.

c Show that the eigen-values of the matrix

H = ( \begin{matrix}2 & 1 & 0 & 0 \\ 1 & 2 & 1 & 0 \\ 0 & 1 & 2 & 1 \\ 0 & 0 & 1 & 2 \end{matrix} ) satisfy (2 − λ)^{4} − 3(2 − λ)^{2} + 1 = 0 .

Solve this equation and infer the nature of the stationary point. [Hint: Consider (\sqrt{5} ± 1)^{2}]

Parts a, b and c carry respectively 28%, 36% and 36% of the marks

\begin{matrix}c \\ ◯ \end{matrix} University of London 1996 Paper 1.9 Page 6 END OF PAPER