5.1  This problem has been moved to the week 5 group activity.  The number has been left in place to maintain parity with the video content.

5.2   (a) Write a Maple procedure which implements the secant method. The procedure should take as its arguments a function f , initial estimates c0  and c1 , a positive number δ and a positive integer max_its. If the procedure finds estimates cj+1  and cj  such that

1cj+1 - cj 1 < δ,

then it should  return cj+1   as its  result.   Otherwise,  if the  number of iterations exceeds max_its, then it should report an error.  Your procedure should store no more than three estimates cj  at any one time.

Use your secant procedure to find all solutions to the following equations.

(i) e-x sin x - 4x2 + 12 = 0,    -5 < x < 5,

(ii) x2 - 5x - 3 - arctan x = 0,    -10 < x < 10.

(iii) 4 + 4 cos x = 0,    0 < x < 5.

Use the value δ = 0.5 × 10-9  in each case. You may find it helpful to

plot graphs of the functions; the Maple command for this is

plot(  f  , x0  . . x1  )

for a function f and real numbers x0  and x1 .

5.3 Apply the Newton– Raphson method to the function f (x) = sin(x) - k , where k is a constant with 1k1 < 1.  Try the resulting iteration scheme in Maple, using different numerical values for k and different initial estimates. Towards what value does it converge?

5.4  Determine whether the following iteration functions produce sequences that converge towards the fixed point x = ^a (with a > 0). For each iteration that converges, find the rate of convergence (i.e. linear, quadratic, etc.). You may assume that x  0.

(a) φ(x) =  ,    (b) φ(x) = 2x -  ,    (c) φ(x) =  x + ．(、) , (d) φ(x) =  3x - ．(、) ,    (e) φ(x) =   -  + 15x．(、) .

5.5 The iteration function

φ(x) = 2x - kx2 + w(x)(kx - 1)2

can be used to find 1/k using only addition, subtraction and multiplication. (a) Show that φ(k-1 ) = k-1  and φ\ (k-1 ) = 0.

(b)  Find w(x) such that φ\\ (k-1 ) = 0, and simplify the expression for φ in this case. Finally, find φ\\\ (k-1 ) and predict the behaviour of the errors in the iteration.

(c)  In Maple, use a do loop to apply the iteration 6 times, using the function w found in part (b). Display the relative error in each iterate and also the prediction for the next error based on the final result of part (b). Try some different values for k and different initial estimates. Do the numerical values support your theoretical prediction?

Hint: you will probably need a large setting for Digits to see the rate of convergence.

5.6 Suppose f (x) has a double root at x = c, that is f (c) = f\ (c) = 0, but f\\ (c)  0.

Use L’Hôpital’s rule to show that

lx   = 0.

Again using L’Hôpital’s rule, find the value of the limit

c = lim _ d   f (x)  ┐

Hint: the product rule for limits, i.e.

lx f (x)g(x) = lx f (x) × lx g(x)

can be used to simplify the algebra (provided the limits on the right-hand side exist).