C3 THEORY OF COMPUTATIONAL DYNAMICS
EXAMPLE SHEET 3
Stars indicate level of difficulty.
1.
C3 May 1994, Q2.
2.
C3 May 1995, Q2.
3.
Discuss the convergence of the Newton method in one dimension when F'(x*) = F''(x*) = 0, but
F'''(x*) ≠ 0. Test your conclusions on F(x) = x3.
4.
Let F be the function
F(x)
=
( )
exp − x1
2
for x ≠ 0, and F(0) = 0 (recall Q4 on Ex. Sheet 1). What would happen if the Newton method
were used to find the root of F at x = 0? In particular show that there is no constant 0 ≤ K < 1
such that |εi+1 | ≤ K|εi| for all sufficiently small εi. Try implementing this on a computer (I know
that in this case you know the root explicitly; but pretend that you don’t).
5.
How fast does the Newton method in one dimension converge when F'(x*) ≠ 0, but F''(x*) = 0
and F'''(x*) ≠ 0? What if all the derivatives from the 2nd to the (r-1) st vanish at x*, but the rth
derivative is non-zero? Test this with F(x) = x + x3, and more generally F (x) = x + xr.
6.
Let G be the function
G(x)
=
( )
x + exp − x1
2
for x ≠ 0, and G(0) = 0 (recall Q4, sheet 1 and Q4 above). What is the rate of convergence to the
root at x = 0 of the Newton method for G. Is this consistent with your answer to Q5? Try running
this on a computer.
7.
Let F be the function
1
F(x)
=
3
3
 − 4
x

Discuss the behaviour of the Newton method for F.
8*. Find a function G whose Newton method is the logistic map xn+1 = λxn(1 - xn).
9*. Find a function G whose Newton method is the tent map
xn+1
=
2x n

2(1 − x n )
0 ≤ xn ≤
1
2
1
2
≤ xn ≤ 1
C3 Exercise Sheet 3
2
10. Let f be the logistic map f(x) = λx(1 - x). Suppose the Newton method is used to find the fixed
points of f. Discuss its behaviour for λ = 1 and λ > 1. Relate this to the dynamics of f.
11. Suppose the Newton method is now used to find the period 2 points of the logistic map f(x) =
λx(1 - x). Discuss its behaviour for λ = 1, λ = 3, 1 < λ < 3 and λ > 3. Relate this to the dynamics of f.
Hint: a fixed point is also a period 2 point; use this to factorize λ2x(1 - x)(1 - λx(1 - x)) - x.
12. Suppose we tried to use the Newton method on the function f 2(x)-x (where f is the logistic map)
to find a fixed point of f at λ = 3. What goes wrong, and why? Why does finding a root of f(x)-x
work without problems, but not of f 2(x)-x? Hint: this is just a different way of looking at Q11.
13. Let f(x,y) = (1 - ax2 + y, bx) be the Hénon map. Discuss the behaviour of the Newton method for
finding fixed points of this map for the case a < -(1-b)2/4, a = -(1-b)2/4 and a > -(1-b)2/4.
14**. Suppose the Newton method is now used to find the period 2 points of the Hénon map.
Discuss its behaviour for a < -3(1+b)2/4, a = -3(1+b)2/4 and a > -3(1+b)2/4. Hint: this is just like
Q11, however you may want to do the calculations using Mathematica.
15. How would you use the Newton method to find the inverse of a function f ?
16. C3 May 1994, Q3 i).
17. C3 May 1994, Q3 ii).
18. C3 May 1995, Q3 i).
19. C3 May 1995, Q3 ii).
20. Suppose that the ith and jth columns of an N×N matrix are the same. Is the matrix invertible?
21. Recall that a quantity f(N) is said to be O(g(N)) if f(N) ≤ Kg(N) for some constant K < ∞. Prove that
n −1
∑i
=
1
2
n(n-1)
=
1
6
n(n-1)(2n-1)
O(n2)
=
i =1
and
n −1
∑i
2
=
O(n3)
i =1
Hint: induction. Deduce that using Gaussian elimination to solve the equation Ax = z, where A
is an N×N matrix, requires O(N 3) arithmetic operations.
22. Check that the product of two lower triangular matrices is again a lower triangular matrix.
23. Verify that the determinant of an N×N triangular matrix is the product of its diagonal elements
(you may use the standard recursion formula). Deduce that if A = LR is the LR-decomposition
described in lectures, then det(A) = det(R).