Math 210 Sec. 201 Assignment 2 - due February 16, 1996

1.(a) When Newton’s method works, it usually converges very rapidly to a solution: typically the

number of correct decimal places doubles at each iteration. Why is this the case?

(b) Use Newton’s method with Digits = 80 and x_{0} = 2 to solve the equation 1 + sin(2x) = x. How many digits are correct in each iteration until x_{n+1} = x_{n} (to the given accuracy)?

(c) Newton’s method converges much more slowly when f′ is 0 at the solution. Why? Consider f(x) = (x − 1)^{3} and start with x_{0} = 2. Find a formula for x_{n}. How does the number of correct digits change with each iteration? How many are needed to get the root correct to 10 decimal places?

(d) For an example with even slower convergence, try

f(x) = \{ \begin{matrix}x^{1/x} & for x > 0 \\ 0 & for x ≤ 0 \end{matrix}

with x_{0} = 1/2. When is x_{n} < 1/10?

2. (a) Find a polynomial for which Newton’s method started with x_{0} = 0 has a stable 3-cycle.

(b) Find the largest interval containing 0 for which you can be sure that Newton’s method, starting with x_{0} in this interval, is attracted to the 3-cycle.

3. Consider the equations

\begin{aligned}x^{3} + 3xy^{3} − y & = 0 \\ x^{2}y^{2} − 2x^{3} − 3y^{3} & = 4 \end{aligned}

(a) Use implicitplot to plot the curves for −5 ≤ x ≤ 5 and −5 ≤ y ≤ 5. Where do the intersections appear to be?

(b) Use fsolve to find numerical values for the intersections you found in (a).

(c) With the help of resultant, find all real solutions of the equations. One of the solutions may be a surprise. Why?

(d) Use solve to find all real solutions of the equations. Compare the results to those of (c). A higher setting of Digits is needed to obtain accurate numerical results this way. Why?

(e) Use Newton’s method to find a solution to the equations. Start at a point whose coordinates are small integers.

(f) Find a point (x_{0}, y_{0}) with |x_{0}| < 1 and |y_{0}| < 1 such that if Newton’s method is started at this point, the next point (x_{1}, y_{1}) will be at (or very close to) the solution of the equations that is farthest from the origin.

4. Complete the “Optimization on a Restricted Domain” example in Lesson 14 by finding

(a) The critical points of f(x, y) in the region g_{1}(x, y) < 0, g_{2}(x, y) < 0.

(b) The critical points of the restriction of f(x, y) to g_{2}(x, y) = 0 with g_{1}(x, y) < 0.

(c) The intersections of the curves g_{1}(x, y) = 0 and g_{2}(x, y) = 0.

Putting it all together, find the maximum and minimum of f(x, y) in the region where g_{1}(x, y) ≤ 0 and g_{2}(x, y) ≤ 0.