gibson:teaching:fall-2013:math445:lab5

Helpful Matlab commands/functions/constructs for this lab:
`while-end`

, `abs`

, `plot`

, `grid on`

, `for-end`

, `\`

, `contour`

, `norm`

, and anonymous functions.

**Problem 1:** Write a `newtonsearch1d`

function that computes a zero of a
1-d function `f`

using the Newton search method, starting from the initial
guess `x`

. Use a `while`

loop to terminate the iteration when either
or when the Newton step is very small: ,
for some suitable choice of .

Use this function to solve the following problems. Check your answers by
plugging the answer `x`

back into `f`

and verifying that `f(x)`

is
approximately zero.

**(a)** Find an `x`

for which .

**(b)** Find the cube root of 54. (Hint: devise an equation whose answer is .)

**(c)** Find an `x`

for which .

To find good initial guesses for the Newton search, plot `f`

versus `x`

and estimate where it crosses the `x`

axis.

**Problem 2:** Write a `newtonsearch2d`

function that finds a zero of
a 2-d function `f`

starting from the initial guess `x`

, where both `x`

and `f(x)`

are two-dimensional vectors. Use this to find a zero of the
nonlinear 2-d function

Use a contour plot of the norm of over to find an initial guess for the search.

**Bonus (10 pts):** Write a `newtonsearchNd`

function that finds a zero of
an N-dimensional function `f`

starting from the initial guess `x`

.
Use this to find a zero of the nonlinear 3d function

Use the initial guess . Verify your answer by applying it to the 3d function. What do you expect to get?

** Bonus (10 points): **
Give a brief explanation for the Newton's Search. Include the answers to the following questions.

** - ** Purpose: What is the Newton's method used for?

** - ** Method: How does it do this? (How is it related to the Taylor Series? Can you explain the equations used in the code?)

gibson/teaching/fall-2013/math445/lab5.txt · Last modified: 2013/10/03 05:42 by gibson