total 2 assignment

Math 105LA Computer Assignment 2

(a) Develop a matlab code to implement the fixed point algorithm.

Algorithm: (Exactly as given in the textbook) START PROGRAM INPUT/INITIALIZATION : Input approximation po; Tolerance TOL; maximum number of iterations No.

OUTPUT : Approximate Solution p or message of failure.

STEP 1 Set i = 1. STEP 2 While i ≤ No repeat Steps 3 - 6 :

STEP 3 Set p = g(po). Compute pi STEP 4 If | p−po |< TOL then

OUTPUT (p) the procedure was successful STOP

STEP 5 Set i = i + 1. STEP 6 Set po = p. Update po

STEP 7 OUTPUT (’ The method failed after No iterations, No =’, No); The procedure was unsuccessful.

END PROGRAM

You may use a ”for” loop or a ”while” loop to implement the iteration. Also output your error in the form a graph to easily visualize the iteration and con- vergence.

(b) Develop a matlab code to implement Newton’s method.

Algorithm: (Exactly as given in the textbook) START PROGRAM INPUT/INITIALIZATION : Input approximation po; Tolerance TOL; maximum number of iterations No.

OUTPUT : Approximate Solution p or message of failure.

STEP 1 Set i = 1.

1

STEP 2 While i ≤ No repeat Steps 3 - 6 : STEP 3 Set p = po −f(po)/f′(po). Compute pi STEP 4 If | p−po |< TOL then

OUTPUT (p) the procedure was successful STOP

STEP 5 Set i = i + 1. STEP 6 Set po = p. Update po

STEP 7 OUTPUT (’ The method failed after No iterations, No =’, No); The procedure was unsuccessful.

END PROGRAM

(c) Use the iteration method, and Newton’s method to find the fixed point of g(x) = x/2 + 1/x in [1, 2] to approximate

√ 2 accurately to within 10−5. Which

method converges faster?

(d) Use your algorithms to solve problem 23 in Sec. 2.2.

Some new commands : Example 1 : Commands for printing in Matlab. output = sprintf(’Any text you want’); disp(output);

This will print the string of characters ‘Any text you want’ on the command window.

Example 2 : To print character strings with number : p = 1.45637798423; output = sprintf(’Any text you want %f’,p); disp(output); When you use %f you instruct matlab to replace with the number in p showing only 6 digits past the decimal point with a round off. This will print on the command window : Any text you want 1.456378

Example 3 : p = 1.45637798423; output = sprintf(’Any text you want %0.10f’,p); disp(output);

This will print on the command window : Any text you want 1.45637798423;

2

When you use %0.10f you instruct matlab to replace with the number in p showing the first 10 significant digits.

Example 3 : p = 0.000012345; output = sprintf(’Any text you want %e’,p); disp(output);

This will print on the command window : Any text you want 1.234500e-5;

3