Numerical Methods Course Notes Version 0.1 (UCSD Math 174, Fall ...

2.4. PLOTTING 21
Exercises
(2.1) What do the following pieces of octave/Matlab code accomplish?
(a) x = (0:40) ./ 40;
(b) a = 2;
b = 5;
x = a + (b-a) .* (0:40) ./ 40;
(c) x = a + (b-a) .* (0:40) ./ 40;
y = sin(x);
plot(x,y);
(2.2) Implement the naïve quadratic formula to find the roots of x 2 + bx + c = 0, for real b, c. Your
code should return
−b ± √ b 2 − 4c
.
2
Your m-file should have header line like:
function [x1,x2] = naivequad(b,c)
Test your code for (b, c) = ( 1 × 10 15 , 1 ) . Do you get a spurious root?
(2.3) Implement a robust quadratic formula (cf. Example Problem 1.9) to find the roots of x 2 +
bx + c = 0. Your m-file should have header line like:
function [x1,x2] = robustquad(b,c)
Test your code for (b, c) = ( 1 × 10 15 , 1 ) . Do you get a spurious root?
(2.4) Write octave/Matlab code to find a fixed point for the cosine, i.e., some x such that x =
cos(x). Do this as follows: pick some initial value x 0 , then let x i+1 = cos(x i ) for i =
0, 1, . . . , n. Pick n to be reasonably large, or choose some convergence criterion (i.e., terminate
if |x i+1 − x i | < 1 × 10 −10 ). Does your code always converge?
(2.5) Write code to implement the factorial function for integers:
function [nfact] = factorial(n)
where n factorial is equal to 1 · 2 · 3 · · · (n − 1) · n. Either use a ‘for’ loop, or write the function
to recursively call itself.