2.1

Question 1

What is the intermediate value theorem?

Answer 1

if f is continuous from a to b and K is any number between f(a) and f(b), then there exists a number, c, between a and b where f(c) = K

Question 2

What is the underlying theorem of the bisection method?

Answer 2

if f is continuous from a to b with f(a) and f(b) of opposite sign, then there exists a root between a and b

Question 3

What is the general algorithm of the bisection method?

Answer 3

Set a1=a, b1=b, p1 = (a1+b1)/2 = a1 + (b1-a1)/2
if f(p1) = 0, then p = p1 STOP
if sign(f(p1)) == sign(f(a))
     p is within (p1, b1). Set a2=p1, b2=b1
else
    p is within (p1, b1). Set a2=a1, b2=p1

Question 4

When are the 2 times the bisection method fails?

Answer 4

1. f(x) is continuous, but has no sign change

2. f(x) is not continuous, but has a sign change

Question 5

In practice, which 2 stopping criteria should we use?

Answer 5

1. Stop after a fixed number of iterations and

2. |pn-p(n-1)| < tol

Question 6

__________ the starting interval decreases the number of operations

Answer 6

Minimizing

Question 7

For the order of convergence, which 2 constants do we use to define the speed?

Answer 7

Lambda and alpha

Lambda affects speed, but alpha affects the order, which affects the speed much more

Question 8

When a = 1, convergence is ______. When a = 2, convergence is _________.

Answer 8

linear, quadratic

Question 9

What order of convergence is the bisection method and what is it’s lambda value?

Answer 9

linear and lambda = 1/2. It is quite slow, but reliable