P2.10 Numerical Methods

Question 1

what is the conclusion for locating roots?

Answer 1

the function f(x) is continuous on the interval [a,b] and f(a) and f(b) have opposite signs, then f(x) has at least one root

Question 2

how do you show a root is a given value to a given degree of accuracy?

Answer 2

use the upper & lower bounds of the given root –> there is a sign change in the interval

Question 3

describe how to solve an equation of the form f(x) = 0 by a rearrangement (iterative) method

Answer 3

rearrange f(x) = 0 into the form x = g(x) & use the iterative formula xn+1=g(xn)
depending on the rearrangement & the starting value of x, the iterations will either converge towards a root or diverge away from a root
NB not all rearrangements of x give all the solutions

Question 4

what makes a good rearrangement?

Answer 4

when the iterations converge towards a root = the increments b/w each value decreases

Question 5

how is the rearrangement method represented graphically?

Answer 5

staircase or cobweb diagrams

Question 6

draw staircase diagram

Answer 6

see pg 278 P2 textbook

Question 7

draw cobweb diagram

Answer 7

see pg 278 P2 textbook

Question 8

how are different roots found using the rearrangement method?

Answer 8

by using different rearrangements & starting values of x

Question 9

what is the Newton-Raphson iterative formula?

Answer 9

x (n+1) = xn - f(xn)/f’(xn)
in FB

Question 10

how does Newton-Raphson work graphically?

Answer 10

using tangent lines to find increasingly accurate approximations of a root
the value of xn+1 is the point at which the tangent to the graph at (xn, f(xn)) intersects the x-axis

Question 11

what is a problem with the Newton-Raphson method/not suitable value to use as first approximation?

Answer 11

at turning point (p,0) where f’(x) = 0
you cannot divide by 0 in N-R formula
graphically, the tangent line will run parallel to x-axis so never intersects