2 - chapter 14 - numerical solutions of eq

Question 1

finding roots of a function

Answer 1

• if the graph is continuous and theres a change in sign in an interval
• then there is a root ( or more than one) in that same interval

Question 2

when change in sign doesnt work

Answer 2

• incorrectly says there is a root - if the function has a vertical asymptote, or any break in the graph eg 1/x
• ignores a root - if the graph is tangent to x axis between the interval, if graph crosses several times in that interval eg x^2 or sinx

Question 3

newton raphson

Answer 3

• draw a tangent at a point and find where it crosses the x axis
• draw another tangent at this new value of x and repeat - if this is repeated gets an accurate root
  (formula in formula booklet)
  xn+1 = xn - f(xn)/f’(xn)
• use ans button to repeat

Question 4

limitiations of newton raphson

Answer 4

if the gradient at one of the points you check is 0 eg f’(xn) = 0 then youd divide by 0 which you cant
- doesn’t work if starting value is a stationary point of f(x)
- if the root is close to a stationary point may initially or permanently move away from the root
- doesn’t work if xn falls outside the domain of the function eg has a vertical asymptote or ln cant be negative

Question 5

fixed point iteration

Answer 5

start guess at x1
then x2 = g(x1)
as solution of x = g(x) is the interception of y = x and y = g(x)
if the sequence xn+1 = g(xn) converges to a limit then the limit is a solution to your eq

Question 6

staircase diagram

Answer 6

• successive terms of an iterative sequence are all increasing or decreasing
  (fixed point iteration)

Question 7

cobweb diagram

Answer 7

• successive terms of an iterative sequence oscillate either side of a value to which the sequence converges
  (fixed point iteration)

Question 8

limitations of fixed point

Answer 8

+ easier to implement
- sequence may diverge or converge to another root - may need to rearrange the equation to lead to a convergent sequence / find another root
can tell if it will converge / what root it will go to from the diagrams

Question 9

fixed point iteration - converges or diverges

Answer 9

converges if the modulus of the gradient <1 near the root and x1 is near the root
diverges if the modulus of the gradient > 1 near the root
if it = 1 or -1 cant tell