what is a numerical method
an approximation for the root or area of a function
what are some numerical methods
iteration
trapezium rule
newton raphson
when is change of sign appropriate
when the eq is continuous
how do you use change in sign
- work in given interval or use bounds
- show their is a change in sign
when does change in sign fail
interval is too large- locates the wrong roots
touches the x axis- this is a root but there is no change in sign
asymptotes- in a discontinous eq may be mistaken for a root
how do you use the newton Raphson’s method?
rearrange into the form f(x) = 0
differentiate to find f’(x)
sub into formula Xn - f(x)/f’(x) = Xn+1
sub in initial and begin iterating
what is the iterative formula?
Xn+1 = g(Xn)
what is x = g(x)
the rearrangement of f (x) to make one of the x values the subject
what does the line of the initial x value look like?
a solid line
how do you draw sketch an iterative graph
drawing a vertical line to the curve then a horizontal one to the line.
dotted line indicates each value
why is Newton Raphson’s often preferred more than iteration?
it converges faster
when does newton Raphson’s fail?
x values with a horizontal tangent (stationery point)
far initial may find a different value
small gradient : slow convergence or divergence
how can you check if an x value is a stationery point?
if f’(x) = 0
x is a stationery point and its tangent won’t intercept the x axis again
What is n in the trapezium rule?
The number of strips
Eg
If Yn is Y4 then n= 4 and not 5!
