Change of Sign Methods
● Sometimes there is no easy way of finding the roots of an equation by
factorising (e.g x3 – 7x + 3 = 0 )
● Alternative is to look at graph to find the interval where the roots lie (e.g
between 2 and 3)
● Then try plugging in values of x between that interval and wait for the
answer to change sign (from + to – or visa versa)
● Repeat until gradually you gain appropriate levels of accuracy (enough
decimal places)
● Can go wrong:
○ If a repeated root occurs (so it touches the x-axis and never goes
below)
○ If there is a discontinuity in the graph (if there’s a change of sign
without a root)
Fixed Point Iteration
● Rearrange f(x) = 0 into x = g(x) and solve to find roots (if done incorrectly
can mean converging to different root or not converging at all)
● This is essentially splitting f(x) into the lines y = x and x = g(x)
● This means the point on the graph where y = x and x = g(x) meet is the
same x-value as the root of the equation f(x) = 0
● You can do this by gradually gaining more accuracy from a start point:
● The sequence converges on a root of the equation, providing x0 is a close
enough approximation and the curve is not too steep close to the root
● The gradient of the curve close to the root must be between -1 and 1
Newton-Raphson Method
● Based on evaluating the gradient of the tangent to the curve y = f(x) at
● Repeating this causes convergence in the same way as fixed point iteration
● This method rarely fails, but if the approximation is close to a turning point
then iterations may diverge or converge to a different root (as f ′ (x) will be
very small)
The Trapezium Rule
● Not all functions can be integrated
● For such functions approximate values for a definite integral can be found
by using the trapezium rule
● This involves dividing the area under the graph into a number of trapezia
and summing the areas of these trapezia
● Using more trapezia will give a more accurate value
● This method can give overestimates or underestimates:
○ If the curve is always concave upwards for the region being
considered then this method will give an overestimate (trapezia will
always lie partially above curve)
○ If the curve is always concave downwards for the region being
considered then this method will give an underestimate (trapezia will
always lie partially below curve)
○ If there is a point of inflection in region being considered, cannot
easily tell if overestimate or underestimate
Upper and Lower Bounds
● Trapezium rule gives approximate value for an integral
● Can often tell from shape of curve if over or underestimate, but not easy to
tell how accurate the estimate is
● Using rectangles instead of trapezia (if the curve has no turning point) can
be useful
● Rectangles that are either above or below the curve can give a lower or
upper bound