Numerical Methods
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:
○ xn+1 = g(xn ), n = 0, 1, 2, 3...
○ Start with an initial approximation, x0 , and find g (x0 )
○ Take this value of g (x0 ) as a new value, x1 , and find g (x1 )
○ Repeat until particular decimal point is constant for two or three
iterations
● 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
Staircase Diagrams
Iterations are on the same side of the root
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:
○ xn+1 = g(xn ), n = 0, 1, 2, 3...
○ Start with an initial approximation, x0 , and find g (x0 )
○ Take this value of g (x0 ) as a new value, x1 , and find g (x1 )
○ Repeat until particular decimal point is constant for two or three
iterations
● 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
Staircase Diagrams
Iterations are on the same side of the root
