Root [0.1 , 0.2]
Root [0.16 , 0.17]
Root [0.169 , 0.170]
Root [0.1690 , 0.1691]
From this method we find that the root lies between [0.16908 , 0.16909]
∴ Root = 0.169085 ± 0.000005
test f (0.16908) = 0.00021
f (0.16909) = -0.000036426
There is a change of sign, hence the root does lie in the interval [0.16908 , 0.16909]
Graphical Illustration
Rearrangement Method (Fixed Point Iteration): -
Solve 2x3 – 5x +1 = 0
Re-arrange this so that “X=…”
2x3 + 1 = 5x
2x3 + 1 = x
5
g(x) = x
Sketch: -
Graph shows that g(x) has 3 roots, which lie at the following intervals [-2,-1] [0,1] [1,2]
Finding root interval between [0,1]
Start X1 = 0 Use Xn+1 = 2xn3 + 1
5
X2 = 2(0)3 + 1 = 0.2
5
X3 = 2(0.2)3 + 1 = 0.216
5
X4 = 2(0.216)3 + 1 = 0.22016
5
X5 = 0.22134
X6= 0.22169
X7 = 0.22179
X8 = 0.22182
X9 = 0.22183 root to 5 s.f (took 9 iterations)
X10 = 0.22183
Failure: -
Solve 2x3 – 5x + 1 = 0
Re-arrange this so that “X=….”
2x3 +1 = 5x
2x3 = 5x – 1
x3 = 5x – 1
2
x = (5x – 1)⅓
2
x = g(x)
Sketch : -