Find `root(3)(48)` using False Position method (regula falsi method)
Solution:
Let `x=48^(1/3)`
`:.x^3=48`
`:.x^3-48=0`
i.e. `f(x)=x^3-48`
Here
| `x` | 0 | 1 | 2 | 3 | 4 |
|---|
| `f(x)` | -48 | -47 | -40 | -21 | 16 |
|---|
`1^(st)` iteration :
Here `f(3) = -21 < 0` and `f(4) = 16 > 0`
`:.` Now, Root lies between `x_0 = 3` and `x_1 = 4`
`x_2 = x_0 - f(x_0) * (x_1 - x_0)/(f(x_1) - f(x_0))`
`x_2=3 - (-21) * (4 - 3)/(16 - (-21))`
`x_2=3.5676`
`f(x_2)=f(3.5676)=3.5676^3-48=-2.5936 < 0`
`2^(nd)` iteration :
Here `f(3.5676) = -2.5936 < 0` and `f(4) = 16 > 0`
`:.` Now, Root lies between `x_0 = 3.5676` and `x_1 = 4`
`x_3 = x_0 - f(x_0) * (x_1 - x_0)/(f(x_1) - f(x_0))`
`x_3=3.5676 - (-2.5936) * (4 - 3.5676)/(16 - (-2.5936))`
`x_3=3.6279`
`f(x_3)=f(3.6279)=3.6279^3-48=-0.2513 < 0`
`3^(rd)` iteration :
Here `f(3.6279) = -0.2513 < 0` and `f(4) = 16 > 0`
`:.` Now, Root lies between `x_0 = 3.6279` and `x_1 = 4`
`x_4 = x_0 - f(x_0) * (x_1 - x_0)/(f(x_1) - f(x_0))`
`x_4=3.6279 - (-0.2513) * (4 - 3.6279)/(16 - (-0.2513))`
`x_4=3.6336`
`f(x_4)=f(3.6336)=3.6336^3-48=-0.0237 < 0`
`4^(th)` iteration :
Here `f(3.6336) = -0.0237 < 0` and `f(4) = 16 > 0`
`:.` Now, Root lies between `x_0 = 3.6336` and `x_1 = 4`
`x_5 = x_0 - f(x_0) * (x_1 - x_0)/(f(x_1) - f(x_0))`
`x_5=3.6336 - (-0.0237) * (4 - 3.6336)/(16 - (-0.0237))`
`x_5=3.6342`
`f(x_5)=f(3.6342)=3.6342^3-48=-0.0022 < 0`
`5^(th)` iteration :
Here `f(3.6342) = -0.0022 < 0` and `f(4) = 16 > 0`
`:.` Now, Root lies between `x_0 = 3.6342` and `x_1 = 4`
`x_6 = x_0 - f(x_0) * (x_1 - x_0)/(f(x_1) - f(x_0))`
`x_6=3.6342 - (-0.0022) * (4 - 3.6342)/(16 - (-0.0022))`
`x_6=3.6342`
`f(x_6)=f(3.6342)=3.6342^3-48=-0.0002 < 0`
Approximate root of the equation `x^3-48=0` using False Position method is `3.6342` (After 5 iterations)
| `n` | `x_0` | `f(x_0)` | `x_1` | `f(x_1)` | `x_2` | `f(x_2)` | Update |
| 1 | 3 | -21 | 4 | 16 | 3.5676 | -2.5936 | `x_0 = x_2` |
| 2 | 3.5676 | -2.5936 | 4 | 16 | 3.6279 | -0.2513 | `x_0 = x_2` |
| 3 | 3.6279 | -0.2513 | 4 | 16 | 3.6336 | -0.0237 | `x_0 = x_2` |
| 4 | 3.6336 | -0.0237 | 4 | 16 | 3.6342 | -0.0022 | `x_0 = x_2` |
| 5 | 3.6342 | -0.0022 | 4 | 16 | 3.6342 | -0.0002 | `x_0 = x_2` |
This material is intended as a summary. Use your textbook for detail explanation.
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