Question

Apply newton raphson method to find root of x^4 - x - 10

Answer 1

Answer:

You need to use a computer or calculator for the computational step.

You could use a spreadsheet.  Some calculators have functionality to make this sort of thing easy, too.  I used python for my numbers.

Here's the set up and the results.

Newton-Raphson method:

1) Pick some value to be the first estimate

2) Create an "improved" estimate according to the rule

x_{n+1} = xₙ - f ( xₙ ) / f' ( xₙ )

3) Keep going until the estimates aren't changing much.

Here

f(x) = x⁴ - x - 10  =>  f' (x) = 4x³ - 1

So the rule is

x_{n+1} = xₙ - ( xₙ⁴ - xₙ - 10 ) / ( 4xₙ³ - 1 )

Especially if using a calculator, might be best to simplify this expression to reduce the number of button presses needed.  Doing that, we get

x_{n+1} = ( 3xₙ⁴ + 10 ) / ( 4xₙ³ - 1 ).

Now pick a starting value.

Let's pick x₀ = 0.

Use the rule above to get x₁ = ( 0 + 10 ) / ( 4 - 1 ) = -10.

And so on...

x₂ ≈ -7.500625

x₃ ≈ -5.628059

x₄ ≈ -4.229136

...

x_11 ≈ -1.697471

x_12 ≈ -1.697471

Not changing anymore.  Here is one root.

Repeat with a different starting position.

x₀ = 1

x₁ = ( 3 + 10 ) / ( 4 - 1 ) ≈ 4.333333

x₂ = 3.290834

...

x₈ ≈ 1.855585

x₉ ≈ 1.855585

Not changing anymore.  Here is another root.

Repeating with other starting points keeps settling in to these two roots.  No others turn up  (just as well, because there are not other real roots!).