In one line
Newton–Raphson improves a guess \(x_n\) by sliding down the tangent line to the \(x\)-axis: \(x_{n+1}=x_n-\dfrac{f(x_n)}{f'(x_n)}\).
Explained simply
Imagine you are standing on a curved slide in the dark and want to reach the ground (where \(f=0\)). You feel the slope under your feet and assume it stays the same all the way down. You jump to where that straight slope would hit the ground. You are not exactly at the root, but much closer. Repeat, and within a few jumps you are there.
Technically, the tangent at \((x_n,f(x_n))\) is \(y-f(x_n)=f'(x_n)(x-x_n)\). Setting \(y=0\) gives the next approximation \(x_{n+1}=x_n-\dfrac{f(x_n)}{f'(x_n)}\). Near a simple root the method converges quadratically: the number of correct decimals roughly doubles each step.
How it works
- Write \(f(x)\) and \(f'(x)\).
- Locate the root and choose \(x_0\) close to it.
- Compute \(x_1,x_2,\dots\) using the formula.
- Stop when two successive values agree to the required decimal places.
- Write the root.
Key formula
\(x_{n+1}=x_n-\dfrac{f(x_n)}{f'(x_n)}\), \(n=0,1,2,\dots\), where \(f'(x_n)\ne0\).
Quick example
Solve \(x^3-2x-5=0\). \(f(2)=-1\), \(f(3)=16\): root in \((2,3)\), close to 2. Take \(x_0=2\). \(f'(x)=3x^2-2\).
| \(n\) | \(x_n\) | \(f(x_n)\) | \(f'(x_n)\) | \(x_{n+1}\) |
|---|---|---|---|---|
| 0 | 2 | \(-1\) | 10 | 2.1 |
| 1 | 2.1 | 0.061 | 11.23 | 2.094568 |
| 2 | 2.094568 | ≈0.000186 | ≈11.1616 | 2.094551 |
Root \(\approx2.0946\) (correct to four decimal places) after only three steps.
Common mistakes
- Starting where \(f'(x_0)\approx0\) (flat tangent); the next value shoots far away.
- Using degrees for trig functions.
- Stopping after one iteration, or rounding too early so successive values never agree.
How MU asks it
Very common 5/6-mark question: "Find a real root of \(x^3-x-1=0\) / \(x\log_{10}x=1.2\) / \(\cos x=xe^x\) by Newton–Raphson method correct to three decimal places". It is also a SCILAB tutorial. For full marks: show \(f\), \(f'\), the sign table, the formula, a neat iteration table, and the final root with the stated accuracy.