Newton-Raphson Method

The Newton-Raphson method is a powerful second-order root-finding algorithm that uses the tangent line of a function $f(x)$ at an initial guess $x_0$ to find successively better approximations of a root.

Formula for Newton-Raphson Method:

x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}

Where:

$x_n$ is the current approximation of the root.
$f(x_n)$ is the value of the function at $x_n$ .
$f'(x_n)$ is the derivative of the function at $x_n$ (must be non-zero).

Example

Given Function

f(x) = x^2 - 5 = 0

Initial Guess $x_0 = 2$

Step 1: Compute Derivative

f'(x) = 2x

Step 2: First Iteration ( $x_1$ )

f(2) = 2^2 - 5 = -1, \quad f'(2) = 2(2) = 4

x_1 = 2 - \frac{-1}{4} = 2 + 0.25 = 2.25

Step 3: Second Iteration ( $x_2$ )

f(2.25) = (2.25)^2 - 5 = 0.0625, \quad f'(2.25) = 4.5

x_2 = 2.25 - \frac{0.0625}{4.5} \approx 2.2361

Conclusion

The Newton-Raphson Method rapidly converges to $\sqrt{5} \approx 2.236068$ in just a few iterations.

Newton-Raphson Method Solver

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Newton-Raphson Method

Formula for Newton-Raphson Method:

x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}

Where:

$x_n$ is the current approximation of the root.
$f(x_n)$ is the value of the function at $x_n$ .
$f'(x_n)$ is the derivative of the function at $x_n$ (must be non-zero).

Example

Given Function

f(x) = x^2 - 5 = 0

Initial Guess $x_0 = 2$

Step 1: Compute Derivative

f'(x) = 2x

Step 2: First Iteration ( $x_1$ )

f(2) = 2^2 - 5 = -1, \quad f'(2) = 2(2) = 4

x_1 = 2 - \frac{-1}{4} = 2 + 0.25 = 2.25

Step 3: Second Iteration ( $x_2$ )

f(2.25) = (2.25)^2 - 5 = 0.0625, \quad f'(2.25) = 4.5

x_2 = 2.25 - \frac{0.0625}{4.5} \approx 2.2361

Conclusion

The Newton-Raphson Method rapidly converges to $\sqrt{5} \approx 2.236068$ in just a few iterations.

Newton-Raphson Method Solver

PreviousIteration Method

Unit 1•4 of 28

NextGauss-Seidel Method

Newton-Raphson Method

Example

Step 1: Compute Derivative

Step 2: First Iteration (x1x_1x1​)

Step 3: Second Iteration (x2x_2x2​)

Conclusion

Newton-Raphson Method Solver

NETZ Engine

Newton-Raphson Method

Example

Step 1: Compute Derivative

Step 2: First Iteration (x1x_1x1​)

Step 3: Second Iteration (x2x_2x2​)

Conclusion

Newton-Raphson Method Solver

Step 2: First Iteration ( $x_1$ )

Step 3: Second Iteration ( $x_2$ )

Step 2: First Iteration ( $x_1$ )

Step 3: Second Iteration ( $x_2$ )