Newton's Method Write A Program to Calculate Square Roots

What is Newton's Method?

Newton's Method is a root-finding algorithm which produces successively better approximations to the roots of a real-valued function. The method uses the idea of linear approximation (using the tangent line to the function at a given point) and repeated linear interpolation to find successively better approximations.

For finding square roots, we can define the function as f(x) = x² - a, where 'a' is the number we want to find the square root of. The derivative f'(x) = 2x.

How to Calculate Square Roots

To calculate square roots using Newton's Method:

1. Choose an initial guess for the square root (x₀). A common choice is x₀ = a/2.
2. Apply the Newton's Method formula repeatedly until the result converges to a desired precision.
3. The iteration stops when the difference between successive approximations is smaller than a specified tolerance.

JavaScript Implementation

Here's a JavaScript function that implements Newton's Method to calculate square roots:

function newtonSquareRoot(a, tolerance = 1e-10, maxIterations = 100) {
    if (a < 0) {
        throw new Error("Cannot calculate square root of negative number");
    }

    if (a === 0) {
        return 0;
    }

    let x = a / 2; // Initial guess
    let previousX;

    for (let i = 0; i < maxIterations; i++) {
        previousX = x;
        x = x - (x * x - a) / (2 * x);

        if (Math.abs(x - previousX) < tolerance) {
            return x;
        }
    }

    throw new Error("Failed to converge within maximum iterations");
}

The function takes three parameters:

• a: The number to find the square root of
• tolerance: The desired precision (default: 1e-10)
• maxIterations: Maximum number of iterations (default: 100)

The function returns the square root of 'a' with the specified precision or throws an error if convergence fails.

Example Calculation

Let's calculate the square root of 25 using Newton's Method:

1. Initial guess: x₀ = 25 / 2 = 12.5
2. First iteration: x₁ = 12.5 - (12.5² - 25) / (2 * 12.5) = 12.5 - (156.25 - 25) / 25 = 12.5 - 131.25 / 25 ≈ 12.5 - 5.251 ≈ 7.249
3. Second iteration: x₂ = 7.249 - (7.249² - 25) / (2 * 7.249) ≈ 7.249 - (52.5 - 25) / 14.498 ≈ 7.249 - 17.5 / 14.498 ≈ 7.249 - 1.209 ≈ 6.040
4. Third iteration: x₃ ≈ 6.040 - (6.040² - 25) / (2 * 6.040) ≈ 6.040 - (36.48 - 25) / 12.08 ≈ 6.040 - 6.88 / 12.08 ≈ 6.040 - 0.569 ≈ 5.471
5. Fourth iteration: x₄ ≈ 5.471 - (5.471² - 25) / (2 * 5.471) ≈ 5.471 - (29.93 - 25) / 10.942 ≈ 5.471 - 2.93 / 10.942 ≈ 5.471 - 0.267 ≈ 5.204
6. Fifth iteration: x₅ ≈ 5.204 - (5.204² - 25) / (2 * 5.204) ≈ 5.204 - (27.07 - 25) / 10.408 ≈ 5.204 - 0.57 / 10.408 ≈ 5.204 - 0.0546 ≈ 5.149

After these iterations, the value converges to approximately 5.000, which is the square root of 25.