Methods

This section explains the three numerical methods used in this project to find the root of a nonlinear equation: Bisection, Newton–Raphson, and Secant.
Each method improves an initial guess step by step until the value of ( x ) becomes close enough to the actual root within a chosen tolerance.

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1. Bisection Method

The Bisection Method is a simple and reliable way to find a root by dividing an interval in half repeatedly and choosing the part where the function changes sign.
It is based on the Intermediate Value Theorem, which states that if ( f(a) ) and ( f(b) ) have opposite signs, then there is at least one root between them.

Procedure:
1. Choose two values ( a ) and ( b ) such that ( f(a) \times f(b) < 0 ).
2. Find the midpoint:

$$ x_{i+1} = \frac{a + b}{2} $$

1. Check the sign of ( f(x_{i+1}) ):
2. If ( f(a) \times f(x_{i+1}) < 0 ), set ( b = x_{i+1} ).
3. Otherwise, set ( a = x_{i+1} ).
4. Continue until the interval ( |b - a| ) is smaller than the chosen tolerance ( \varepsilon ).

Stopping Criterion:

$$ |f(x_{i+1})| < \varepsilon $$

Advantages:
- Always converges if ( f(a)f(b) < 0 ).
- Simple and easy to apply.

Limitations:
- Convergence is slow.
- Requires that the initial interval contains a sign change.

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2. Newton–Raphson Method

The Newton–Raphson Method uses the tangent line at a point to find where the function crosses the x-axis.
It requires the derivative ( f'(x) ) and a good initial guess ( x_0 ).

Formula:

$$ x_{i+1} = x_i - \frac{f(x_i)}{f'(x_i)} $$

Procedure:
1. Choose a starting value ( x_0 ).
2. Use the formula to calculate a new value ( x_{i+1} ).
3. Repeat until ( |x_{i+1} - x_i| < \varepsilon ).

Advantages:
- Very fast (quadratic) convergence if the first guess is close to the root.
- Needs fewer iterations than other methods.

Limitations:
- Requires the derivative ( f'(x) ).
- May fail if ( f'(x_i) = 0 ) or if the first guess is not near the root.

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3. Secant Method

The Secant Method is similar to Newton–Raphson but does not require a derivative.
Instead, it estimates the slope using two recent points.

Formula:

$$ x_{i+1} = x_i - f(x_i) \frac{x_i - x_{i-1}}{f(x_i) - f(x_{i-1})} $$

Procedure:
1. Choose two initial guesses ( x_0 ) and ( x_1 ).
2. Use the formula to find ( x_{i+1} ).
3. Repeat until ( |x_{i+1} - x_i| < \varepsilon ).

Advantages:
- Faster than Bisection and does not need the derivative.
- Easier to apply than Newton–Raphson.

Limitations:
- May fail if ( f(x_i) ) and ( f(x_{i-1}) ) are almost the same.
- Convergence is slower than Newton–Raphson.

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Summary of Methods

Method	Requires Derivative	Convergence Speed	Reliability	Notes
Bisection	No	Slow	Always convergent	Simple and reliable
Newton–Raphson	Yes	Fast	Depends on initial guess	Needs ( f'(x) )
Secant	No	Moderate	Depends on initial guesses	Approximates derivative