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Study · Background & methods

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.

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.

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.

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.

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