Comparison of Methods

This section compares the three numerical methods used in this project — Bisection, Newton–Raphson, and Secant — based on how they find the root, their rate of convergence, and overall behavior.
Although all three methods aim to solve ( f(x) = 0 ), they differ in how the next value ( x_{i+1} ) is calculated from previous iterations.

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5.1 Bisection Method

• The Bisection Method starts with two values ( a ) and ( b ) such that ( f(a) \times f(b) < 0 ).
• The midpoint is found by:
  $$ x_{i+1} = \frac{a + b}{2} $$
• The new interval is then selected based on the sign of ( f(x_{i+1}) ).
• This process repeats until the difference between ( a ) and ( b ) becomes very small.

Main Points:
- Converges slowly but always works if ( f(a)f(b) < 0 ).
- Does not need derivatives.
- Each step cuts the interval in half.

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5.2 Newton–Raphson Method

• The Newton–Raphson Method uses the tangent line to find where the function crosses the x-axis.
• Starting from an initial guess ( x_0 ), it applies the formula:
  $$ x_{i+1} = x_i - \frac{f(x_i)}{f'(x_i)} $$
• The next value ( x_{i+1} ) is where the tangent line at ( (x_i, f(x_i)) ) meets the x-axis.

Main Points:
- Very fast convergence if the first guess is close to the actual root.
- Needs the first derivative ( f'(x) ).
- Might diverge if ( f'(x_i) = 0 ) or if the first guess is poor.

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5.3 Secant Method

• The Secant Method is similar to Newton–Raphson but uses two previous points instead of a derivative.
• The formula is:
  $$ x_{i+1} = x_i - f(x_i) \frac{x_i - x_{i-1}}{f(x_i) - f(x_{i-1})} $$
• It uses the slope of the secant line between ( (x_{i-1}, f(x_{i-1})) ) and ( (x_i, f(x_i)) ).

Main Points:
- Faster than Bisection but a bit slower than Newton–Raphson.
- No derivative is required.
- Can fail if ( f(x_i) ) and ( f(x_{i-1}) ) are nearly the same.

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5.4 Summary of Comparison

Feature	Bisection	Newton–Raphson	Secant
Derivative Required	No	Yes	No
Initial Guesses	Two points ( a, b )	One point ( x_0 )	Two points ( x_0, x_1 )
Formula	( x_{i+1} = \frac{a + b}{2} )	( x_{i+1} = x_i - \frac{f(x_i)}{f'(x_i)} )	( x_{i+1} = x_i - f(x_i) \frac{x_i - x_{i-1}}{f(x_i) - f(x_{i-1})} )
Convergence Speed	Slow	Fast	Moderate
Stability	Always convergent	May diverge	May diverge
Ease of Use	Very Easy	Moderate	Easy

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5.5 Final Remarks

• The Bisection Method is the safest and always converges but is the slowest.
• The Newton–Raphson Method is the fastest and most accurate if the derivative is available.
• The Secant Method provides a balance — it is faster than Bisection and simpler than Newton–Raphson.

Overall, the Newton–Raphson method performed best in this project because it reached the correct root in fewer iterations, while all three methods produced the same final result (x = 2).