In many engineering and scientific problems, we often encounter equations that cannot be solved analytically or directly using algebraic techniques. These equations may be either algebraic (involving polynomial expressions) or transcendental (involving trigonometric, exponential, or logarithmic functions). To find their roots (solutions), numerical methods become essential tools. This topic introduces various iterative techniques to approximate the roots of such equations with desired accuracy. These methods are particularly useful when exact solutions are difficult or impossible to find.

1. Algebraic Equations
2. Equations formed using algebraic operations (addition, subtraction, multiplication, division, and exponentiation with rational numbers).
3. Example: 𝑥³− 4𝑥 + 1 = 0
4. Transcendental Equations
5. Equations involving transcendental functions like sin(x), log(x), or e^x.
6. Example: 𝑒ˣ = 3𝑥, 𝑥sin(𝑥) = 1

• Principle: Repeatedly bisect the interval [𝑎,𝑏] where the function changes sign, and narrow down the root.
• Condition: Function 𝑓(𝑥) must be continuous in [𝑎,𝑏] and 𝑓(𝑎)𝑓(𝑏) < 0.
• Formula:
𝑥ₘ𝑖𝑑 = \( \frac{𝑎 + 𝑏}{2} \)
• Steps:
a. Compute 𝑓(𝑎) and 𝑓(𝑏)
b. Check if root lies between 𝑎 and 𝑥ₘ𝑖𝑑 or 𝑥ₘ𝑖𝑑 and 𝑏
c. Repeat until desired accuracy.
• Pros: Simple and reliable
• Cons: Slow convergence

• Principle: Uses linear interpolation between two points to estimate the root.
• Formula:
𝑥 = \( \frac{𝑎𝑓(𝑏)−𝑏𝑓(𝑎)}{𝑓(𝑏)−𝑓(𝑎)} \)
• Improvement over Bisection: Approximates root more intelligently using the function values.
• Steps:
a. Select 𝑎 and 𝑏 such that 𝑓(𝑎)𝑓(𝑏) < 0.
b. Calculate new root using the formula.
c. Replace the interval based on the sign of 𝑓(𝑥).

• Principle: Uses tangents to approximate root.
• Formula:
𝑥ₙ₊₁ = 𝑥ₙ - \( \frac{𝑓(𝑥ₙ)}{𝑓′(𝑥ₙ)} \)
• Steps:
a. Choose an initial guess 𝑥₀.
b. Evaluate 𝑓(𝑥₀) and 𝑓′(𝑥₀).
c. Update 𝑥 iteratively.
• Pros: Fast convergence
• Cons: Requires derivative; fails if 𝑓′(𝑥) is zero or very small.

• Principle: Similar to Newton-Raphson but doesn't require derivative.
• Formula:
𝑥ₙ₊₁ = 𝑥ₙ - \( 𝑓(𝑥ₙ)\frac{𝑥ₙ - 𝑥ₙ₋₁}{𝑓(𝑥ₙ)−𝑓(𝑥ₙ₋₁)} \)
• Pros: Doesn’t require 𝑓′(𝑥).
• Cons: Requires two initial guesses.

• Form: Rearrange the equation into 𝑥 = 𝑔(𝑥).
• Formula:
𝑥ₙ₊₁ = 𝑔(𝑥ₙ).
• Condition: |g'(x)| < 1 for convergence.
• Pros: Easy implementation.
• Cons: May diverge if not properly chosen.

Initial Derivative
Method Guess Required Speed Reliability
Bisection Two No Slow Always converges
Regula Falsi Two No Faster than Sometimes slow Bisection
Newton-Raphson One Yes Very Fast May fail
Secant Two No Fast Less stable
Fixed Point One No Depends on Can diverge function

Iteration is stopped when any of the following are satisfied:
• |f(xₙ)| < 𝜖 (function value is close to 0)
• |xₙ - xₙ₋₁| < 𝜖 (change in root is small)
• Fixed number of iterations reached

• Solving complex circuit equations
• Engineering simulations
• Structural analysis
• Optimization problems
• Fluid dynamics and heat transfer models

• Algebraic and transcendental equations often arise in engineering but are not always solvable by analytical methods.
• Numerical techniques provide approximate but efficient solutions to these equations.
• Methods like Bisection, Regula Falsi, Newton-Raphson, Secant, and Fixed Point Iteration each have their own use-cases.
• The choice of method depends on the nature of the function, required accuracy, and available information (e.g., derivative).