Numerical Root Finding Methods Overview
Bisection Method
The bisection method is a numerical technique used to find the roots of a continuous function. It is a type of bracketing method, which means it requires two initial guesses that bracket (i.e., surround) the root. The function must change signs over this interval, indicating that there is at least one root in that interval (by the Intermediate Value Theorem).
Advantages
It’s a simple, straightforward algorithm that’s easy to understand and implement. Its robustness makes it suitable for solving complicated engineering problems. It has definite convergence, ensuring a solution will be found if one exists within the given interval.
Disadvantages
It can be relatively slow, especially when compared with other numerical methods such as the Newton-Raphson method or Secant method. It requires the function to be continuous in the interval of interest, which may not always be the case. It provides an approximate solution and multiple iterations are required for higher precision.
Secant Method
Secant method is also a recursive method for finding the root for the polynomials by successive approximation. It’s similar to the Regular-falsi method but here we don’t need to check f(x1)f(x2) again and again after every approximation. In this method, the neighbourhoods roots are approximated by secant line or chord to the function f(x). It’s also advantageous of this method that we don’t need to differentiate the given function f(x), as we do in Newton-Raphson method.
Secant Method’s Algorithm:
- Start
- Obtain the values of e, the stopping conditions, x0, x1, 𝑥0, 𝑥1 and e.
- Calculate f(x0)𝑓(𝑥0) and f(x1)𝑓(𝑥1)
- Calculate x2=x1-f(x1)x1-x0f(x1)-f(x0)𝑥2=𝑥1-𝑓(𝑥1)𝑥1-𝑥0𝑓(𝑥1)-𝑓(𝑥0)
- Check the x2𝑥2 accuracy. //∣∣x2-x1x2∣∣>e|𝑥2-𝑥1𝑥2|>𝑒 ///x0=x1𝑥0=𝑥1 and x1=x2𝑥1=𝑥2
- Show the source that is needed.
- Stop
Advantages
The speed of convergence of secant method is faster than that of Bisection and Regula falsi method. It uses the two most recent approximations of root to find new approximations, instead of using only those approximations which bound the interval to enclose root
Disadvantages
The Convergence in secant method is not always assured. If at any stage of iteration this method fails. Since convergence is not guaranteed, therefore we should put limit on maximum number of iterations while implementing this method on computer.
Regula-Falsi Method
The regula-falsi method is the oldest method of finding the approximate numerical value of a real root of an equation f(x) = 0. This method is also known as method of false position. The method used to estimate the roots of a polynomial f(x). In this method we suppose that x1 and x2 are two points where f(x1) and f(x2) are of opposite sign. Let f(x1) 2) > 0. Hence the root of the equation f(x) = 0 lies between x1 and x1 and so, f(x1)f(x2)
The Regula Falsi formula 
Find f(x3) is positive or negative. If f(x3) > 0 then root lies between x3 and x1 or if f(x3) 3 and x1 similarly we calculate x4. Proceed in this manner until the desired accurate root is found.
Newton-Raphson Method
The Newton-Raphson method is a powerful and widely used numerical technique for finding approximate solutions to real-valued functions. It is particularly effective for finding roots (or zeros) of a real-valued function ( f(x) ). The method uses an iterative approach and requires an initial guess ( x_0 ).
Method Outline :- Given a function ( f(x) ) and its derivative ( f'(x) ), the Newton-Raphson method follows these steps: 1. Initial Guess: Start with an initial guess ( x_0 ). 2. Iteration: Compute the next approximation using the formula: [ x_{n+1} = x_n – frac{f(x_n)}{f'(x_n)} ] 3. Convergence Check: Repeat step 2 until the change in ( x ) is smaller than a predetermined tolerance level, or until a maximum number of iterations is reached.
Advantages
The method has one of the fastest convergences to the root. It is easy to program as it has a simple formula. It is used to further improve a root found by other methods. It requires only one guess to find the root. Derivation of the method is more intuitive, so it is easier to understand its behaviour, like when it is to converge and when it is to diverge.
Disadvantages
Derivative Required: Needs the first derivative of the function, which can be difficult to compute. Initial Guess Dependent: The choice of the initial guess is crucial; a poor guess can lead to divergence or slow convergence. Convergence Issues: May not converge for poorly behaved functions and can get stuck at non-root stationary points. Complexity for Multivariable Functions: Requires Jacobian matrix calculation and solving linear systems, which can be complex. Numerical Stability: Sensitive to small derivative values, leading to potential numerical instability.