CME

1. Bisection Method
1What is bisection method?/statement
Bisection method is based on the intermediate value theorem,
which states that if a continuous function f(x) changes sign between
two points a and b, then there exists a point c between them such
that f(c) is zero. The method works by dividing the interval [a,b] into
two equal subintervals, and checking which one contains the root.
Then, it repeats the process with the chosen subinterval, until the
desired accuracy is reached.
.
Bisection Method
Formula is: x2 = (x0 + x1) / 2
Bisection Method Procedure
To solve bisection method problems, given below is the step-by-step
explanation of the working of the bisection method algorithm for a given function
f(x):

2. Step 1: Choose two values, a and b such that f(a) > 0 and f(b) < 0 .
Step 2: Calculate a midpoint c as the arithmetic mean between a and b such
that c = (a + b) / 2. This is called interval halving.
Step 3: Evaluate the function f for the value of c.
Step 4: The root of the function is found only if the value of f(c) = 0.
Step 5: If (c) ≠ 0, then we need to check the sign:
a. we replace a with c if f(c) has the same sign as f(a) and we
keep the same value for b
b. we replace b with c if f(c) has the same sign as f(b), and we
keep the same value for a
To get the right value with the new value of a or b, we go back to step 2 And
recalculate c.
Advantages of Bisection Method
1. Convergence is guarenteed: Bisection method is bracketing
method and it is always convergent.
2. Error can be controlled: In Bisection method, increasing number
of iteration always yields more accurate root.
3. Does not involve complex calculations: Bisection method does
not require any complex calculations. To perform Bisection
method, all we need is to calculate average of two numbers.
4. Guaranteed error bound: In this method, there is a guaranteed
error bound, and it decreases with each successive iteration. The
error bound decreases by ½ with each iteration.
5. Bisection method is very simple and easy to program in
computer.
6. Bisection method is fast in case of multiple roots.
Disadvantages of Bisection Method
1. Slow Rate of Convergence: Although convergence of
Bisection method is guaranteed, it is generally slow.

3. 2. Choosing one guess close to root has no
advantage: Choosing one guess close to the root may result
in requiring many iterations to converge.
3. Can not find root of some equations. For example: f(x) =
x2 as there are no bracketing values.
4. It has linear rate of convergence.
5. It fails to determine complex roots.
6. It can not be applied if there are discontinuities in the guess
interval.
7. It can not be applied over an interval where the function takes
values of the same sign.
Q: How to compare bisection method with
other numerical methods?
When searching for the root of a function, there are many numerical
methods to choose from, such as Newton's method, secant method,
regula falsi method, and fixed point iteration method. The best
choice depends on the nature and behavior of the function, the
accuracy of the initial guess and derivative, if needed, the desired
precision and speed of convergence, and the computational cost
and complexity of the method. To compare bisection method with
other numerical methods, criteria like the number of iterations and
evaluations required to reach a certain accuracy or tolerance, the
rate of convergence and order of convergence, and the error
analysis and bounds can be used. These can help determine how
close the approximate solution is to the exact solution and how
much it can vary due to errors.

4. Comparison between Bisection Method and Newton Raphson
Method
Sr.
No. Bisection Method Newton Raphson Method
1.
In the Bisection Method, the rate of
convergence is linear thus it is slow.
In the Newton Raphson method, the
rate of convergence is second-order
or quadratic.
2.
In Bisection Method we used following
formula
x2= (x0 + x1) / 2
In Newton Raphson method we
used following formula
x1 = x0 – f(x0)/f'(x0)
3.
In this method, we take two initial
approximations of the root in which the
root is expected to lie.
In this method, we take one initial
approximation of the root.
4.
The computation of function per
iteration is 1.
The computation of function per
iteration is 2.
5.
The initial approximation is less
sensitive.
The initial approximation is very
sensitive.
6.
In the Bisection Method, there is no
need to find derivatives.
In the Newton Raphson method,
there is a need to find derivatives.
7.
This method is not applicable for
finding complex, multiple, and nearly
equal two roots.
This method is applicable for
finding complex, multiple, and
nearly equal two roots.
8. Accuracy is less than Newton raphson Accuracy is more than bisection
Advantages of Newton Raphson Method
o The method has one of the fastest convergences to the
root.
o It is easy to program as it has a simple formula

5. o It is used to further improve a root found by other
methods.
o It requires only one guess to find the root.
o 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.
Differences between Bisection Method and Regula False Method
Basis Bisection Method Regula Falsi Method
Definition
In mathematics, the bisection
method is a root-finding
method that applies to
continuous function for which
knows two values with opposite
signs.
In mathematics, the false
position method is a very old
method for solving equations
with one unknown this method
is modified form is still in use.
Simplicity
it is simple to use and easy to
implement.
Simple to use as compared to
Bisection Method
Computational
Efforts
Less as compared to Regula
Falsi Method
More as compared to Bisection
Method
Iteration
required
In the bisection method, if one
of the initial guesses is closer to
the root, it will take a large
number of iterations to reach
the root.
Less as compared to Bisection
Method. This method can be
less precise than bisection – no
strict precision is guaranteed.
Convergence
The order of convergence of
the bisection method is slow
and linear.
This method faster order of
convergence than the bisection
method.

6. Basis Bisection Method Regula Falsi Method
General
Iterative
Formula
Formula is : X3 =( X1 + X2)/2
Formula is : X3 = X1(fx2) –
x2(fx1)/ f(x2) -f(x1)
Other Names
It is also known as the Bolzano
method, Binary chopping
method, half Interval method.
It is also known as the False
Position method.
Advantages of Secant Method:
 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 of Secant Method:
 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.