Numerical Methods Lab - Shakil Anower Samrat

Mymensingh Engineering College
Department of Computer Science & Engineering
Course Name: Numerical Methods Lab
Course Code: CSE-3212
Submitted by:
Name: Md. Shakil Anower Samrat
Roll: 307
Reg No: 2850
Batch: CSE-3rd
Session: 2019-20
Submitted to:
Bidhan Krisna Roy
Lecture, Department of CSE
Mymensingh Engineering College
Submission Date:
Signature:

Index
Course: Numerical Methods Lab
Code: CSE-3212
SL Experiment Name Date Signature
1 Study Of Bisection Method
2 Study of False Position Method
3 Study of Newton Raphson
Method
4 Study of Secant Method
5 Study of the Gauss Elimination
Method
6 Study of the Linear Regression
Method
7 Study of Trapezoidal Rule
8 Study of the Simpson’s Rule
9 Study of the Runge Kutta
Method
Experiment No. 1

Experiment Name: Study of Bisection Method.
Objective: The major goal of this exercise is to examine and comprehend the
Bisection Method, a numerical approach for determining the roots of a real-valued
function. Participants in this study want to acquire knowledge into the method's iterative
character, investigate its convergence behavior, and improve their ability to use the
Bisection Method to solve basic equations.
Theory: The Bisection Method is a root-finding algorithm that operates on
continuous functions. Given an interval [a, b] where the function f(x) changes sign, the
Bisection Method iteratively narrows down the interval containing the root until a
sufficiently accurate approximation is obtained. The steps of the method are as follows:
Algorithm: Bisection method can be easily programmed using the following
computational method:
1. Initial Interval:
 Start with an initial interval [a, b] where f(a) and f(b) have opposite signs,
indicating that the root lies within the interval.
2. Midpoint Calculation:
 Calculate the midpoint c of the interval: c=
a+b
2
3. Sign Analysis:
 Determine the sign of f(c) and replace either a or b with c such that the
new interval [a, b] still contains the root.
4. Iteration:
 Repeat the process iteratively, narrowing down the interval in each step
until the desired level of accuracy is achieved.
Code/Program:

Discussion:
Advantages of the bisection method:
1) The bisection approach converges in all cases. The method is guaranteed to
converge because it brackets the root.
1) The interval is cut in half as the iterations are carried out. Thus, it is possible to
ensure that the equation's answer contains an error.
Disadvantages of bisection method:
1) Because the bisection approach is reliant on half the gap, its convergence is
sluggish.
2) It will require more iterations to get to the root if one of the first estimates is closer to
the actual root.

Experiment No. 2
Experiment Name: Study of False Position Method.
Objective: The primary objective of this experiment is to investigate and comprehend
the False Position Method as a numerical technique for finding the roots of a real-
valued function. The experiment aims to provide participants with a practical
understanding of the method's iterative process, convergence characteristics, and its
application to solve equations.
Theory: The False Position Method, also known as the Regula Falsi method, is an
iterative root-finding algorithm similar to the Bisection Method, it operates on an interval
[a, b] where the function f(x) changes sign. The steps of the method include:
Algorithm: False Position method can be programmed using the following
computational steps:
1. Initial Interval:
 Begin with an initial interval [a,b] where f(a) and f(b) have opposite signs,
indicating the presence of a root in the interval.
2. Linear Interpolation:
 Interpolate the next approximation c using linear interpolation based on
the points (a, f(a)) and (b, f(b)):
 c=
af (b)−bf (a)
f (b)−f (a)
3. Sign Analysis:
 Determine the sign of f(c) and update the interval [a, b] such that it still
contains the root.
4. Iteration:
 Repeat the process iteratively, adjusting the interval and updating the
approximation until the desired level of accuracy is achieved.
The False Position Method offers faster convergence than the Bisection Method, but
careful consideration of the function's behaviour is required to avoid convergence
issues.

Discussion:
Advantages of False Position Method:
1) The computation of the derivative is not necessary.
2) This approach is linearly convergent, meaning it has a first order rate of convergence.
It converges in all cases.
Disadvantages of the False Position Method:
1) Since this method relies on trial and error, it can occasionally take a long time to
compute the correct root, which slows down the procedure.
2) It is applied to the computation of a single equation's unknown.

Experiment No. 3
Experiment Name: Study of Newton Raphson Method.
Objective: The primary objective of this experiment is to comprehensively study the
Newton-Raphson Method as a numerical technique for finding the roots of a real-
valued function. Participants aim to gain a practical understanding of the method's
iterative nature, convergence characteristics, and the factors influencing its
effectiveness in solving equations.
Theory: The Newton-Raphson Method is an iterative root-finding algorithm that
utilizes the derivative of a real-valued function to approximate its roots. Given an initial
guess 0
𝑥 x0, the method iteratively refines the approximation using the formula:
xn+1 = xn −
f (xn)
f '(xn)
where:
 xn+1 is the next approximation of the root.
 xn is the current approximation of the root.
 f ¿) is the function value at xn.
 f ' (xn) is the derivative of the function at xn.
Algorithm: Newton Raphson method can be programmed using the following
computational steps:
Step 1: Find points a & b, such that a<b and f (a)⋅ f (b)<0.
Step 2: Take the interval [a,b] and find next value x0=
a+b
2
.
Step 3: Find f (x0) & f '
( x¿¿0)¿. Then determine,
x1=x0−
f (x0)
f
'
(x0)
Step 4: If f (x1)=0, then x1 is an exact root;
Else x0=x1.
Step 5: Repeat steps 3 & 4 until f (xi)=0 or, |f (xi)|≤accuracy.

Discussion:
Advantages of Newton Raphson Method:
1) One of the quickest approaches that gets to the root immediately.
2) Converges quadratically on the root, meaning that the rate of convergence is 2.
3) The number of significant digits roughly doubles with each step as we get closer to
the root.
4) Simple to translate into various dimensions.
Drawbacks of Newton Raphson Method:
1) Newton Raphson's slow convergence rate and potential for hundreds of repetitions
around a critical point are its primary drawbacks.
2) To apply this strategy, we need to find the derivative.
3) Unsatisfactory features of global convergence.

Experiment No: 4
Experiment Name: Study of Secant Method.
Objective:
The primary objective of this experiment is to conduct a comprehensive study of the
Secant Method as a numerical technique for finding the roots of a real-valued
function. Participants aim to gain practical insights into the method's iterative process,
convergence characteristics, and its application to solving equations, particularly in
cases where the derivative is not readily available.
Theory:
The Secant Method is an iterative root-finding algorithm that approximates the roots of a
real-valued function using a finite difference quotient. Given two initial guesses xn−1
and xn the method iteratively refines the approximation using the formula:
xn+1=xn −
f (xn).(xn−xn−1 )
f (xn)−f (xn−1)
where:
 xn+1 is the next approximation of the root.
 xn andxn−1 are the current and previous approximations of the root.
 and f (xn) and f (xn− 1) are the function values at xnand xn−1, respectively.
Algorithm: Second method can be easily programmed using the following
computational method:
1. Input:
 f(x): The function.
 xn−1and xn Initial guesses.
 TOL: Tolerance.
2. Iteration:
 While ∣ f (xn) >
∣ TOL:
 Calculate xn+1=xn −
f (xn).(xn−xn−1 )
f (xn)−f (xn−1)
 Update xn−1=¿ xn , xn=¿ xn+1
 Increment n

3. Output:
 Final xn when ∣ f (xn) ≤
∣ TOL or after a set number of iterations.
Code/Program:
Output:

Discussion:
Advantages of Secant Method:
(1) The secant approach converges more quickly than the bisection and regular falsi
methods.
(2) Rather of relying just on the approximations that bind the interval to encompass the
root, it finds additional approximations using the two most recent approximations of the
root.
Disadvantages of Secant Method:
(1) Because there is no guarantee of convergence, we should set a maximum number
of iterations while using this method on a computer. If at any stage of iteration this
method fails.
(3) When using a computer to implement this method, we need set a maximum
restriction on the number of repetitions because convergence is not guaranteed.

Numerical Methods Lab - Shakil Anower Samrat

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