Numerical Analysis

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Satyabrata Sadangi

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Core Courses

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Course Name : Numerical Analysis

Code(Credit) : CUTM1518(3-2-1)

Course Objectives

To understand the limitations of analytical methods and the need for numerical methods and the ability to apply these numerical methods to obtain the approximate solutions to engineering and mathematical problems. Ability to decide and to derive appropriate numerical methods for approximating the solutions of various types of problems in engineering and science and analyze the error incumbent in any such numerical approximation. Ability to report analysis, solution and results in a standard engineering format.

Learning Outcomes

Perform error analysis to select an appropriate numerical model and to estimate errors in numerical solution of a given problem. Derive a variety of numerical algorithms/methods & compare the viability of different approaches to the numerical solutions of various mathematical problems arising in roots of linear and non-linear equations, interpolation and approximation, numerical differentiation and integration, system of linear algebraic equations and differential equations. Analyze and evaluate the accuracy of common numerical methods.

Course Syllabus

Module – I (5hr+6hr+0hr)

Introduction to Numerical Analysis, Concept of Number system: Binary numbers, octal numbers, hexa-decimal numbers. Floating point Arithmetic, K – digit Arithmetic, Dealing with Errors: Relative,absolute,roundoff,truncation.

Practice 1: Calculate the sum 1+1/2+1/3+1/4+------------+1/N. using python

Practice 2: To find the absolute value of an integer using python

Practice 3: Enter 100 integers into an array and sort them in an ascending order using python

Module – II (7hr+8hr+4hr)

Numerical Solution of non-linear equations: method of bisection, Regula-falsi method, Secant method, Newton-Rapshon method, fixed point iteration method.

Practice 4: To find the approximate value of a polynomial using Bisection method.

Practice 5:   To find the approximate value of a polynomial using Newton Rapshon method

Practice 6: To find the approximate value of a polynomial using Secant method

Practice 7: To find the approximate value of a polynomial using Regula Falsi method

Project 1: Numerical Methods for Solving Systems of Nonlinear Equations

Project 2: A Study on Transcendental and Polynomial equations.

Module – III (3hr+2hr+2hr)

Polynomial Interpolation, Existence and uniqueness of interpolating polynomials, Lagrange’s interpolating polynomial.

Practice 8: To find the value of a function using Lagrange interpolation

Project 3: A study on Lagrange interpolation

Module – IV (4hr+2hr+4hr)

Newtons Divided Difference Interpolating Polynomial, Forward and backward Difference operators. Newton’s Forward and backward Difference Interpolation formula

Practice 9: Solving problems on Newton forward/backward difference formula using python

Project 4: A study on Newton’s Divided Difference Interpolation Formula

Project 5: A study to apply Newton forward difference/backward formula on solving numerical problems with MATLAB program

Module – V ( 5hr+4hr+0hr)

Numerical Integration, Newton-cotes rule, Trapezoidal Rule, Simpsons 1/3 rd rule, Simpsons 3/8 th

Practice 10: Solving a problem on trapezoidal rule using python

Practice 11: Solving a problem on Simpson’s 1/3 rd rule using python

Module – VI (2hr+0hr+0hr)

Compound Quadrature rules, Gauss-Legendre Rules,

Module – VII (4hr+2hr+2hr)

Numerical solution of Differential Equation, Euler’s method, Runge-Kutta  4th order method

Practice 12: Solving Problems on 4th order Runge-Kutta method using python

Project 6: A Study on Runge-Kutta method

 

Text Book:

  1. A Course on Numerical Analysis : B. P. Acharya & R. N. Das, Kalyani Publisher

Chapters: 1, 2 (2.1 – 2.3, 2.6 - 2.9), 3 (3.1 to 3.4, 3.6 to 3.8). 6 (6.1- 6.3, 6.5, 6.10, 6.11), 7 (7.1, 7.3,7.7)

 

Reference Books:

  1. K. Jain, S.R.K. Iyengar and R.K. Jain, Numerical Methods For Scientific And Engineering Computation, 6th Ed., New Age International Publisher, India, 2007.
  2. John H. Mathews and Kurtis D. Fink, Numerical Methods Using MatLab, 4th Ed., Phi Learning Private Limited, 2012.

Session 3

Binary numbers, octal numbers, hexa-decimal

https://www.youtube.com/watch?v=5sS7w-CMHkU

Session 4

Floating point Arithmetic, K – digit Arithmetic

https://youtu.be/4R5quPNgxdg

Session 6 & 7

Practice 1:

Calculate the sum 1+1/2+1/3+1/4+------------+1/N using python

https://www.youtube.com/watch?v=20-VzykSbXM

Session 8 & 9

Practice 2:

To find the absolute value of an integer using python

https://www.youtube.com/watch?v=RpEbPV7jziY

Session 10 & 11

Practice 3:

Enter 100 integers into an array and sort them in an ascending order using python

https://www.youtube.com/watch?v=9XkUshhnH7E

Session 12

Session 13

Solved problems on bisection method

https://www.youtube.com/watch?v=CSQh_vSOhms

Session 16

Session 17

Solved problems on Newton-Raphson method

https://www.youtube.com/watch?v=QvFYgkYNwS4

Session 18

Fixed point iteration method

https://www.youtube.com/watch?v=OLqdJMjzib8

Session 19 & 20

Practice 4:

To find the approximate value of a polynomial using Bisection method.

https://www.youtube.com/watch?v=QcuVPbN4_Vk

Session 21 & 22

Practice 5:

To find the approximate value of a polynomial using Newton Rapshon method

https://www.youtube.com/watch?v=hZ4l5wXRV4g

Session 23 & 24

Practice 6:

To find the approximate value of a polynomial using Secant method

https://www.youtube.com/watch?v=-JY0oavOhfw

Session 25 & 26

Practice 7

To find the approximate value of a polynomial using Regula Falsi method

https://www.youtube.com/watch?v=6vh8QP1GliY

Session 27 & 28

Project 1:

Numerical Methods for Solving Systems of Nonlinear Equations

https://www.lakeheadu.ca/sites/default/files/uploads/77/docs/RemaniFinal.pdf

Session 29 & 30

Project 2:

A Study on Transcendental and Polynomial equations.

https://cims.nyu.edu/~donev/Teaching/NMI-Fall2010/Lecture6.handout.pdf

Session 31

Polynomial Interpolation, Existence and uniqueness of interpolating polynomials

https://www.youtube.com/watch?v=qBFGFDrwtas

Session 32

Lagrange’s interpolating polynomial

https://www.youtube.com/watch?v=1wKUMzljZa8

Session 33

Solved Problems on Lagrange’s interpolating polynomial

https://www.youtube.com/watch?v=yAQ-7NPVnW4

Session 34 & 35

Practice 8:

To find the value of a function using Lagrange interpolation

https://www.youtube.com/watch?v=dTGqOj1NZwY&t=492s

Session 36 & 37

Project 3:

A study on Lagrange interpolation

https://ccrma.stanford.edu/~jos/pasp/Lagrange_Interpolation.html

Session 38

Newtons Divided Difference Interpolating Polynomial

https://www.youtube.com/watch?v=j8XGxPyyoPw

Session 39

Forward and backward Difference operators

https://www.youtube.com/watch?v=96i2cxP6E64&t=106s

Session 40

Newtons Forward Difference Interpolating Polynomial

https://www.youtube.com/watch?v=A1CpBcDkiWY

Session 41

Newtons Backward Difference Interpolating Polynomial

https://www.youtube.com/watch?v=7AgZoAV4PPs

Session 42 & 43

Practice 9:

Solving problems on Newton forward/backward difference formula.

https://www.youtube.com/watch?v=MehlwxYs9Kk

Session 44 & 45

Project 4:

A study to apply Newton forward difference formula on solving numerical problems with MATLAB program

https://myclassbook.org/newtons-forward-interpolation-formula-matlab-program/

Session 46 & 47

Project 5:

A study to apply Newton backward difference formula on solving numerical problems.

https://mat.iitm.ac.in/home/sryedida/public_html/caimna/interpolation/nbdf.html

Session 48

Session 49

Session 50

Session 51

Session 52

Session 53 & 54

Practice 10:

Solving a problem on trapezoidal rule using python

https://www.youtube.com/watch?v=mCPWtXVyzFg

Session 55 & 56

Practice 11:

Solving a problem on Simpson’s 1/3 rd rule using python

https://www.youtube.com/watch?v=zy1XgE-XjRM

Session 57

Session 58

Session 60

Session 61

Solved Problems on Euler’s method

https://www.youtube.com/watch?v=sdfMGA60jss

Session 62

4th order runge-kutta method

https://www.youtube.com/watch?v=1YZnic1Ug9g

Session 63 & 64

Practice 12:

Solving Problems on 4th order Runge-Kutta method using python

https://math.okstate.edu/people/yqwang/teaching/math4513_fall11/Notes/rungekutta.pdf

Session 65 & 66

Project 6:

A Study on Runge-Kutta method

https://math.okstate.edu/people/yqwang/teaching/math4513_fall11/Notes/rungekutta.pdf

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    Our Main Teachers

    Mr. Satyabrata Sadangi has received his M.Sc. (Mathematics) from National Institute of Technology(NIT) Rourkela, Orissa, India and B.Ed. from PRSU University, C.G, India. Mr. Sadangi has more than ten years of teaching experience in different educational institutes. Presently he is working as an Assistant Professor, Department of Mathematics, School of Applied Sciences, Centurion University of […]