 # Numerical Analysis Teacher Category

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# 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)

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

## Session 4

Floating point Arithmetic, K – digit Arithmetic

https://youtu.be/4R5quPNgxdg

## 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

## Session 13

Solved problems on bisection method

## Session 17

Solved problems on Newton-Raphson method

## Session 18

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.

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

## Session 31

Polynomial Interpolation, Existence and uniqueness of interpolating polynomials

## Session 32

Lagrange’s interpolating polynomial

## Session 33

Solved Problems on Lagrange’s interpolating polynomial

## Practice 8:

To find the value of a function using Lagrange interpolation

## Project 3:

A study on Lagrange interpolation

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

## Session 38

Newtons Divided Difference Interpolating Polynomial

## Session 39

Forward and backward Difference operators

## Session 40

Newtons Forward Difference Interpolating Polynomial

## Session 41

Newtons Backward Difference Interpolating Polynomial

## Practice 9:

Solving problems on Newton forward/backward difference formula.

## 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/

## 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

## Practice 10:

Solving a problem on trapezoidal rule using python

## Practice 11:

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

## Session 61

Solved Problems on Euler’s method

## Session 62

4th order runge-kutta method

## Practice 12:

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

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

## Project 6:

A Study on Runge-Kutta method

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

## Case Studies 