Shaligram Prajapat Ph.D.

Computer-Oriented Numerical Methods (CONM)

Numerical Analysis and Design (NAD)

Computer Oriented Numerical and Otimization Techniques

IT614 | IT 503 Syllabus | IC-504 | IT-402

Preamble:This is M.Tech.(IT)-V Semester's core subject of Computer Science. It is termed by various names like:

Numerical Analysis and Design (NAD)
Design and Analysis of Numerical Algorithms ( D&A of NM )
Design of Numerical Algorithms ( DNA )
Computer-Oriented Numerical Methods (CONM )
Computer-Oriented Numerical and Statistical Methods ( CONSM )
Numerical Analysis ( NA )

These subjects are taught in various courses with different perspectives. In this class, the subject will be discussed from the perspective of the development and Analysis of algorithms for Numerical Problems.The Theoretical base/Concept will be discussed in detail with the approach of developing algorithms and the result of the algorithm will be analysed by various alternatives. A comparative study of these algorithms will be attempted with the focus of efficient algorithms.The Sequence in which each method will be discussed is:

Derivation, Geometrical Interpretations
Development of Algorithms/Flowchart
Development of program/Coding
Numerical problems
Efficiency comparisons with peer algorithms

UNIT-1: Numeric Information Representation :

Integer Representations
- - Unsigned integer representations
  - One's Complement Method
  - Two's Complement Method
Real Number Representations
- - Fixed Point Notations
  - Normalisation and Floating Point Representations
Arithmetic using floating point representations
Consequences of floating-point representations
Concept of Zeros
Errors in handling numerical quantities and Measures and types of errors.
Problems and Solutions

UNIT-2 Roots of Polynomials Concept of roots, Formula approach v/s Iterative approach, Iterative Methods:

Bisection Method/Bolzano's Method/Midpoint Subdivision Method
Method of False Position/Regula Falsi Method
Secant Method
Newton-Raphson Method
Fixed Point Iteration Method
Convergence Order Theorem
Order of Convergence of iterative Methods: Secant Method, Newton-Raphson Method
Comparison among various iterative methods.

UNIT-3: Finite Difference Calculus

Function with discrete arguments, Basics of Finite difference calculus.
Forward difference operator, Forward difference table,
Algorithm/program/flowchart for construction of forward difference tables.
Backward difference operator,Backward difference table, algorithm/program/flowchart for construction of Backward difference tables.
Divided difference operator,Divided difference table, algorithm/program/flowchart for construction of Divided difference tables.
Shift Operator, Central difference operator, Averaging operator
Relationship among various operators.

UNIT-4: Interpolations

Concepts of interpolation,
Purpose of interpolation,
Newton Gegory Forward interpolation formula, Newton Gregory Backward interpolation formula, Newton's divided difference interpolation formula,
Lagrangee's interpolation formula,Inverse interpolation.Numerical problems and algorithms/ Programs.
Inverse Interpolation

UNIT-5: Numerical Differentiation and Integration

Newton Cote's formula,General Quadrature Formula,
Derivation of Trapezoidal rule, Simpson's rule (1/3 and 3/8), Weedle's Rule,(Algorithms, programs, numerical)
Gaussian Quadrature Formulla
Numerical Differentiation using Finite Difference Methods

UNIT-6: Solutions of Ordinary Differential

Solution of differential Equations using:
- Euler’s Method,
- Euler’s Modified Method
- Runge - Kutta Methods,
- Picard’s Method,
- Taylor’s Method,
- Predictor Corrector Methods
- Statbility of Solutions

Unit-7: Approximation , Hypothesis and Curve Fitting

Taylor series representation
Chebyshev representation.
t-Test
z-Test
Testing of Hypothesis
Curve Fitting and principle of least square
Linear, Quadratic & Nonlinear Curve Fitting

(Algorithms, Programs , Numericals, Derivations)

Assessment

LAB Assignment and Learning Activity

References
[1]. V. Rajaraman, Computer Oriented Numerical Methods, 3rd ed. New Delhi, India: Pearson Education, 2007.
[2]. M. K. Jain, S. R. K. Iyengar, and R. K. Jain, Numerical Methods for Scientific and Engineering Computation, 2nd ed. New Delhi, India: New Age International, 2007.
[3]. C. F. Gerald and P. O. Wheatley, Applied Numerical Analysis, 7th ed. Boston, MA, USA: Addison-Wesley, 2004.
[4]. B. S. Grewal, Numerical Methods in Engineering and Science. New Delhi, India: Khanna Publishers, 2010.
[5]. T. Veerarajan and T. Ramachandran, Theory and Problems in Numerical Methods. New Delhi, India: Tata McGraw-Hill, 2008.
[6]. P. Niyogi, Numerical Analysis and Algorithms. New Delhi, India: Tata McGraw-Hill, 2009.
[7]. F. Scheid, Numerical Analysis. New York, USA: McGraw-Hill, 1988.
[8]. S. S. Sastry, Introductory Methods of Numerical Analysis, 5th ed. New Delhi, India: Pearson Education, 2012.
[9]. C. B. Gupta and V. Gupta, Introduction to Statistical Methods. New Delhi, India: Vikas Publishing, 2014.
[10]. M. Goyal, Computer-Based Numerical and Statistical Techniques. New Delhi, India: Firewall Media, 2008.
[11]. M. Schäfer, Computational Engineering: Introduction to Numerical Methods, 2nd ed. Cham, Switzerland: Springer, 2022.
[12]. G. V. Davis, Numerical Methods in Engineering and Science. Dordrecht, Netherlands: Springer, 1986.

Online Calculators
Root Finding techniques

System of equation

Interpolation Method

Forward difference interpolation
Backward difference interpolation
Divided difference interpolation
Lagrangee's interpolation

LAB Assignment

Resources/ Links

Refer following OER and Reference material to boost your studies:

https://nptel.ac.in/courses/111107062/
https://nptel.ac.in/courses/122102009/
https://nptel.ac.in/courses/122106033
Attendance: Section A and Section B
July December 2014:Test-2(Solution by Preeti Sharma, Anjeeneelu -28/10/2014)

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