Preamble:

This subject is a 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 developing and analyzing algorithms for Numerical Problems.

 The theoretical basis/Concept will be discussed in detail using an approach focused on developing algorithms, and the algorithm's results will be analysed using various alternatives. A comparative study of these algorithms will be attempted with a focus on 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  and Associated Errors

• 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-Gregory forward interpolation formula, Newton-Gregory backward interpolation formula, Newton's divided difference interpolation formula, 

• Lagrange's interpolation formula, Inverse interpolation.Numerical problems and algorithms/ Programs.

• Inverse Interpolation

Resources

UNIT-5: Numerical Differentiation and Integration

• Newton-Cotes formula, General Quadrature Formula 

• Derivation of Trapezoidal rule, Simpson's rule (1/3 and 3/8), Weedle's Rule,(Algorithms, programs, numerical)

• Gaussian Quadrature Formula

• Numerical Differentiation using Finite Difference Methods

UNIT-6: Solutions of  Ordinary Differential Equations

• • 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 Squares 

• Linear, Quadratic & Nonlinear Curve Fitting 

 (Algorithms, Programs, Numericals, Derivations)

LAB Assignment and Learning Activity

• Assignment-1

• Assignment-2

• Assignment-3

• link to Submit assignment

• Quiz-1

• Quiz-2

• Video and Poster 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.