CMS-A-CC-3-6-TH: Computational Mathematics Core Course- 6: Theory, Credits: 04, Contact hours: 60.

 Introduction

Set Theory: Finite and Infinite Sets, Uncountable Infinite Sets, Relations: Properties of

Binary Relations, Closure, Partial Ordering Relations, Equivalence, Functions: definition,

one-to-one, onto and invertible, Mathematical Functions: Exponential and Logarithmic,

Counting: Mathematical Induction, Pigeonhole Principle, Permutation and Combination,

Binomial Theorem, Principle of Inclusion and Exclusion.

10 hours

Introduction to Probability

Elementary events, Sample space, Classical and Axiomatic definition of Probability,

Theorems on Total Probability, Conditional Probability, Bernoulli Trials and Binomial

Distribution, Bayes’ Theorem, Random Variables, Expectation, Variance, Standard

Deviation.

10 hours

Growth of Functions

Asymptotic Notations, Standard notations and common functions with simple examples.

04 hours

Recurrences

Relations, Generating Functions, Linear Recurrence Relations with Constant Coefficients

and their solution, Substitution Method, Recurrence Trees.

06 hours

Numerical Methods (Algorithmic Approach)

Errors: Approximate and Rounding of Numbers, Significant digits, Errors and their types,

Propagation of errors.

Interpolation: Newton Forward and Backward interpolation, Lagrange interpolation.

Solving a Set of Linear Equations: Gaussian Elimination, Gauss–Jordan, Iteration methods

a n d t h ei r convergence conditions, Gauss-Seidel, Gauss-Jacobi Iterative Methods.

Solving Non-linear equations: Bisection, Regula-falsi, Secant and Newton-Raphson, their

order of convergence.

Solving Differential Equations: Euler, Runge-Kutta second and fourth order methods.

Numerical Integration:

Trapezoidal and Simpson’s 1/3rd rules.

Curve fitting :

Least square approximation, Linear regression, Polynomial regression, Fitting Exponential

and Trigonometric functions.

Graph Theory

Basic Terminology, Models and Types, Multi graphs and Weighted graphs, Graph

Representation, Graph Isomorphism, Connectivity, Euler and Hamiltonian Paths and

Circuits, Planar Graphs, Trees and their basic terminologies and properties.