# Numerical Analysis

By Prof. Madhumangal Pal   |   Vidyasagar University
Learners enrolled: 2298

The numerical analysis/ method is a very important and common topic for computational mathematics and hence studied by students from many disciplines like mathematics, computer science, physics, statistics and other subjects of physical sciences and engineering. The numerical analysis/method is an interdisciplinary course used by students/ teachers/ researchers from several branches of science and technology, particularly mathematics, statistics, computer science, physics, chemistry, electronics, etc. This subject is also known as computational mathematics. To design several functions of a computer and to solve a problem by computer numerical method is essential. It is not possible to solve any large-scale problem without the help of numerical methods. Numerical methods also simplify the conventional methods to solve problems, like definite integration, solution of equations, solution of differential equations, interpolation from the known to the unknown, etc. To explore complex systems, mathematicians, engineers, and physicists require computational methods since mathematical models are only rarely solvable algebraically. The numerical methods based on computational mathematics are the basic algorithms underpinning computer predictions in modern systems science. After completing the course, the students can design algorithms and program codes to solve real-life problems. In each module, an exercise is provided to test the students’ performance. Also, some more references are added in the learn more section to investigate the subject more thoroughly and to learn more topics of numerical analysis. The entire course is divided into nine chapters and thirty-six modules. It is a 15 weeks one-semester course including assignments, discussions and evaluation. Almost all Indian universities offer this course as a core course.

Summary
 Course Status : Completed Course Type : Core Duration : 15 weeks Category : Mathematics Credit Points : 4 Level : Postgraduate Start Date : 16 Jan 2023 End Date : 30 Apr 2023 Enrollment Ends : 15 Mar 2023 Exam Date :

### Course layout

Week 1: Errors in Numerical Computations
1.      Ch 1 Mod 1:  Error in Numerical Computations.
2.      Ch 1 Mod 2:  Propagation of Errors and Computer Arithmetic.

Week 2:

Interpolation - I
3.      Ch 1 Mod 3:  Operators in Numerical Analysis.
4.      Ch 2 Mod 1:  Lagrange’s. Interpolation.
5.      Ch 2 Mod 2:  Newton’s Interpolation Methods.
6.      Ch 2 Mod 3:  Central Difference Interpolation Formulae.

Week 3:

Interpolation - II
7.      Ch 2 Mod 4:  Aitken’s and Hermite’s Interpolation Methods.
8.      Ch 2 Mod 5:  Spline Interpolation.
9.      Ch 2 Mod 6:  Inverse Interpolation.
10.    Ch 2 Mod 7:  Bivariate Interpolation.

Week 4:
Approximation of Functions
11.    Ch 3 Mod 1:  Least Squares Method.
12.    Ch 3 Mod 2:  Approximation of Function by Least Squares Method.
13.    Ch 3 Mod 3:  Approximation of Function by Chebyshev Polynomials.

Week 5:
Solution of Algebraic and Transcendental Equation
14.    Ch 4 Mod 1:  Newton’s Method to Solve Transcendental Equation.
15.    Ch 4 Mod 2:  Roots of a Polynomial Equation.
16.    Ch 4 Mod 3:  Solution of System of Non-linear Equations.

Week 6:

Solution of System of Linear Equations-I
17.    Ch 5 Mod 1:  Matrix Inverse Method.
18.    Ch 5 Mod 2:  Iteration Methods to Solve System of Linear Equations.
19.    Ch 5 Mod 3:  Methods of Matrix Factorization.

Week 7:
Solution of System of Linear Equations-II
20.    Ch 5 Mod 4:  Gauss Elimination Method and Tri-diagonal Equations.
21.    Ch 5 Mod 5:  Generalized Inverse of Matrix.
22.    Ch 5 Mod 6:  Solution of Inconsistent and Ill Conditioned Systems.

Week 8:

Assessment

Week 9:
Eigenvalues and Eigenvectors of Matrices
23.    Ch 6 Mod 1:  Construction of Characteristic Equation of a Matrix.
24.    Ch 6 Mod 2:  Eigenvalue and Eigenvector of Arbitrary Matrices.
25.    Ch 6 Mod 3:  Eigenvalues and Eigenvectors of Symmetric Matrices.

Week 10:

Differentiation and Integration-I
26.    Ch 7 Mod 1:  Numerical Differentiation.
27.    Ch 7 Mod 2:  Newton-Cotes Quadrature.

Week 11:

Differentiation and Integration-II
28.    Ch 7 Mod 3:  Gaussian Quadrature.
29.    Ch 7 Mod 4:  Monte-Carlo Method and Double Integration.

Week 12:

Ordinary Differential Equations-I
30.    Ch 8 Mod 1:  Runge-Kutta Methods.
31.    Ch 8 Mod 2:  Predictor-Corrector Methods.

Week 13:

Ordinary Differential Equations-II
32.    Ch 8 Mod 3:  Finite Difference Method and its Stability.
33.    Ch 8 Mod 4:  Shooting Method and Stability Analysis.

Week 14:
Partial Differential Equations
34.    Ch 9 Mod 1:  Partial Differential Equation: Parabolic.
35.    Ch 9 Mod 2:  Partial Differential Equations: Hyperbolic.
36.    Ch 9 Mod 3:  Partial Differential Equations: Elliptic

Week 15:

Final examination

### Books and references

Jain, M.K., Iyengar, S.R.K., and Jain, R.K. Numerical Methods for Scientific and Engineering Computation. New Delhi: New Age International (P) Limited, 1984.

Pal, M.
Numerical Analysis for Scientists and Engineers: Theory and C Programs. New Delhi: Narosa, Oxford: Alpha Sciences, 2007.

Krishnamurthy, E.V., and Sen, S.K.
Numerical Algorithms. New Delhi: Affiliated East-West Press Pvt. Ltd., 1986.

Mathews, J.H.
Numerical Methods for Mathematics, Science, and Engineering, 2nd ed., NJ: Prentice-Hall, Inc., 1992.

Danilina, N.I., Dubrovskaya, S.N., and Kvasha, O.P., and Smirnov, G.L.
Computational Mathematics. Moscow: Mir Publishers, 1998.

Hildebrand, F.B.
Introduction of Numerical Analysis. New York: London: McGraw-Hill, 1956.

### Instructor bio

Vidyasagar University

Prof. Madhumangal Pal is currently a Professor of Applied Mathematics, at Vidyasagar University. He has received Gold and Silver medals from Vidyasagar University for ranking first and second in M.Sc. and B.Sc. examinations, respectively. Also, he received the “Computer Division Medal” from the Institute of Engineers (India) in 1996 for best research work. Prof. Pal has successfully guided 39 research scholars for Ph.D. degrees and has published more than 390 articles in international and national journals. His specializations include Algorithmic and Fuzzy Graph Theory, Fuzzy Matrices, and Genetic and Parallel Algorithms. He has evaluated more than 160 Ph.D. theses from Indian and Abroad. Prof. Pal is the author of nine textbooks, including Numerical Analysis and two edited books published in India, the United Kingdom and the USA. He has published 23 chapters in several edited books. Prof. Pal completed three research projects funded by UGC and DST and is ongoing. Prof. Pal is the Editor-in-Chief of two journals and area editor of  SCIE Indexed journal, and a member of the Editorial Boards of several journals. Also, he has visited China, Greece, London, Taiwan, Malaysia, Thailand, Hong Kong, Dubai and Bangladesh for academic purposes. He is also a member of the American Mathematical Society, USA; Calcutta Mathematical Society; Advanced Discrete Mathematics and Application; Neutrosophic Science International Association, USA; Ramanujam Mathematical Society, India, etc. As per Google Scholar, the citation of Prof. Pal is 10303, the h-index is 53 and the i10-index is 252, as of 02.12.2022. He was a member of several selection committees and several administrative and academic bodies at Vidyasagar University and other institutes. He is recently included in the list of the top 2% of scientists among all disciplines prepared by Stanford University.

### Course certificate

Certificate policy
The course is free for all to enroll. But, for getting a certificate, learner have to register and deposit the registration fee (amount is to be declared latter).
The final examination will be held in first week of May 2021.
Registration url: Announcements will be made when the registration form is open for registrations.
The online registration form for examination has to be filled and the certification examination fee has to be paid. More details will be made available when the examination registration form is published. Any changes will be informed accordingly.
Please check the form for more details on the exam cities and other information.

Criteria to get a certificate

Assignment score = 30% among best 8 assignments out of the total 13 assignments given in the course.
Exam score = 70% of the proctored certification exam score out of 100.
Final score = Assignment score + Exam score.
Assignment score and final score must be at least 40% separately.
Final score must be at least 40% to get a certificate.

How to get certificate

Certificate contains learner name, photograph and the score in the final exam with the breakup.
Only the e-certificate will be made available.