# Numerical Analysis

By Prof. S. Baskar   |   IIT Bombay
Learners enrolled: 196
Numerical analysis is a branch of mathematics that deals with the construction and analysis of numerical methods to various mathematical problems. For a given suitable mathematical problem, the aim is
• to develop numerical methods; and
• to develop mathematical tools to study the error involved in the numerical solution when compared to the exact solution of the mathematical problem.
In this course, we learn to meet the above two goals through some simple and familiar mathematical problems. We will also learn to implement some simple numerical methods as computer code.

PREREQUISITES: Calculus, Linear Algebra at UG level. Some experience in programming in any language may be desirable.
Summary
 Course Status : Upcoming Course Type : Elective Duration : 12 weeks Start Date : 23 Jan 2023 End Date : 14 Apr 2023 Exam Date : 29 Apr 2023 IST Enrollment Ends : 30 Jan 2023 Category : Mathematics Credit Points : 3 Level : Undergraduate/Postgraduate

### Course layout

Week 1:
• Motivations, Preliminaries, Order of convergence.
• Error Analysis: Floating-point approximations, Significant digits, Stability in computation.
Week 2:
• Tutorial Session for Week 1.
• Introducing Python coding.
• Direct Methods for Linear System of Equations: Gaussian elimination method, Partial pivoting, LU factorization, Operation counting.
Week 3:
• Matrix Norms: Subordinate matrix norms, Condition number of an invertible matrix.
• Iterative Methods for Linear Systems: Jacobi Method
• Tutorial Session for Week 2 and Python implementation.
Week 4:
• Iterative Methods (continued): Gauss-Seidel method,  Residual corrector method, Conjugate gradient method.
• Tutorial Session for Week 3 and Python implementation.
Week 5:
• Methods for Computing Eigenvalue and Eigenvectors: Power method, inverse power method, Gerschgorin’s theorem.
• Tutorial Session for Week 4 and Python implementation.
Week 6:
• Nonlinear Equations: Bisection method, Regula-falsi method, Secant method, Newton-Raphson method, Stopping criteria. Convergence analysis
• Tutorial Session for Week 5 and Python implementation.
Week 7:
• Nonlinear Equations (Continued): Fixed-point iteration method, Newton’s method for system of nonlinear equations
• Tutorial Session for Week 6 and 7, and Python implementation.
Week 8:
• Polynomial Interpolation: Lagrange form, Newton’s form, divided differences, error analysis.
Week 9:
• Piecewise Interpolation: Piecewise polynomial interpolation, Hermite interpolation, Cubic spline interpolations.
• Tutorial Session for Week 8 and 9, and Python implementation.
Week 10:
• Numerical Integration: Rectangle rule, Mid-point rule, Trapezoidal rule, Simpson’s rule, Gaussian quadrature rule, error analysis
• Numerical Differentiation: Method of undetermined coefficients, error analysis.
Week 11:
• Tutorial Session for Week 10 and Python implementation.
• Numerical Methods for ODEs: Initial Value Problems for First Order ODEs - Euler methods, Modified Euler methods, error analysis, Runge-Kutta method of order 2 and 4.
Week 12:
• Numerical Methods for ODEs: Numerical methods for two-point boundary value problems.
• Numerical Methods for PDEs: Numerical methods for linear advection equation.
• Stability analysis
• Tutorial Session for Week 11 and 12, and Python implementation.

### Books and references

Study material will be given.
Atkenson, K. E. Introduction to Numerical Analysis (2-edition)

### Instructor bio

IIT Bombay
Prof. S. Baskar has 16 years of teaching experience in the Department of Mathematics, IIT Bombay. He has been teaching several UG and PG courses in mathematics. Especially he has taught numerical analysis more than 15 times and also taught other mathematics courses like ordinary differential equations, partial differential equations, multivariable calculus, probability theory, and derivative pricing. Baskar received the IIT Bombay excellence in teaching award for the year 2020 and the departmental excellence in teaching award in 2018.

### Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
Date and Time of Exams: 29 April 2023 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
Registration url: Announcements will be made when the registration form is open for registrations.
The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Bombay. It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team