To explore complex systems, physicists, engineers, financiers and mathematicians require computational methods since mathematical models are only rarely solvable algebraically. Numerical methods, based upon sound computational mathematics, are the basic algorithms underpinning computer predictions in modern systems science. The course will cover the classical fundamental topics in numerical methods such as, approximation, numerical linear algebra, solution of nonlinear algebraic systems, matrix decomposition, interpolation and unstable systems. The viewpoint will be modern, with connections made between each topic and a variety of applications.

*Intended ** Learning Outcomes: *Upon successful completion of the course, a student will be able to:

- Derive numerical methods for various mathematical operations and tasks, such as interpolation, differentiation, the solution of linear and nonlinear equations, and the solution through polynomials.
- Apply the numerical methods (such as Bisection, False position, Newton - Raphson) to solve nonlinear equations.
- Recognize Iterative methods (for example, Jacobi, Gauss Seidel)
- Implement a variety of numerical algorithms using appropriate programming (for example, Maple),
- Effectively use numerical tools to solve problems in their own field of interest.

**Prerequisites:** Linear Algebra

*Contents*

- Computer arithmetic,
- Approximations and errors,
- Linear and non-linear equations,
- Methods for solution of linear equations,
- Gaussian elimination method,
- Gauss-Jordan method,
- Crout’s method,
- Cholesky’s method,
- Doolittle’s method,
- LU-factorization and Matrix inversion,
- Iterative methods,
- Jacobi,
- Gauss-Seidel,
- Error analysis for iterative methods,
- Methods for the solution of nonlinear equations,
- Bisection method,
- Regula-falsi method,
- Fixed point iteration method,
- Newton-Raphson method,
- Secant method
- Interpolation: polynomial approximation,
- Forward, backward and centered difference formulae,
- Lagrange interpolation,
- Newton’s divided difference formula,
- Interpolation with a cubic spline,
- Hermite interpolation,
- Least squares approximation,
- Numerical differentiation, Richardson’s extrapolation,
- Maple Programming to implement above mentioned topics.

*Recommended Books*

- R. L. Burden and J. D. Faires:
*Numerical Analysis*, (Latest edition, PWS Pub. Co) - C. F. Gerald and P.O. Wheatley,
*Applied Numerical Analysis*, (Pearson Education, Singapore, 2005)

*Suggested Books*** **

- S. C. Chapra and R. P. Canale:
*Numerical Methods for Engineers*, (6th edition, McGraw Hill. Sankara K. 2005) *Numerical Methods for Scientists and Engineers*.(2nd ed. New Delhi: Prentice Hall)- J. H. Mathews,
*Numerical Methods for Mathematics*, (Latest Edition, Prentice HallInternational)

**RESEARCH PROJECT /PRACTICALS/LABS/ASSIGNMENTS**

Research project/assignment based on numerical techniques and algorithms often involve features which require their solutions can be awarded to students during their course work period. Examples of features are

- Accelerating convergence of numerical methods.
- Finding solution of complex root finding problems
- Improving existing numerical techniques

**Assessment Criteria**

- Sessional: 20 (Presentation / Assignment 10, Attendance 05, Quiz 05)
- Mid-Term Exam: 30 (online MCQS)
- Final-Term Exam: 50 (or Viva + online)

*Key Dates and Time of Class Meeting*

Monday 08:00--09:30, 09:30--11:00, 11:00--12:30 and 14:00--15:30

Tuesday 08:00--09:30 and 14:00--15:30

Wednesday 09:30--11:00

Friday 14:00--15:30

Commencement of Classes October 12, 2020

Mid Term Examination December 14-18, 2020

Final Term Examination February 08-12, 2021

Declaration of Result February 19, 2021