Definitions, Sources and Propagation of errors, Floating-point arithmetic and rounding errors. Definitions of Stability and convergence. Root finding of nonlinear equations: Method of Incremental search, Fixed point iteration method, Bisection method, Regula-falsi method, Newton Raphson method and Secant method. Convergence analysis of these methods. Numerical evaluation of multiple roots.

Finite differences, Polynomial interpolation, Spline interpolation. Least Square methods, Numerical integration : Trapezoidal and Simpson's rules, Newton-Cotes formulae, Gaussian quadrature, Richardson Extrapolation.

Numerical methods in linear algebra: Linear system of equations - Gauss elimination, LU-Factorization, Matrix inversion, Solution by iteration, Matrix Eigenvalue problems: Inclusion of matrix Eigenvalues, Eigenvalues by iteration, Tri-diagonalization.
Newton's method for non-linear system of equations.

IVP: Taylor series method, Picard’s method, Euler and modified Euler methods, Runge-Kutta methods, Adam’s-Bashforth method, Adam’s- Moulton method.
PDE: Numerical solution of simple Elliptic, parabolic and hyperbolic PDEs.

To understand a broad range of numerical methods for solving mathematical problems that arise in Science and Engineering.

To know advantages of using various numerical methods

To have knowledge of solving various governing equations with respect to different practical problems via numerical methods.

Theoretical understanding of the numerical methods and their convergence, stability and computational efficiency

This will help to choose, develop and apply the appropriate numerical techniques for problem solving, interpret the results, and assess accuracy. The problems cover (i) systems of linear and nonlinear equations, (ii) eigenvalue calculation (iii) interpolation, approximation, and integration of functions (iv) initial values problems governed by ordinary differential equations (v) ODEs and PDEs.

C. F. Gerald and P. O. Wheatley, Applied Numerical Analysis, Pearson Education India , 2007

R.B. Bhat and S. Chakraverty, Numerical Analysis in Engineering, Narosa Publishing House/Alpha Science Int. Ltd. (U.K.) , 2004/2007

R. L. Burden, J. Douglas Faires, Numerical Analysis, Cengage Learning , 2011

S. Chakraverty, Nisha Rani Mahato, Perumandla Karunakar and Tharasi Dilleswar Rao, Advanced Numerical and Semi Analytical Methods for Differential Equations, Wiley , 2019