Sources of errors, Propagation of errors, Stability in numerical analysis, Root finding of nonlinear for equations, The numerical evaluation of multiple roots, Brent’s root finding algorithm, Hermite interpolation, Piecewise polynomial interpolation, The minimax and near minimax approximations, Numerical integration, Asymptotic error formulas, and their applications, Adaptive numerical integration, Numerical methods for differential equations: Multistep method, Derivation of higher-order multistep methods, Iterative methods for linear systems: Classical iterative methods (Jacobi, Gauss-Seidel and successive overrelaxation (SOR) methods), Krylov subspace methods GMRES, Conjugate-gradient, biconjugate-gradient (BiCG), BiCGStab methods, preconditioning techniques, parallel implementations.

This course is an introduction to a broad range of numerical methods for solving mathematical problems that arise in Science and Engineering.

The goal is to provide a basic understanding ofthe derivation, analysis, and use of these numerical methods, along with a rudimentary under-standing of finite precision arithmetic and the conditioning and stability of the various problems and methods.

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

K. E. Atkinson, An Introduction to Numerical Analysis, Wiley India Pvt. Limited , 2008

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

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