Error and computer arithmetic: Floating point numbers (accuracy, rounding and chopping), Consequences for programming of floating-point arithmetic, Errors (definition, sources and examples), Propagation of error
Root finding: Bisection method, Fixed point-iteration, Newton’s method and its extensions, Error analysis, Rate of convergence
Solution of systems of linear equations: Direct (Gauss elimination, Factorization) and Iterative (Gauss Jordan, Gauss-Seidel) methods, Relaxation techniques, Errors in solving linear systems
Interpolation and approximation: Polynomial interpolation (Divided Differences, Hermite Interpolation, Cubic spline interpolation, Lagrange interpolation), Error analysis, Least square approximations
Numerical Differentiation and integration: Differentiation using interpolation and Taylor series, Method of undetermined coefficients, Application to solving differential equations, Trapezoidal and Simpson’s rules, Gaussian quadrature, Method of steepest descent, Error analysis
Solution of ordinary differential equations: Initial value problems (Taylor’s, Picard’s, Euler’s and Runge-Kutta methods), Consistency, Order, Stability, and Convergence, Error prediction and control, Predictor-Corrector methods, Boundary value problems (shooting methods, Finite difference approximations)
Lab Sessions: Algorithms and Computer programs to implement the above numerical techniques.