Introduction to Numerics

Elements of floating point arithmetics. Representation of numbers with normalized mantissa and characteristic. Set of floating-point numbers. Overflow and underflow. Truncation, rounding and related errors. Machine precision. Propagation of round-off error in floating-point operations.

Elements of iterative methods for linear systems. General form of an iterative method for linear systems. Jacobi and Gauss-Seidel iterations. Convergence.

Elements of iterative methods for nonlinear equations. Bisection method. Local linearization. Regula falsi, secant method, Newton-Raphson method. Fixed point iterations. Convergence.

Elements of polynomial interpolation. Construction of the interpolation polynomial: construction via Vandermonde matrices; Lagrange interpolation polynomial, fundamental Lagrange polynomials. Estimation of interpolation error.  

Elements of numerical quadrature. Interpolatory quadrature, computation of weights and nodes. Newton-Cotes formulae. Trapezoidal quadrature rule and error estimate. Cavalieri-Simpson quadrature rule and error estimate. Degree of precision. Idea of composite quadrature.