This course introduces the basic numerical methods used in many applications areas in computational science and engineering.

Topics Covered: 

• Introduction: Floating point arithmetics, sources of errors
• Systems of Linear Equations: Gaussian elimination, pivoting, norms, condition numbers
• Linear Least Squares: Normal equations method, orthogonalization methods for full rank problems
• Solution of Nonlinear Equations: Bisection and secant methods, fixed point iteration, Newton's method
• Interpolation: Lagrange interpolation, Newton interpolation, Chebyshev polynomials, Hermite interpolation, Splines, Fast Fourier Transformation
• Numerical Differentiation and Integration: Trapezoidal rule, Simpson's rule, Newton-Cotes quadrature, Gaussian quadrature, adaptive quadrature, finite difference, Richardson extrapolation
• Numerical Solutions of Ordinary Differential Equations: initial value problems, systems of equations, Euler method, Runge-Kutta method
• Optimization (if the schedule allows): Existence of solutions, Optimization in one dimension, unconstrained and constrained optimizations, optimality conditions, Newton's method, Steepest descent, Conjugate gradient method