Representation of numbers, computer arithmetic, overflow and underflow, rounding error, cancellation error, truncation error, algorithms, stopping conditions. Solving f(x) = 0, bisection, secant, Newton methods. Interpolation, Lagrange polynomials, Hermite polynomials, divided differences, splines. Solving Ax = b, Gaussian elimination algorithm, pivoting strategies, LU decomposition, complexity, special matrices (diagonally dominant, positive definite, banded). Iterative methods, Jacobi, Gauss-Seidel, SOR methods. Eigenvalues and eigenvectors, power method, Householder's method, QR-algorithm. Non-linear systems, Newton's and steepest descent methods. Approximation theory, orthogonal polynomials, Legendre polynomials, Chebyshev polynomials. Numerical differentiation. Numerical integration, simple and composite rules, error estimates, Gaussian quadrature, adaptive quadrature methods, improper integrals, multiple integrals. Numerical methods in ODE, Runge-Kutta, stiff ODE's, predictor-corrector and adaptive techniques. Numerical solutions to partial differential equations.