Review Questions

• Truncation vs. round-off errors: origins and characterization.
• Similarities and differences between the fixed-point iteration, Newton's method and the secant method for solution is nonlinear equations.
• Convergence analysis of different root-finding methods; the role of the fixed-point theorem.
• Criteria and techniques of multidimensional minimization.
• Solution of a general approximation problem in a Hilbert space - cases with orthogonal and non-orthogonal bases.
• Decomposition in a non-orthogonal basis.
• Different techniques of polynomial interpolation: direct approach, Lagrange polynomials and divided differences.
• What are the origins of the Runge phenomenon in polynomial interpolation? What does the interpolation error depend on?
• Given a set of points (x_i,y_i), i=1,...,N and an approximating function in the form y= Ax + B, use the least-squares method to derive a system of equations characterizing the unknown parameters A and B; how do solutions of approximation and interpolation problem differ?
• Numerical differentiation: one-sided (forward / backward) and central difference formulas for the first and second derivative. Estimates of numerical differentiation errors.
• Complex step derivative.
• Derive an estimate for the order of accuracy of the trapezoidal rule for numerical integration. Local vs. global error.
• Why are the Gaussian quadratures more accurate than the Newton-Cotes quadratures when the integrand function is a polynomial of a given order? What are the disadvantages of the Gaussian quadratures?
• Chebyshev polynomials: definition, relation to trigonometric polynomials, applications.
• Applications of orthogonal polynomials: Gaussian integration, Chebyshev interpolation, Fourier analysis.