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.