**Review Questions**

What are the origins of the Runge phenomenon in polynomial interpolation?
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?
Derive an estimate for the order of accuracy of the trapezoidal rule for
numerical integration.
Why are the Gaussian quadratures more accurate than the Newton-Cotes quadratures
for a given order of the interpolating polynomial? What are the disadvantages of
the Gaussian quadratures?
Compare the Runge-Kutta and multistep methods for numerical integration
of ODEs.
Analyze implementation of Dirichlet and Neumann boundary conditions in finite-difference
solutions of boundary value problems (both for ODEs and elliptic PDEs).
How to solve efficiently sparse algebraic systems depending on the number of their
diagonals
Discuss the criteria determining the choice of time step for explicit and implicit
integration schemes for parabolic and hyperbolic PDEs
Derive and compare explicit and implicit (in time) schemes for numerical
solution of parabolic PDEs
Compare the different variants of the Finite Element Method for solution
of a given ODE or PDE.