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• Section 8.2: Iterative Solutions of Linear Equations
  • Be familiar with, and know how to compute, the l₁, l₂, and l_infinity vector and matrix norms.
  • Be familiar with the concept of, and know how to compute, the condition number of a matrix. In particular, you should be able to explain why it can be useful to know the condition number of a matrix.
  • Know why it can be desirable to use a method that provides an iterative solution to a system of linear equations, as opposed to a method that directly gives a solution to the system.
  • Be able to describe the basic approach used to define an iterative method for the solution of a system of linear equations (see pages 322-323 of our textbook).
  • Be familiar with the equation formulation of the Jacobi, Gauss-Seidel and SOR iterative methods (pp. 323-324), and be able to use it to determine successively more accurate estimates (x⁽¹⁾, x⁽²⁾, etc.) of the exact solution vector x for a system of linear equations Ax = b, given an initial estimated solution vector x⁽⁰⁾.
  • Given a system of linear equations Ax = b, be able to determine the non-singular matrix Q, the iterative matrix B = I - Q^-1A, and the constant vector h = Q^-1b that enable obtaining successive estimates of the solution vector x using Jacobi iteration. Also, know how to obtain the corresponding matrices and constant vectors used in Gauss-Seidel and SOR iteration.
  • Be able to describe the similarities/differences/advantages/disadvantages and applicability of these three methods, including the role played by the factor ω in the SOR method.
  • Be able to determine, for a given system, whether the Jacobi, Gauss-Seidel and SOR methods are guaranteed to converge. Specifically:
    • You should be able to correctly answer questions such as:
      • True or False: if A is not strictly diagonally dominant, then there is no starting vector x⁽⁰⁾ for which the Jacobi method will converge
      • True or False: if A is not strictly diagonally dominant, then it is guaranteed that you will be able to find some starting vector x⁽⁰⁾ for which the Jacobi method will not converge
      • True or False: if A is not strictly diagonally dominant, then the Jacobi method may still converge for all possible starting vectors x⁽⁰⁾
    • You should also be able to use the spectral radius theorem to determine whether the Jacobi, Gauss-Seidel and SOR methods for the solution of a given system Ax = b will converge for all initial iterates x⁽⁰⁾ (see pp.348-349)
  • Be able to easily and quickly compute the determinant of a 2x2 matrix. (Hint: The determinant of a matrix A containing the elements A[i,j], where i denotes the row number and j denotes the column number, and A[1,1] = a, A[1,2] = b, A[2,1] = c, A[2,2] = d, is given by: ad - cb).
  • Be able to easily and quickly compute the determinant of a 3x3 matrix. (Hint: The determinant of a matrix A containing the elements A[1,1] = a, A[1,2] = b, A[1,3] = c, A[2,1] = d, A[2,2] = e, A[2,3] = f, A[3,1] = g, A[3,2] = h, A[3,3] = i is given by: a(ei-fh)-b(di-fg)+c(dh-eg)).

• Section 9.1: First-Degree and Second-Degree Splines
  • Be able to explain, in your own words, what a spline is.
  • Know the definition of a first-degree spline, and be able to determine if a given expression represents a first-degree spline.
  • Know the definition of a second-degree spline, and be able to determine if a given expression represents a second-degree spline.
  • Be able to determine the first-degree spline S(x) that interpolates a given set of points.
  • Be able to determine the second-degree spline Q(x) that interpolates a given set of points and that has zero first derivative at the first point.
  • Be able to determine the Subbotin quadratic spline Q(x) that is defined by a given set of knots, nodes, and values.
  • Be able to determine, for a given function f(x), a bound on the inaccuracy of approximating f(x) by a first-degree spline p(x) in terms of the maximum distance h between any successive pair of nodes x_i and x_i+1 (hint: see theorem 2 on pp.375).

• Section 9.2: Natural Cubic Splines
  • Know the definition of a spline of degree k
  • Be able to determine whether a given expression represents a spline of degree k (for arbitrary k).
  • Know the definition of a natural cubic spline.
  • Know that other types of splines can be defined by setting the two "extra" constraints in a different way, resulting in curves that have different characteristic properties.
  • Be able to determine the natural cubic spline that interpolates a given set of points.
  • Be able to discuss the relative advantages and disadvantages of interpolating a table of values using a spline versus using a single polynomial function.

• Section 9.3: B-Splines and Bezier Splines
  • Know how a parametric Bezier spline segment of degree 2 or 3 can be defined by a collection of three or four control points. [see the notes]
  • Know how a uniform, parametric, approximating B spline of degree 1, 2, or 3 can be defined by a collection of control points. [see the notes]

• Chapter 10: Methods for Solving Ordinary Differential Equations
  • Be able to exaplin what an initial value problem is.
  • Know how to use Euler's method (Taylor series method of order 1) to solve an initial value problem {dx/dt = f(t,x(t); x(a)=s} to obtain an approximate value for x(b).
  • Be familiar with the 2^nd order Runge-Kutta method for solving an initial value problem {dx/dt = f(t,x(t)); x(a)=s} to obtain an approximate value for x(b).
  • Be familiar with the classical 4^th order Runge-Kutta method for solving an initial value problem {dx/dt = f(t,x(t)); x(a)=s} to obtain an approximate value for x(b).

• material from section 8.3 of our textbook
• material from section 8.4 of our textbook
• interpolating B-splines (from section 9.3 of our textbook)
• asking you to solve a specific, given initial value problem using Runge-Kutta methods