Study Outline for Final Exam

The exam will consist of 4 problems. You will required to do an LU decomposition with virtual scaling and partial pivoting. This will be 40 points. The additional 60 points will come from 3 out of at least 5 problems. Problems will come from all parts of the course.

Final exam study outline:

• Trapizodal rule, Simpson's rule, Gauss rule.
• Computing integrals numerically with each rule.
• Derivation of Gauss 2 point rule.
• Using RK-4 to solve for the solution to a first order initial value ODE.

• Norms, Condition Numbers

  p norms of general vectors

  definition of induced matrix p norms

  p=infinity and p=1 matrix norms, actual computation.

  definition of p=2 matrix norm. (Do not need to compute it.)

• Definition of Condition number of a function.

  Definition of the Condition number of a matrix.

  Error propagation thru condition numbers.

  How can the relative roundoff error in b (=A*x) be small with the roundoff error in x still being large?

• Iterative methods for solutions of linear systems

  Jacobi method.

  Gauss-Seidel method.

  Given a system of linear equations recast it in the proper form so that the G-S or J method is sure to converge. Compute several iterates.

  Definition of a convergent matrix.

  A sufficient condition to insure that S or G-J converge (Diagonal dominance.)

• Finite differences.

  Computation of L_ij(x) (=1 when X_i=X_j, =0 when x_i!=x_j)

  Computation of divided difference table.

  Numerically best form of the Newton divided difference approximating polynomial for expansion about a specific given point. e.g. given the value of f(x) ant x=0,1.2,3.4,5.0, and 7.3 find a Newton divided difference polynomial of the third degree which interpolates the data at the point x=1.4.

  Facts about f[x_0,x_1,x_2,...,x_n].

  Constant Differences.

  Why interpolations of high degree (>7) are undesireable.

• Spline aproximations

  Definition of a Natural cubic spline.

  Defintion of parabolic and linear splines.

  How to extend a given function as a cubic spline.

  Cubic spline end conditions:

  1. Natural end conditions.
  2. Parabolic end conditions.
  3. Not a knot end conditions.

  Computation of a natural cubic spline going through three points (x_0,y_0), (x_1,y_1) and (x_2,y_2) with natural end conditions.

  Special cases of splines which are easy to compute.

  Note: you are NOT responsible for the general formula to compute the spline ging through n points.

• Theory of numerical differentiation

  Truncation error.

  Roundoff error.

  Simple approximations to f'(x) which have O(h) and O(h^2) error.

  Construction of how to chose the optimal value of h to minimize the error in computation.

• Eigenvlaues

  Compute the most dominant eigenvalue by the power method.

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