Final Exam Review

The final exam will consist of questions selected from the following collection of questions and topics.

$\hangindent\parindent \newcount\qnum\newcount\qprt\qnum=0\qprt=0 \def\qn{\global\qprt=0\global\advance\qnum by1\bigskip \par\noindent\item{\bf\the\qnum.}} \qn The definition of relative and absolute error.$

$\qn The difference between generated and propagated error.$

$\qn Definition of forward, backwords and central differencing operators.$

$\qn Make a difference table and explain the expected round-off error for the $n$-th column by means of the table $$ \vbox{\halign{#\hfil&&\quad\hfil$#$\cr Order of difference&1&2&3&4&5&6\cr Expected error limits&\pm1&\pm2&\pm3&\pm6&\pm12&\pm22\cr }} $$$

$\qn Taylor's series, the remainder term, and use of the remainder term.$

$\qn Newton's forward and backwords interpolating polynomials.$

$\qn Lagrange interpolation formula.$

$\qn Describe, as an algorithm, interval bisection, the secant method, and Newton's method for solving a non-linear equation.$

$\qn Convergence criterion for Newton's method.$

$\qn Show the rate of convergence for Newton's method is quadratic and in particular that the error $E_n$ satisfies $$ E_{n+1}=\alpha-x_{n+1}= -{E_n^2 f''(\xi_n)\over 2 f'(x_n)} \approx -{E_n^2 f''(\alpha)\over 2 f'(\alpha)}. $$$

$\qn Definition of conditioning number for a matrix and what it means.$

$\qn Show that if $Ax=b$ and $A\bar x=\bar b$ then $$ {\|x-\bar x\|_\infty\over \|x\|_\infty} \le{\rm cond}(A){\|b-\bar b\|_\infty\over \|b\|_\infty}.$$$

$\qn Describe, as an algorithm, the Gauss-Seidel iterative method for solving a system of linear equations.$

$\qn Derive the normal equations for the linear least squares minimization problem.$

$\qn Define the pseudo inverse $A^+$ of $A$.$

$\qn Definition of a Householder matrix.$

$\qn Explain in terms of numerical roundoff error why and how to choose the pivots in Gauss-Jordan elimination.$

$\qn Definition of diagonally dominant.$

$\qn Definition of the weak min-max property.$

$\qn Explain what is meant by the order of a integration method.$

$\qn Find the truncation error.$

$\qn Explain the idea behind Romberg integration.$

$\qn Compare taking a smaller step size to using a higher order method to attain increased accuracy when for solving an initial value problem.$

$\qn Find conditions on $h$ so that the central difference method for solving a two point boundary problem has a unique solution.$

Last updated: Sat Dec 7 07:12:05 PST 2002