====== Math 753/853 final exam topics ======

Wed, Dec 14, 2016 10:30am-12:30pm Kingsbury N343 

Floating point numbers
   * binary representation
   * how #s of bits in mantissa and exponent lead to # digits in same
   * floating point arithmetic: expected accuracy of arithmetic operations
   * what is machine epsilon?

Solving 1d nonlinear equations
   * bisection: the algorithm, the required conditions, the convergence rate
   * Newton: the algorithm, the required conditions, the convergence rate
   * when to use bisection, when to use Newton

Gaussian elimination / LU decomposition
   * the LU algorithm: what are the formulae for computing the multipliers $\ell_ij$ of $L$?
   * be able to compute the LU decomp of a small matrix by hand
   * backsubstitution, forward substitution
   * using LU to solve $Ax=b$
   * pivoting --what is it, why is it a practical necessity?
   * what form does the LU decompostion take with pivoting? How do you use this form to solve $Ax=b$? 

QR decomposition
   * what is a QR decomposition?
   * what algorithm do you know for computing the QR decomposition?
   * what are the formulae for the elements $r_ij$ of $R$ and the column vectors $q_j$ of $Q$?
   * what is an orthogonal matrix? 
   * how to use QR decomp to solve a square $Ax=b$ problem
   * how to use QR decomp to find a least-squares solution to an oblong $Ax=b$ problem ($m \time n$ matrix $A$, with $M>n$)

Polynomials 
   * Horner's method: be able to rearrange a polynomial into Horner's form, and understand why you'd do that
   * Lagrange interpolating polynomial: be able to write down the Lagrange interpolating polynomial passing through a set of data points $x_i, y_i$, and understand why the formula works
   * Newton divided differences: know how to use this technique to find the interpolating polynomial through a set of data points $x_i, y_i$
   * Chebyshev points: what are they, what are they good for, why do we need them?

Least-squares models
   * Understand how to set up least-squares $Ax=b$ problems to find the best fit for functions of the following forms to $m$ pairs of datapoints $t_i, y_i$
       * an $n$th order polynomial 
       * an exponential $y=c e^{at}$ 
       * a power law $y=c t^a$ 
       * a curve of the form $y = c t e^{at}$ 

Finite differencing and quadrature
    * be able to approximate the first & second derivatives of a function $y(x)$ from the values $y_i = y(x_i)$ where the $x_i$ are evenly spaced gridpoints $x_i = x_0 + i h$
    * provide error estimates of those approximate derivatives
    * be able to approximate the integral $\int_a^b y(x) dx$ of the function $y(x)$ from evenly space gridpoint values $y_i = y(x_i)$, using the Trapeziod Rule and Simpson's rule
    * provide error estimates for those approximate integrals

Ordinary differential equations
    * what is an initial value problem?
    * why do we need to solve initial value problems numerically?
    * what are the timestepping formulae for 
        * Forward Euler
        * Midpoint Method (a.k.a. 2nd order Runge-Kutta)
        * 4th-order Runge-Kutta
        * Backwards Euler
        * Adams-Moulton
    * what are the global error estimates of the above timestepping formulae?
    * what is a global error estimate versus a local error estimate, and how are the two related?
    * what's the difference between an explicit method and an implicit method?
    * what's a stiff differential equation? what kind of method do you use for a stiff  equation?
    * how do you convert an $n$th order differential equation in one variable to a system of first order differential equations in $n$ variables?