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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 $M>n$)


  • 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?
gibson/teaching/fall-2016/math753/finalexam.txt · Last modified: 2016/12/12 19:00 by gibson