Math 753/853 final exam topics

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

Floating point numbers

Solving 1d nonlinear equations

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?