Math 753/853 final exam topics
Wed, Dec 14, 2016 10:30am-12:30pm Kingsbury N343
Floating point numbers
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

of

?
be able to compute the LU decomp of a small matrix by hand
backsubstitution, forward substitution
using LU to solve

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

?
QR decomposition
what is a QR decomposition?
what algorithm do you know for computing the QR decomposition?
what are the formulae for the elements

of

and the column vectors

of

?
what is an orthogonal matrix?
how to use QR decomp to solve a square

problem
how to use QR decomp to find a least-squares solution to an oblong

problem (

matrix

, with

)
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

, 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

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

problems to find the best fit for functions of the following forms to

pairs of datapoints

an

th order polynomial
an exponential
a power law
a curve of the form
Finite differencing and quadrature
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
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

th order differential equation in one variable to a system of first order differential equations in

variables?