gibson:teaching:fall-2016:math753:finalexam

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 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

- be able to approximate the first & second derivatives of a function from the values where the are evenly spaced gridpoints
- provide error estimates of those approximate derivatives
- be able to approximate the integral of the function from evenly space gridpoint values , 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 th order differential equation in one variable to a system of first order differential equations in variables?

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