Study guide for Examlet 2

Here is a non-exhaustive list of questions you should be able to answer as you prepare for the examlet.

Past chapters

See the study guide for examlet 1. Recall from the course policies that our examlets are cumulative. However, the focus will be on new material.

What is interpolation? What are interpolation nodes?
What is a Vandermonde matrix?
What is the monomial basis?
How does one determine the coefficients of a polynomial interpolant?
What is a generalized Vandermonde matrix?
How can interpolation be used to predict interpolant integrals or derivatives based on function values at a set of nodes?
What is the asymptotic (big-O) behavior of the error in interpolation? What does $h$ represent in the error term?
For what types of functions do you expect interpolation to work well? For which is that not the case?

What is a random variable?
What is a distribution function? What requirements does it satisfy?
What is a sample?
What is a sample mean?
What is an expected value? What is variance?
How does taking a function of a set of samples (equivalently, a random variable) change the probability density function of their distribution?
What assumptions are needed on a set of samples to guarantee convergence of their average to the mean of a random variable?
How do you approximate an expected value of one random variable based on a sample of another?
How does one use Monte Carlo for integration?
What is the appropriate scaling factor if Monte Carlo integration is done based on samples from a larger region than the integration domain?
What is the asymptotic (big-O) behavior of the error in sampling?
In relative terms, for what types of problems is Monte Carlo effective and for what types of problems is it not?

What are absolute and relative errors?
What does it mean for a result to have $n$ accurate digits?
What are common sources of error in numerical methods?
How does the number of accurate digits relate to rounding?
What is a condition number?
What can you say about a condition number given data points with relative errors on inputs and outputs?

What is fixed point arithmetic? How are numbers represented in fixed point?
What is the significand? the exponent? of a floating point number?
What is floating point arithmetic? How do the significand and exponent define a floating point number?
What relative error does a floating point number representation of a real number have?
What numbers can be more accurately represented in fixed point than floating point? What about the other way around?