Study guide for Midterm 1

Here is a non-exhaustive list of questions you should be able to answer as you prepare for the first midterm. The midterm will cover chapter 1-2.

Scientific Computing

What is posedness and conditioning of problems?
What are absolute and relative errors?
What are forward and backward errors?
What are the categories of sources of error in numerical methods?
What does it mean for a result to have $n$ accurate digits?
How does the number of accurate digits relate to rounding?
What is the relative and absolute condition number of evaluating a function for a given input? over a domain of inputs?
What are the main components of floating point numbers?
What is the typical relative accuracy in a floating point representation of a real number?
What are normalized floating point numbers?
Why are subnormal numbers used?
How many digits can be lost during addition and multiplication of floating point numbers?

What is a vector norm? a normalized vector? a unit ball?
What defined a matrix norm? what are induced vector norms? what is the Frobenius norm?
What is the matrix condition number?
What is the conditioning of solving a linear system? of matrix-vector multiplication?
How are the propagated data error, forward error, and backward error related? in terms of conditioning?
How does one solve a triangular linear system? What is the cost?
What is LU factorization, when does it exist and when is it unique?
Why is pivoting necessary and what type of pivoting strategies are possible?
What is the cost of Gaussian elimination?
How can Gaussian elimination be done with elementary elimination and permutation matrices?
How do you solve a linear system given a pivoted LU factorization?
What is the SVD of a matrix? What are the properties of the 3 matrices obtained from SVD of a matrix?
What are the Cholesky and $LDL^T$ factorizations? When can they be used and what are their advantages?
How can we solve a rank-1 perturbed problem via the Sherman Morrison formula?
How can we take advantage of tridiagonal or banded structure in solving a linear system?