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 1-4.

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?
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 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?

What is the conditioning of a linear system?
How can you solve a (square) linear system using the SVD?
What are the norms and condition number using the SVD? what are they for a rectangular matrix?
What is the reduced SVD?
Why is the SVD helpful for (tall-and-skinny) least-squares system using the SVD? What is the residual in such a problem?
How can you solve a least-squares problem using the SVD?
Given an SVD of the matrix and a right-hand side, how would you find the 2-norm of the residual of a least-squares problem?
How would you use the SVD to solve a (short-and-fat/tall-and-skinny matrix) least-squares problem?
How can least squares problems be solved via the normal equations? What are the advantages and disadvantages of that?
What is QR factorization, when does it exist and is it unique?
What is a projection matrix?
What are the classical and modified Gram-Schmidt processes? What can you say about their stability?
What is the Householder QR factorization algorithm, and what can you say about its stability?
What is a Householder reflector matrix, what properties does it have?
What is a Givens rotation matrix, what properties does it have?
How can Givens rotations be used to factorize a sparse matrix?
How does one solve linear least squares problems using a QR factorization?

What is an eigenvector? an eigenvalue of a matrix? (i.e. know the definition)
What is a similarity transformation?
What is the relationship between the SVD and the eigenvalue decomposition?
When are eigenvectors linearly independent?
What are the Jordan and Schur forms?
What is a normal matrix?, a defective matrix? a diagonalizable matrix?
What is an eigenvalue multiplicity? what is a complex eigenvalue pair?
What is the relationship between the SVD and the eigenvalue decomposition?
What is power iteration?
What can be obtained using power iteration?
What is normalized power iteration? What problem does it address?
Given an approximate eigenvector, how can you estimate eigenvalues? What is the Rayleigh Quotient? What is inverse iteration? Rayleigh Quotient iteration?
What is the conditioning of eigenvalues and eigenvectors in an eigenvalue problem?
What is orthogonal iteration? QR iteration? how are they related?
How can one reduce a matrix to Hessenberg form and why is it helpful?
How can one incorporate shifting into QR iteration?
What is a Krylov subspace? how is it related to a Companion matrix?
What is the Arnoldi method? the Lanczos method?