Study guide for Examlet 6

$\let\b=\mathbf$ Here is a non-exhaustive list of questions you should be able to answer as you prepare for the examlet.

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(Recall from the course policies that our examlets are cumulative.)

Given point values of a function, how can you use interpolation to compute an approximation of the derivative of that function at the same or different points?
How do finite difference formulas make this process more efficient?
If you shorten the distance between points from, say, $h$ to $h/2$, how will the finite difference formula change?
If you shift a finite difference formula from, say $3+h$ to, say, $4+h$, how does the formula change?
What is the order of accuracy of this process? (I.e. how does the error depend on $h$?)

Given point values of a function, how can you use interpolation to compute an approximation of the definite integral (over some interval) of that function?
How do quadrature rules make this process more efficient?
If you shorten the distance between points from, say, $h$ to $h/2$, how will the quadrature rule change?
If you shift a quadrature rule from, say $3+h$ to, say, $4+h$, how does the formula change?
What is the order of accuracy of this process? (I.e. how does the error depend on $h$?)

What is an eigenvector? an eigenvalue of a matrix? (i.e. know the definition)
How do eigenvectors/eigenvalues change under Shift? Inversion? Taking the $n$th power? Taking the inverse? Applying a similarity transform? (What is a similarity transform?)
When is a matrix diagonalizable? Are all matrices diagonalizable?
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?
How does the error in power iteration behave?
Under what circumstances will power iteration converge? When can we not guarantee that it will?
What is inverse iteration?
What is Rayleigh quotient iteration?

How can the power method be applied to find the equilibrium distribution of a Markov chain?
How can the power method be applied to understand the behavior of a system given as $d\mathbf y/dt = A \mathbf y$, for a given matrix $A$ and an unknown vector $\mathbf y$?

What is the singular value decomposition?
What are left/right singular vectors? singular values?
How can the SVD be computed?
Given a non-square matrix, what shape do the component matrices of the SVD have? In the 'full' case and the 'reduced' case?

How can you solve a (square) linear system using the 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 code to solve a (short-and-fat/tall-and-skinny) least-squares problem? (Quiz 25 provides practice for the short/fat case.)