Study guide for Examlet 5

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

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All topics for examlet 4 study guide
How do eigenvectors/eigenvalues change under Shift? Inversion? Taking the $n$th power? Taking the inverse?
When is a matrix diagonalizable? Are all matrices diagonalizable?
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?

What is the singular value decomposition?
What are left/right singular vectors with respect to $A^TA$ and $AA^T$? singular values?
What properties do the singular vectors and singular values satisfy?
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 the SVD be used for low-rank approximation?
What is the pseudoinverse? What properties does it satisfy?

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 matrix) least-squares problem?

What are the drawbacks of equispaced nodes in interpolation? How are those addressed by edge-clustered nodes?
What are the drawbacks of monomials as an interpolation basis? How are those addressed by orthogonal polynomials?
What does it mean for two functions to be orthogonal?
What does it mean for two polynomials to be orthogonal?
How are the Legendre polynomials defined? and the Chebyshev polynomials?
What are the Chebyshev interpolation nodes?

How can a first/second/third derivative be computed using interpolation? How would that be expressed using Vandermonde matrices?
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?
What are finite difference formulas?
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 $h$?)