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

What do forward/backward substitution accomplish? How? At what computational cost?
What are elimination matrices?
How can they be inverted? multiplied by one another?
What happens when you multiply an elimination matrix by another matrix? a vector?
What is LU factorization? How does it work? What is its computational cost?

Does the LU factorization always exist? Why is pivoting needed?
What is partial pivoting? What is its purpose? How does it work?
What is a permutation matrix? How does it help realize partial pivoting?
What is the form of an LU factorization with pivoting? (can be $PA=LU$ or $A=\bar PLU$--note that $\bar P=P^T$)
What is the cost of LU factorization with pivoting?

How is LU factorization used to solve a linear system of equations $A\b x=\b b$?
How is LU factorization used to solve many linear systems of equations $A\b x_i=\b b_i$ with many different right-hand sides?
How is LU factorization used to solve a matrix equation $AX=B$?
Be able to (write down algorithms to) solve more complicated matrix equations involving triangular/orthogonal/other matrices.
How is LU factorization used to compute determinants?

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 is the Newton basis for interpolation? What are its advantages? its drawbacks?
What does it mean for two polynomials to be orthogonal?
What are the Legendre polynomials? 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?