Study guide for Examlet 3
Here is a non-exhaustive list of questions you should be able to answer as you prepare for the examlet.
Past chapters
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(Recall from the course policies that our examlets are cumulative.)
Floating Point
- What is fixed point arithmetic? How are numbers represented in fixed point?
- What is floating point arithmetic? How are numbers represented in floating point?
- What is the significand? the exponent? of a floating point number?
- What is machine epsilon?
- How can you quantify the least possible amount of rounding error that floating point arithmetic introduces with every operation?
- How are floating point numbers stored? What is the 'implicit one' in the significand?
- How is zero represented in floating point?
- What are subnormal numbers? What is (gradual and non-gradual) underflow? overflow?
- What can we say about error in the subnormal representation of numbers?
- How is floating point addition performed?
- What is catastrophic cancellation? How can you estimate the relative error (/number of digits) in the result of a calculation that incurs catastrophic cancellation?
Computational Linear Algebra
- How are images represented as vectors? What does addition/scalar multiplication mean for them?
- How are sound clips represented as vectors? What does addition/scalar multiplication mean for them?
- How are shapes represented as vectors? What does addition/scalar multiplication mean for them?
- How do matrices operate on bases? How can linear operations on a basis be expressed using a basis?
Graphs and Sparsity
- What is an adjacency matrix?
- What is a Laplacian matrix?
- How do these representaitons change for directed or weighted graphs?
- What does matrix-vector multiplication with an adjacency matrix mean?
- What is a Markov chain? What is the Markov property?
- What is a transition matrix/graph? What is a steady state?
- What is a sparse matrix?
- How does CSR format for the representation of sparse matrices work?
- What would matrix-vector multiplication with CSR matrices look like?
- What is the computational cost of sparse-matrix vector multiplication?
Vector and Matrix Norms
- What criteria does a vector norm have to satisfy?
- What is the triangle inequality?
- What are the $p$-norms?
- What is the "unit ball" of a norm?
- What is an induced matrix norm?
- What is the Frobenius matrix norm?
- What does an induced matrix norm imply about the amplification of a vector norm during matrix-vector multiplication?
- How can the matrix norm of a diagonal matrix be computed?
- How can an induced matrix norm be estimated by sampling?