Mar 1 Eckart-Young for F-norm, POD. This file contains all updated SVD notes without my writing.
Mar 3 review of GS algorithm, QR, and modified QR.
Mar 13 preview. Done Modified GS for QR (from the projection perspective, I corrected the index mistake in class), flops and operation counts.
Mar 15 operation count of the modified GS for QR, construction of Q_k (of the Household reflector), Household triangularization.
Mar 17 Householder algorithm, implicit computation of Q^*b and Qx
Mar 20 least square problems, normal equation, QR to compute least square, SVD and least square
Mar 22 least square problem examples, relative conditioning number, examples of well/ ill-conditioned problem, polynomial root finding is ill-conditioned.
Mar 27 conditioning of Ax given perturbation in x (special case when A is square invertiable, A is rectangle but has full col rank), conditioning of Ax = b given perturbation in A.
Mar 29 midterm 2 summary, floating point, machine epsilon
Mar 31 algorithm accuracy, backward stability, condition plus bw stable = accuracy, sin(x) when x = \pi/2-\delta.
April 3 backward error and numerical verification of QR backward stability. The kaggle code of the class is here: QR.
April 5 proof of backward stability of QA where Q is an unitary matrix.
April 7 complete notes of conditioning and backward stability.
April 7 backward stability of solving Ax = b by QR (I corrected a small error I made in the first page of the handwritten notes), and the algorithm accuracy.
April 10 Rayleigh quotient, power iteration (from eigenvector approx to eigenvalue by Rayleigh quotient), inverse iteration for eigenvalue problems.
April 12 Rayleigh quotient iteration, reduction to Hessenberg form.
April 17 QR iteration for eigenvalue, proof of A^k = Q^{(k)}R^{(k)}
April 19 connection of QR with Power iteration, shifted QR, preprocessing.