## CSci 5304 - Fall 2021 - Lecture Notes

• Topics to be asked about on the second take-home exam (scheduled for Nov 18) marked in magenta.

• Topics to be asked about on the third take-home exam (scheduled for Dec 13) marked in red.

1. LecN1.pdf; LecN1_R.pdf; 09/08/2021
Introduction; Types of problems seen in this course ;
Math. background; Matrices; Eigenvalues and eigenvectors;
Null space and range; Rank;
Types of matrices; Special matrices.
Background ; READ GvL: 1.1--1.3, 2.1.

2. LecN2.pdf; LecN2_R.pdf; 09/13/2021
Inner products and norms; Vector norms;
Convergence of vector sequences; Matrix norms.

3. LecN3.pdf; LecN3_R.pdf; 09/15/2021
Solving Linear Systems; Background;
Gaussian Elimination (review); Gauss-Jordan;
The LU factorization; Pivoting.

4. LecN4.pdf; LecN4_R.pdf; 09/22/2021
Concepts in floating point arithmetic; Error analysis
Forward and backward errors; Errors in inner products;
Application to linear systems.

5. LecN5.pdf; LecN5_R.pdf; 09/22/2021
Perturbation theory for linear systems; sensitivity analysis;
Condition numbers; Error bounds; Norm-wise error analysis;
Estimating cond. numbers;
Estimating forward errors from residual norms.

6. LecN6.pdf; LecN6_R.pdf; 10/11/2021
Positive Definiteness; Symmetric Positive Matrices;
The LDLT and Cholesky factorizations.

7. LecN7.pdf; LecN7_R.pdf; 10/25/2021
The Gram-Schmidt algorithms and the QR Factorization;
Least-squares problems; Applications; Data fitting;

8. LecN8.pdf; LecN8_R.pdf; 10/27/2021
The Householder QR; the rank Deficient case; Computational cost;
Solving Least-squares problems with the householder QR;
Givens rotations and the Givens QR.

9. LecN9.pdf; LecN9_R.pdf; 10/27/2021
Orthogonal subspaces & orthogonal projectors; Orthogonal decomposition;
The 4 fundamental subspaces; the URV decomposition.
Introduction to the Singular Value Decomposition.
The singular value decomposition and its properties.

10. LecN10.pdf; LecN10_R.pdf; 11/01/2021
The Pseudo inverse; Application to least-squares; Moore-Penrose Pseudo-inverse.
Ill-conditioned systems and the SVD; Numerical rank and the SVD;

11. LecN11.pdf; LecN11_R.pdf; 11/03/2021
A few applications of the SVD; Regularization; Information retrieval;
Principal Component Analysis (PCA); Dimension reduction.

12. LecN12.pdf; LecN12_R.pdf; 11/10/2021
Eigenvalue problems; Brief background; the Schur form;
Perturbation analysis; Gerschgorin; conditioning of a simple eigenvalues;
The power method and related techniques.

13. LecN13.pdf; LecN13_R.pdf; XX/XX/2021
Eigenvalue problems (continued); The QR algorithm; Practical variants;
Symmetric eigenvalue problems -Min-max theorem; The law of inertia;
The QR algorithm for symmetric matrices; The Jacobi algorithm.