CSci 5304 - Fall 2021 - Lecture Notes

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     • 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.
   Matrices ; READ GvL: 2.1

2. LecN2.pdf; LecN2_R.pdf; 09/13/2021
   Inner products and norms; Vector norms;
   Convergence of vector sequences; Matrix norms.
   Norms ; READ GvL 2.2-2.3;

3. LecN3.pdf; LecN3_R.pdf; 09/15/2021
   Solving Linear Systems; Background;
   Gaussian Elimination (review); Gauss-Jordan;
   The LU factorization; Pivoting.
   Systems ; READ GvL 3.\{1,3,5\}

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.
   READ GvL 2.7
   READ GvL 2.7
   

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.
   READ GvL 3.5
   

6. LecN6.pdf; LecN6_R.pdf; 10/11/2021
   Positive Definiteness; Symmetric Positive Matrices;
   The LDLT and Cholesky factorizations.
   READ GvL 4
   

7. LecN7.pdf; LecN7_R.pdf; 10/25/2021
   The Gram-Schmidt algorithms and the QR Factorization;
   Least-squares problems; Applications; Data fitting;
   READ GvL 5, 5.3

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.
   READ GvL 5.1
   

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;
    READ GvL 2.4, 5.4-5
    

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

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.
    READ GvL 7.1-7.4,7.5.2
    

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.
    READ GvL 8.1-8.2.3
    

14. LecN14.pdf; LecN14_R.pdf; XX/XX/2021
    [Last set of notes] Large Sparse eigenvalue problems;
    Rayleigh-Ritz projection; Subspace iteration;
    Lanczos algorithms; Loss of orthogonality;
    Golub-Kahan-Lanczos bidiagonalization.
    READ Gvl4 10.1,10.5.1
    

15. LecN15.pdf; LecN15_R.pdf; XX/XX/2021
    A few applications. Graph partitioning;
    Page rank; ...