# Numerical Linear Algebra, August-December 2018

Credits 9 credits
When F slot (Tuesday,Wednesday, Thursday, Friday)
Where Humanities and Sciences Block, Room 264

### Instructor

Sivaram Ambikasaran,

### Detailed Syllabus

Matrix and vector norms, floating points arithmetic, forward and backward stability of algorithms, conditioning of a problem, perturbation analysis, algorithmic complexity, Solving linear systems, Gaussian elimination, LU factorization, Pivoting, Cholesky decomposition, Iterative refinement, QR factorization, Gram-Schmidt orthogonalization, Projections, Householder reflectors, Givens rotation, Singular Value Decomposition, Least squares and least norm solution of linear systems, pseudoinverse, normal equations, Eigenvalue problems, Gershgorin theorem, Similarity transform, Eigenvalue & eigenvector computations and sensitivity, Power method, Schur decomposition, Jordan canonical form, QR iteration with & without shifts, Hessenberg transformation, Rayleigh quotient, Symmetric eigenvalue problem, Jacobi method, Divide and Conquer, Computing the Singular Value Decomposition, Golub-Kahan-Reinsch algorithm, Chan SVD algorithm, Rank and matrix approximations, image compression using SVD, Generalized SVD, Generalized and Quadratic eigenvalue problems, generalized Schur decomposition (QZ decomposition), Iterative methods for large linear systems: Jacobi, Gauss-Seidel and SOR, convergence of iterative algorithms, Krylov subspace methods: Lanczos, Arnoldi, MINRES, GMRES, Conjugate Gradient and QMR, Pre-conditioners, Approximating eigenvalues and eigenvectors, Structured matrix computations, Designing matrix algorithms on modern computer architectures

### Textbooks

• Numerical Linear Algebra and Applications, 2nd Edition, by Biswa Nath Datta from now on abbreviated as BD
• Applied Numerical Linear Algebra, by James W. Demmel from now on abbreviated as JD
• Numerical Linear Algebra, by Lloyd N. Trefethen & David Bau III from now on abbreviated as TD
We will be drawing material from all three books. Notes will be posted as and when required.

• Matrix Computation, by Gene H. Golub & Charles F. Van Loan
• Functions of Matrices: Theory and Computation, by Nicholas Higham
• Accuracy and Stability of Numerical Algorithms, by Nicholas Higham

### Other references

 Evaluation Homework Quiz-I Quiz-II Final Exam Points 30 20 20 30

### Homework

There will be a total of $$15$$ homework due roughly weekly. Homework will be posted on this website and will be due on Monday before 5 PM. Late homework will not be accepted. The homework will consist of both written questions and MATLAB/C++ exercises. Students are strongly encouraged to typeset their solutions using $$\LaTeX/\TeX$$ (10% bonus points for submitting in $$\LaTeX$$/$$\TeX$$). To save trees, the students need to send their homework through email to with the subject reading 2018_NLA_HW_#_firstname, where # needs to be replaced with the homework number (between $$1$$ and $$15$$) and firstname is to be replaced with your first name in lower case. Details on how to submit the computing part of the homework will be elaborated in the homework itself. No collaboration is allowed for homework.

Grading: The grader will expect you to express your ideas clearly, legibly, and completely, often requiring complete English sentences rather than merely just a long string of equations or unconnected mathematical expressions. This means you could lose points for poorly written proofs or answers. Clear exposition is a crucial ingredient of technical communication. Clarity of thought and presentation is more important in mathematics & sciences than any other field. The only way to master exposition is by repeated practicing.

#### $$\LaTeX$$

To use $$\LaTeX$$, try any of the following online LaTeX compilers.

### Computational requirement

Each homework will have couple of computational exercises. Students must be comfortable with programming and are expected to have working knowledge in C++ & MATLAB. If not, they should be able to learn and immediately pick it up.

#### C++ compilers

For C++ compilers, try any of the following (there any many other options if you google for "online compiler"). Pick your favorite one.

### Exams

 Exam Date Quiz-I August $$31$$, Friday Quiz-II October $$12$$, Friday End Term November $$27$$, Tuesday

All exams will be closed book, closed notes, closed calculator, closed computer, closed tablet, closed internet. Did I miss any thing else? Please have an open mind though.

### Calendar

Below is a tentative calendar, which will be updated as we make progress in the course.
 Week Tuesday Wednesday Thursday Friday Homework Jul 30- Aug 3 Motivation & Chapter 1 from BD HW1: Due Aug 6 Aug 6-10 HW2: Due Aug 13 Aug 13-17 Extra class Holiday HW3: Due Aug 20 Aug 20-24 Extra class Holiday HW4: Due Aug 27 Aug 27-31 Quiz-I, August $$31$$ HW5: Due Sep 3 Sep 3-7 HW6: Due Sep 10 Sep 10-14 Extra class Holiday HW7: Due Sep 17 Sep 17-21 Extra class Holiday HW8: Due Sep 24 Sep 24-28 HW9: Due Oct 1 Oct 1-5 HW10: Due Oct 8 Oct 8-12 Quiz-II, October $$12$$ HW11: Due Oct 15 Oct 15-19 Extra class Holiday HW12: Due Oct 22 Oct 22-26 HW13: Due Oct 29 Oct 29-Nov 2 HW14: Due Nov 5 Nov 5-9 Last day of class HW15: Due Nov 12 Nov 26-30 EndSem, November $$27$$