Numerical Linear Algebra, August-December 2016
When Tuesday and Thursdays morning from 8:30 AM to 10:00 AM
Where Department of Computational & Data Sciences, SERC building, Room 102, Seminar Hall
Office: Department of Computational & Data Sciences, SERC building, Room 205.
Office hours: Tuesday 3:00 PM-5:00 PM or by appointment.
Office: Department of Computational & Data Sciences, SERC building, Room 208.
Office hours: Tuesday 2:00 PM-3:00 PM.
Matrix and vector norms, floating points arithmetic, forward and backward stability of algorithms, conditioning of a problem, perturbation analysis, algorithmic efficiency, Structured matrices, 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, Rank and matrix approximations, image compression using SVD, 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, 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.
We will attempt to cover the book by Biswa Nath Datta. Some material (especially the later part of the course) will be drawn from Trefethen & Bau as well. Notes will be posted as and when required.
- Numerical Linear Algebra and Applications, 2nd Edition, by Biswa Nath Datta
- Numerical Linear Algebra, by Lloyd N. Trefethen & David Bau III
Other useful texts to possess:
- Applied Numerical Linear Algebra, by James W. Demmel
- 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
There will be a total of 15 homework due roughly weekly. Homework will be posted on this website and will be due on Wednesday before 5 PM. Late homework will not be accepted. The homework will consist of 3-4 written questions and couple of 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 NLA_2016_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.
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 mathematical 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.
To use \(\LaTeX\), try any of the following online LaTeX compilers.
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. Depending on the comfort level of students with coding, I may reduce/remove the coding part from the subsequent assignments or might have them as extra credit/bonus.
For C++ compilers, try any of the following (there any many other options if you google for "online compiler"). Pick your favorite one.
There will be roughly a dozen short (3-5 minutes) surprise quiz in class, which will used for marking attendance.
Midterm : October 4th, Tuesday in class.
Final exam : December 5th, Monday, from 3:00 PM to 5:00 PM.
There will be no make up exams. All exams will be closed book, closed notes. No internet/calculator allowed.
Below is a tentative calendar, which will be updated as we make progress in the course. The chapters mentioned below are from the book by Biswa Nath Datta (unless and otherwise stated explicitly).