# Numerical Linear Algebra, August-December 2018

Credits 9 credits
When F slot (Tuesday,Wednesday, Thursday, Friday)
Where Humanities and Sciences Block, Room 266

### Instructor

Sivaram Ambikasaran,
Office hours: 2PM-4PM, Mondays, HSB 241C

### Teaching Assistant

Vaishnavi Gujjula, ma16d301@smail.iitm.ac.in
Office hours: 2PM-4PM, Wednesdays

### Assignments

• Total of $$15$$ assignments due on Wednesdays before 5 PM.
• Late assignments will be marked zero.
• There will be both written and computational part in the assignments.
• We will be using AutoGradr to evaluate the computational part of the assignment. Register as a student on AutoGradr.
• Students are strongly encouraged to typeset their solutions using $$\LaTeX/\TeX$$ (10% bonus points for submitting in $$\LaTeX$$/$$\TeX$$).
• Students need to drop their assignments (both written and code) via the dropbox link. Each assignment has its own dropbox link. You need to have your roll number as the folder name (all characters in small case), compress it (either zip or rar or tar or any compression format), when you upload it.
• Any copying on assignments will give you a zero on the assignment.
• We will be using JPlag to detect similarities among multiple sets of source code files.
• 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.
• Assignment grades can be found here.

Lecture Summary

### Calendar

First Class will be on August 1, 2018, Wednesday. Below is a tentative calendar, which will be updated as we make progress in the course.
 Week Tuesday Wednesday Thursday Friday Assignment Jul 30- Aug 3 No class Motivation Chapter 1; BD Chapter 1; BD Assign-1: Due Aug 8 Drop Assignment; Solution Aug 6-10 Chapter 1,2; BD Holiday Chapter 2; BD No class Assign-2: Due Aug 15 Drop Assignment; Solution Aug 13-17 No class Class on Monday; Chapter 3, BD Chapter 3, BD Gaussian elimination and the need for pivoting Assign-3: Due Aug 22 Drop Assignment; Solution Aug 20-24 No Class Holiday Stability LU factorization Assign-4: Due Aug 29 Drop Assignment; Solution Aug 27-31 No Class LU factorization and asymptotically optimal algorithms Partial Pivoted LU Quiz-I, August $$31$$ Quiz-I solutions; Performance summary Assign-5: Due Sept 5 Drop Assignment; Solution Sep 3-7 No class No class No class Complete pivoting, cost of pivoting, Cholesky Assign-6: Due Sept 12 Drop Assignment; Solution Sep 10-14 No class Cholesky, Least squares Holiday No class Assign-7: Due Sept 26 Drop Assignment; Solution Sep 17-21 Least squares, Least norm (Monday) Holiday Assign-8: Due Sept 30 Drop Assignment; Solution Sep 24-28 Oct 1-5 Assign-9: Due October 21 Drop Assignment Oct 8-12 Quiz-II, October $$12$$, Quiz-II solutions Oct 15-19 Holiday Oct 22-26 Oct 29-Nov 2 Nov 5-9 Last day of class Assign-10: Due November 14Drop Assignment Nov 26-30 EndSem, November $$27$$

### Grading

 Evaluation Assignments Quiz-I Quiz-II Final Exam Points 30 15 15 40

### 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

We will be drawing material from all three books. Notes will be posted as and when required.

Other useful texts to read:

• 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

### Exams

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