# Matrix Solver

By Prof. Somnath Roy   |   IIT Kharagpur
Learners enrolled: 755
The objective of the course is to teach students about different algorithms for efficiently solving large matrix problems. One focus is discussing the mathematical background behind these schemes and the other focus is showing their implementations. The course will first try to present a basic understanding of fundamental issues of linear algebra relevant to matrix solutions. This will also discuss direct solution schemes like TDMA which has utility in scientific computing codes. The later half of the course will focus on iterative schemes for large matrices. Issues with convergence of the solvers will also be discussed. Students will be introduced to Krylov space based fast solvers. Implementations will be demonstrated using working codes. Techniques for improving convergence like preconditioning and multi-grid will also be briefly introduced.
As an outcome of the course, a student will build an understanding of the matrix equations and suitable solution algorithms for them. This will help him to develop his own solver as well as to appreciate the open-source/commercial libraries of linear algebra and to utilise them efficiently.

INTENDED AUDIENCE: Basic science, Engineering

PREREQUISITES: PG course but senior UG students may credit it
Summary
 Course Status : Completed Course Type : Elective Duration : 12 weeks Category : Mathematics Credit Points : 3 Level : Postgraduate Start Date : 25 Jul 2022 End Date : 14 Oct 2022 Enrollment Ends : 08 Aug 2022 Exam Date : 29 Oct 2022 IST

Note: This exam date is subjected to change based on seat availability. You can check final exam date on your hall ticket.

### Course layout

Week 1: Introduction to matrix and equation systems (1) Symmetric matrix and transpose (1) Determinant and rank(1) , Gauss elimination(2), Row permutation (1)
Week 2: Inverse of Matrix (1), Gauss Jordon Method for Finding Inverse(2) Matrix form of difference equations (1) Tridiagonal matrix algorithm (2)
Week 3:  Introduction to vector space (1) Column space and row space (1) Null space from solving Ax=0(3) Solving Ax=b (1)
Week 4:  Linear independence and spanning (1) Basis and dimension (1) Four fundamental subspaces and solvability of a matrix equation (2) Linear transformation (2)
Week 5:  Gram-Schmidt orthogonalization(3) QR Factorization for Normal equation (1) Eigen values, eigen vectors, spectral radius (2)
Week 6:  Mid term exam
Week 7:  Introduction to iterative methods, Gauss-Siedel, Jacobi and SOR (4) Convergence of iterative methods (2)
Week 8: Introduction to programming of matrix algorithms and code demonstration (2) Steepest descent algorithm and its variants (4)
Week 9:  Introduction to Krylov subspaces(2) Krylov subspace for Av, Arnoldi’’s Algorithm (2) GMRES (2)
Week 10:  Lanczos method for symmetric matrix (1) Conjugate gradient method (1) Krylov subspace for ATv, biorthogonalization and Biconjugate gradient and BiCG-STAB(4)
Week 11:  Lanczos method for symmetric matrix (1) Conjugate gradient method (1) Krylov subspace for ATv, biorthogonalization and Biconjugate gradient and BiCG-STAB(4)
Week 12: Block relaxation schemes (2) Introduction to Preconditioner and Multigrid Methods (4)

### Books and references

1. Gilbert Strang, “Linear Algebra and Its Applications”, 4th edition, Thomson Brooks/Cole, India (2006)
2. Yousef Saad, Iterative Methods for Sparse Linear Systems, SIAM (2003)
3. Gene H. Golub and Charles F Van Loan, Matrix Computations, The John Hopkins University Press, 4th Edition, (2013)

### Prof. Somnath Roy

IIT Kharagpur
My primary area of work is computational fluid dynamics (CFD). I have been working on several applications involving heat transfer, mixing and turbulence. I also investigate CFD problems involving high computational cost and try to propose high performance computing (HPC) methodologies to address them using multi-core clusters and GPGPU platforms. In last few years, I have been mostly involved in addressing flow problems with moving boundaries. My group works on developing immersed boundary method (IBM) based computationally efficient algorithms to solve moving boundary problems and we have utilized these implementations to predict flow and heat transfer in engineering and biological applications. Over last eight years I have also been involved in teaching several courses like Fluid Mechanics, Thermodynamics, Aerodynamics Advanced Engineering Mathematics, Matrix Computing and High Performance Scientific Computing to students at different levels (UG and PG). I have earlier offered an NPTEL MOOC course titled Matrix Solvers.

### Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
Date and Time of Exams: 29 October 2022Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
Registration url: Announcements will be made when the registration form is open for registrations.
The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Kharagpur. It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team