Course11110511160 lectures
Matrix Solver
Prof. Somnath Roy · IIT Kharagpur
About
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
Lectures
Week 1
- Lecture 01 : Introduction to Matrix Algebra - I
- Lecture 02 : Introduction to Matrix Algebra - II
- Lecture 03 : System of Linear Equations
- Lecture 04 : Determinant of a Matrix
- Lecture 05 : Determinant of a Matrix (Contd.)
Week 2
- Lecture 06 : Gauss Elimination
- Lecture 07 : Gauss Elimination(Contd.)
- Lecture 08 : LU Decomposition
- Lecture 09 : Gauss-Jordon Method
- Lecture 10 : Representation of Physical Systems as Matrix Equations
Week 3
- Lecture 11 : Tridiagonal Matrix Algorithm
- Lecture 12 : Equations with Singular Matrices
- Lecture 13 : Introduction to Vector Space
- Lecture 14 : Vector Subspace
- Lecture 15 : Column Space and Nullspace of a Matrix
Week 4
- Lecture 16 : Finding Null Space of a Matrix
- Lecture 17 : Solving Ax=b when A is Singular
- Lecture 18 : Linear Independence and Spanning of a Subspace
- Lecture 19 : Basis and Dimension of a Vector Space
- Lecture 20 : Four Fundamental Subspaces of a Matrix
Week 5
- Lecture 21 : Left and right inverse of a matrix
- Lecture 22 : Orthogonality between the subspaces
- Lecture 23 : Best estimate
- Lecture 24 : Projection operation and linear transformation
- Lecture 25 : Creating orthogonal basis vectors
Week 6
- Lecture 26 : Gram-Schmidt and modified Gram-Schmidt algorithms
- Lecture 27 : Comparing GS and modified GS
- Lecture 28 : Introduction to eigenvalues and eigenvectors
- Lecture 29 : Eigenvlues and eigenvectors for real symmetric matrix
- Lecture 30 : Positive definiteness of a matrix
Week 7
- Lecture 31 : Positive definiteness of a matrix (Contd.)
- Lecture 32 : Basic Iterative Methods: Jacobi and Gauss-Siedel
- Lecture 33 : Basic Iterative Methods: Matrix Representation
- Lecture 34 : Convergence Rate and Convergence Factor for Iterative Methods
- Lecture 35 : Numerical Experiments on Convergence
Week 8
- Lecture 36 : Steepest Descent Method: Finding Minima of a Functional
- Lecture 37 : Steepest Descent Method: Gradient Search
- Lecture 38 : Steepest Descent Method: Algorithm and Convergence
- Lecture 39 : Introduction to General Projection Methods
- Lecture 40 : Residue Norm and Minimum Residual Algorithm
Week 9
- Lecture 41 : Developing computer programs for basic iterative methods
- Lecture 42 : Developing computer programs for projection based methods
- Lecture 43 : Introduction to Krylov subspace methods
- Lecture 44 : Krylov subspace methods for linear systems
- Lecture 45 : Iterative methods for solving linear systems using Krylov subspace methods
Week 10
- Lecture 46 : Conjugate gradient methods
- Lecture 47 : Conjugate gradient methods(Contd.)
- Lecture 48 : Conjugate gradient methods(Contd.) and Introduction to GMRES
- Lecture 49 : GMRES (Contd.)
- Lecture 50 : Lanczos Biorthogonalization and BCG Algorithm
Week 11
- Lecture 51 : Numerical issues in BICG and polynomial based formulation
- Lecture 52 : Conjugate gradient squared and Biconjugate gradient stabilized
- Lecture 53 : Line relaxation method
- Lecture 54 : Block relaxation method
- Lecture 55 : Domain Decomposition and Parallel Computing
Week 12
Syllabus and materials
At NPTEL, as the course publishes them.
Credits and licence
- Institution
- IIT Kharagpur
- Department
- Management
- Teacher
- Prof. Somnath Roy
- Level
- Graduate
- Language
- English
- Subjects
- Business and management
- Source
- NPTEL
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
- Who can watch
- NPTEL gives no rating. Invisible Network shows it to Tweens profiles and older, judged from what the source says about it. How courses are placed