Swayam Central

# Numerical Linear Algebra

By Prof. P. N. Agarwal, Prof. D. N. Pandey   |   IIT Roorkee
This course is a basic course offered to UG/PG students of Engineering/Science background. It contains basics of matrix algebra, computer arithmetic, conditioning and condition number, stability of numerical algorithms, vector and matrix norms, convergent matrices, stability of non-linear systems, sensitivity analysis, singular value decomposition (SVD), algebraic and geometric properties of SVD, least square solutions, Householder matrices and applications, QR method, Power method and applications, Jacobi method for finding the eigenvalues of a given matrix. This course has tremendous applications in diverse fields of Engineering and Sciences such as control theory, image processing, numerical analysis and dynamical systems etc.

INTENDED AUDIENCE  : UG and PG students of technical institutions/ universities/colleges
PRE-REQUISITES : None
INDUSTRY SUPPORT : None

Learners enrolled: 1262

### SUMMARY

 Course Status : Ongoing Course Type : Elective Duration : 12 weeks Start Date : 27 Jan 2020 End Date : 17 Apr 2020 Exam Date : 26 Apr 2020 Enrollment Ends : 03 Feb 2020 Category : Mathematics Level : Undergraduate/Postgraduate
This is an AICTE approved FDP course

### COURSE LAYOUT

Week 1: Matrix operations and type of matrices, Determinant of a Matrix, Rank of a matrix, Vector Space-I, Vector Space-II
Week 2: Linear dependence and independence, Bases and Dimensions – I, Bases and Dimension - II, Linear Transformation - I,
Linear Transformation - II
Week 3: Orthogonal subspaces, Row space, column space and null Space, Eigenvalues and Eigenvectors-I, Eigenvalues and
Eigenvectors-II, Diagonalizable Matrices
Week 4: Orthogonal Sets, Gram Schmidt orthogonalization and orthonormal bases, Introduction to Matlab, Sign integer representation
Computer representation of numbers
Week 5: Floating point representation, Round-off error, Error propagation in computer arithmetic, Addition and multiplication of floating
point numbers, Conditioning and condition numbers-I
Week 6: Conditioning and condition numbers-II, Stability of numerical algorithms-I, Stability of numerical algorithms-II, Vector
norms - I, Vector norms - II
Week 7: Matrix  Norms - I, Matrix Norms-II, Convergent Matrices - I, Convergent Matrices - II, Stability of non-linear system
Week 8: Condition number of a matrix: Elementary properties, Sensitivity analysis-I, Sensitivity analysis-II, Residual theorem, Nearness to singularity
Week 9: Estimation of the condition number,  Singular value decomposition of a matrix – I, Singular value decomposition of a
matrix - II, Orthogonal Projections, Algebraic and geometric properties of  matrices using SVD
Week 10: SVD and their applications, Perturbation theorem for singular values, Outer product expansion of a matrix, Least
square solutions-I, Least square solutions-II
Week 11: Psudeo - inverse  and least square solution, Householder  matrices  and their applications, Householder QR factorization –I,
Householder QR factorization –II, Basic theorems on eigenvalues and QR method
Week 12: Power method, Rate of convergence of  Power method, Applications of Power method with shift, Jacobi method-I, Jacobi method-II

### BOOKS AND REFERENCES

1. V. Sundarapandian, Numerical Linear Algebra, PHI, 2008.
2. Biswa Nath Dutta, Numerical Linear Algebra and Applications, SIAM, 2010.
3. Roger A. Horn and Charles R. Johnson, Matrix Analysis, Cambridge University Press, 1994.
4. William Ford, Numerical Linear Algebra with Applications, Academic Press, 2014..

### Prof. P. N. Agarwal

IIT Roorkee
Dr. P. N. Agarwal is a Professor in the Department of Mathematics, IIT Roorkee. His area of research includes approximation Theory and Complex Analysis. He delivered 13 video lectures on Engineering Mathematics in NPTEL Phase I and recently completed Pedagogy project on Engineering Mathematics jointly with Dr. Uaday Singh in the same Department. Further he has completed online certification course “Mathematical methods and its applications” jointly with Dr. S.K. Gupta of the same department. He taught the course on “Integral equations and calculus of variations” several times to MSc (Industrial Mathematics and Informatics) students. He has supervised nine Ph.D. theses and has published more than 187 research papers in reputed international journals of the world. Currently, he is supervising eight research students.

### Prof. D. N. Pandey

Dr. D. N. Pandey is an Associate Professor in the Department of Mathematics, IIT Roorkee. Before joining IIT Roorkee, he worked as a faculty member in BITS-Pilani Goa campus and LNMIIT Jaipur. His area of expertise includes semigroup theory and functional differential equations of fractional and integral orders. He has already prepared e-notes for the course titled “Ordinary Differential Equations and Special Functions” under e-Pathshala funded by UGC. Also, he has published a book titled “Nonlocal Functional Evolution Equations: Integral and fractional orders, LAP LAMBERT Academic Publishing AG Germany”. He has delivered several invited talks at reputed institutions in India and abroad. He has guided three PhD theses and has published more than 60 papers in various international journals of repute. Currently, he is supervising five research students.

### 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: 26th April 2020, Morning 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 Roorkee. It will be e-verifiable at nptel.ac.in/noc.
• Only the e-certificate will be made available. Hard copies will not be dispatched.