C6.1 Numerical Linear Algebra - Material for the year 2020-2021

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Prof. Yuji Nakatsukasa
General Prerequisites: 

Only elementary linear algebra is assumed in this course. The Part A Numerical Analysis course would be helpful, indeed some swift review and extensions of some of the material of that course is included here.

Course Term: 
Course Lecture Information: 

16 lectures

Course Weight: 
1.00 unit(s)
Course Level: 

Assessment type:

Course Overview: 

Linear Algebra is a central and widely applicable part of mathematics. It is estimated that many (if not most) computers in the world are computing with matrix algorithms at any moment in time whether these be embedded in visualization software in a computer game or calculating prices for some financial option. This course builds on elementary linear algebra and in it we derive, describe and analyse a number of widely used constructive methods (algorithms) for various problems involving matrices.

Numerical Methods for solving linear systems of equations, computing eigenvalues and singular values and various related problems involving matrices are the main focus of this course.

Learning Outcomes: 

Students should understand the Singular Value Decomposition and its wide uses, state-of-the art algorithms for eigenvalue computation and core algorithms for solving linear systems, including in particular iterative solution methods of Krylov subspace type and randomised algorithms.

Course Synopsis: 

Common problems in linear algebra. Matrix structure, singular value decomposition. QR factorization, the QR algorithm for eigenvalues. Direct solution methods for linear systems, Gaussian elimination and its variants. Iterative solution methods for linear systems.

Chebyshev polynomials and Chebyshev semi-iterative methods, conjugate gradients, convergence analysis, preconditioning.

Reading List: 
  1. L. N. Trefethen and D. Bau III, Numerical Linear Algebra (SIAM, 1997).
  2. J. W. Demmel, Applied Numerical Linear Algebra (SIAM, 1997).
  3. A. Greenbaum, Iterative Methods for Solving Linear Systems (SIAM, 1997).
  4. G. H. Golub and C. F. van Loan, Matrix Computations (John Hopkins University Press, 3rd edition, 1996).
  5. H. C. Elman, D. J. Silvester and A. J. Wathen, Finite Elements and Fast Iterative Solvers (Oxford University Press, 1995), only chapter 2.