Advanced Topics of Numerical Linear Algebra

We discuss advanced numerical linear algebra problems like: matrix equations, matrix functions, tensor problems.

Dates:       Wednesday, 9:00-11:00, G12-201, Thursday 13:00-15:00, G05-314.

Last exercise on Thursday, July 5.  

Exams: please write an E-mail to kuerschner@mpi-...... to schedule a time.

Topics:

• Brief recapitulation of important concepts from (numerical) linear algebra, especially regarding linear systems of equations and linear eigenvalue problems.
• Matrix equations
  • Theory, applications
  • Methods for small / dense linear and quadratic equations
• Matrix functions
  • Theory, applications
  • Computing functions of small, dense matrices
  • Approximating matrix function times a vector
• Randomized Algorithms
  
• Brief Outlook to Multilinear (Numerical) Algebra

Seminar Advanced Topics in Numerical Linear Algebra

in parallel to the lecture, there will be a seminar about further topics in Numerical Linear Algebra. It will be held on September 18, 2018 at the Max Planck Institute, seminar room Prigogine - V0.05-2+3.

Although the seminar is supplementary, it is advised for lecture participants to attend the seminar even without giving an own talk because interesting additional problems will be discussed.

Problem sheets

• Exercise 1 for Thursday, April 12.
• Exercise 2 for Thursday, April 26. Data for problem 1, problem 3.
• Exercise 3 for Thursday, May 17. Data for problem 2.
• Exercise 4 for Thursday, May 31.
• Exercise 5 for Thursday, June 14.
• Exercise 6 for Thursday, July  5 ( corrected problem 4). Data for problem 3.

Handouts

• Bartels-Stewart Algorithm
• Projection methods for large Lyapunov equations
• Low-rank ADI
• Schur-Parlett Algorithm for f(A)
• Scaling & Squaring methods for exp(A), log(A)

Literature recommendations

• Basics:
  • Saad: Iterative Methods for Sparse Linear Systems, SIAM, 1996.
  • G. Golub, C. Van Loan: Matrix Computations, The John Hopkins University Press, 1996 / 2013.
• Matrix equations:
  • D. A. Bini, B, Iannazzo, B. Meini: Numerical Solution of Algebraic Riccati Equations, SIAM, 2012.
  • Chapter 13 (free) in A. J. Laub: Matrix Analysis for Scientists and Engineers
  • Chapters 1+4 in P. Benner, M. Bollhöfer, D. Kressner, C. Mehl, T. Stykel: Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory, Springer, 2014.
• Matrix functions:
  • N. J. Higham: Functions of Matrices, Theory and Computation, SIAM, 2008
  • C. Moler, C. Van Loan: Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later, SIAM Rev. 45(1), 3–49, 2003.
  • N. J. Higham: The Scaling and Squaring Method for the Matrix Exponential Revisited, SIAM J. Matrix Anal. & Appl., 26(4), 1179–1193, 2005.
• Randomized Algorithms:
  • N. Halko, P. G. Martinsson, and J. A. Tropp: Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions, SIAM Rev. , 53(2), 217–288.
• Brief Outlook to Multilinear (Numerical) Algebra
  • L. De Lathauwer, B. De Moor, and J. Vandewalle: A Multilinear Singular Value Decomposition, SIAM J. Matrix Anal. & Appl. , 21(4), 1253–1278.