Numerical linear algebra: algorithms, analysis and applications

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In order to solve many scientific, engineering, business and other problems using the computer, we usually convert the problems to subproblems involving matrices --- rectangular arrays of numbers.  Because of this, matrix computations and its theory (often called numerical linear algebra) lies at the heart of most scientific computing.  The key to solving many smaller matrix problems is to find some appropriate factorizations of matrices --- so for example a given matrix can be described as the product of two or more such factors.  However many problems are large or complex, and the resulting matrix problems are extremely large --- a million rows and a million columns is not at all unusual.  And such large matrices are nearly always very sparse --- the vast majority of elements being zero.  Factorization methods can then become too costly, or sometimes simply impossible, and iterative methods may be necessary for solving such problems.  This research develops numerically reliable and efficient algorithms for solving both classes of problems.  It carries out analyses of such algorithms to prove their efficiency and reliability, and sensitivity analyses to show what effects changes in the data (for example caused by finite precision computation, or uncertainty in the data) will have on the final results. These analyses lead to greater understanding of individual problems and their computed answers, and improved general purpose and specific area algorithms. The results of the research are useful for all practitioners in scientific computation.  Some application areas of particular importance in Canada that use these techniques are the aerospace industry (via global positioning systems, aircraft body design, etc.), power generation and distribution, and resource discovery, extraction and transportation.