id: 06035286
dt: b
an: 2012f.01027
au: Laub, Alan J.
ti: Computational matrix analysis.
so: Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM)
(ISBN 978-1-611972-20-7/pbk). xiii, 154~p. (2012).
py: 2012
pu: Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM)
la: EN
cc: N15 N35
ut: floating point arithmetic; finite arithmetic; rounding error; Gaussian
elimination; eigenvalues; QR algorithm; singular value decomposition;
textbook; LU and QR factorization; least squares solution; generalized
inverse; software; numerical stability; ill conditioning; complexity;
perturbation analysis; conditioning; matrix function; matrix equation;
numerical examples
ci:
li:
ab: This booklet is not a numerical linear algebra course. It assumes that the
basic notions and algorithms are known (Gaussian elimination, LU and QR
factorization, least squares solution, generalized inverse,
eigenvalues, singular value decomposition (SVD), etc.). It only adds to
this the particular aspects of the computations when the algorithms are
implemented in finite arithmetic. Obviously, the use of Matlab or the
likes is necessary for a proper assimilation of the material, but it
does not introduce the code either. What it does contain are the
notions and the formulas to write good software, or rather to analyse
existing software and understand why it is implemented as it is. The
first chapters are devoted to generalities like floating point
arithmetic, numerical stability of a method and ill conditioning of a
problem, complexity by counting flops, and the importance of row or
column oriented processing of a matrix. After that the particularities
of the algorithms are analysed. Gaussian elimination and the solution
of linear systems, (generalized) linear least squares (QR,
Gram-Schmidt, Householder, Givens), eigenvalues (double-shift QR), and
the Golub-Kahan-Reinsch SVD algorithm. For the latter more advanced
algorithms, the reader is referred to the literature for further
details, but the basics are given together with an analysis of the
condition (perturbation analysis). Of particular interest is the
statistical condition estimation, as opposed to worst case
conditioning. The book ends with some applications; computing a matrix
function, in particular $e^A$, and the solution of some matrix
equations (Lyapunov, Sylvester, Riccati). Because the analysis is never
very deep, it should be accessible to many students from mathematics,
sciences, and engineering. It stays on a basic level throughout, so
there is no discussion of large-scale problems, sparsity, advanced
structured matrices, etc. This lines up with the detailed step-by-step
treatment of the first chapters illustrating the steps with numerical
examples. However, in the more advanced algorithms of the later
chapters, many of these details are skipped and the reader should spend
some time looking up the literature for further details, which seems to
be a bit in disproportion with the start of the booklet. Each chapter
has some exercises. The text is not suitable for a stand-alone course.
It is best used to interlace with or to accompany some (numerical)
linear algebra course (from basic to more advanced), with computer
assignments for the students to fully assimilate the yeast of it.
rv: Adhemar Bultheel (Leuven)