Numerical Algorithmic Science and Engineering (NAS&E), or more
compactly, Numerical Algorithmics, is the theoretical and empirical
study and the practical implementation and application of
algorithms for solving finite-dimensional problems of a numeric
nature. The variables of such problems are either discrete-valued,
or continuous over the reals, or, and as is often the case, a
combination of the two, and they may or may not have an underlying
network/graph structure. This re-emerging discipline of numerical
algorithmics within computer science is the counterpart of the now
well-established discipline of numerical analysis within
mathematics, where the latter's emphasis is on
infinite-dimensional, continuous numerical problems and their
finite-dimensional, continuous approximates. A discussion of the
underlying rationale for numerical algorithmics, its foundational
models of computation, its organizational details, and its role, in
conjunction with numerical analysis, in support of the modern modus
operandi of scientific computing, or computational science &
engineering, is the primary focus of this short monograph. It
comprises six chapters, each with its own bibliography. Chapters 2,
3 and 6 present the book's primary content. Chapters 1, 4, and 5
are briefer, and they provide contextual material for the three
primary chapters and smooth the transition between them.
Mathematical formalism has been kept to a minimum, and, whenever
possible, visual and verbal forms of presentation are employed and
the discussion enlivened through the use of motivating quotations
and illustrative examples. The reader is expected to have a working
knowledge of the basics of computer science, an exposure to basic
linear algebra and calculus (and perhaps some real analysis), and
an understanding of elementary mathematical concepts such as
convexity of sets and functions, networks and graphs, and so on.
Although this book is not suitable for use as the principal
textbook for a course on numerical algorithmics (NAS&E), it
will be of value as a supplementary reference for a variety of
courses. It can also serve as the primary text for a research
seminar. And it can be recommended for self-study of the
foundations and organization of NAS&E to graduate and advanced
undergraduate students with sufficient mathematical maturity and a
background in computing. When departments of computer science were
first created within universities worldwide during the middle of
the twentieth century, numerical analysis was an important part of
the curriculum. Its role within the discipline of computer science
has greatly diminished over time, if not vanished altogether, and
specialists in that area are now to be found mainly within other
fields, in particular, mathematics and the physical sciences. A
central concern of this monograph is the regrettable, downward
trajectory of numerical analysis within computer science and how it
can be arrested and suitably reconstituted. Resorting to a biblical
metaphor, numerical algorithmics (NAS&E) as envisioned herein
is neither old wine in new bottles, nor new wine in old bottles,
but rather this re-emerging discipline is a decantation of an
age-old vintage that can hopefully find its proper place within the
larger arena of computer science, and at what appears now to be an
opportune time.
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