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Linear programming attracted the interest of mathematicians
during and after World War II when the first computers were
constructed and methods for solving large linear programming
problems were sought in connection with specific practical problems
for example, providing logistical support for the U.S. Armed Forces
or modeling national economies. Early attempts to apply linear
programming methods to solve practical problems failed to satisfy
expectations. There were various reasons for the failure. One of
them, which is the central topic of this book, was the inexactness
of the data used to create the models. This phenomenon, inherent in
most pratical problems, has been dealt with in several ways. At
first, linear programming models used "average" values of
inherently vague coefficients, but the optimal solutions of these
models were not always optimal for the original problem itself.
Later researchers developed the stochastic linear programming
approach, but this too has its limitations. Recently, interest has
been given to linear programming problems with data given as
intervals, convex sets and/or fuzzy sets. The individual results of
these studies have been promising, but the literature has not
presented a unified theory. Linear Optimization Problems with
Inexact Data attempts to present a comprehensive treatment of
linear optimization with inexact data, summarizing existing results
and presenting new ones within a unifying framework."
Linear programming attracted the interest of mathematicians during
and after World War II when the first computers were constructed
and methods for solving large linear programming problems were
sought in connection with specific practical problems for example,
providing logistical support for the U.S. Armed Forces or modeling
national economies. Early attempts to apply linear programming
methods to solve practical problems failed to satisfy expectations.
There were various reasons for the failure. One of them, which is
the central topic of this book, was the inexactness of the data
used to create the models. This phenomenon, inherent in most
practical problems, has been dealt with in several ways. At first,
linear programming models used average values of inherently vague
coefficients, but the optimal solutions of these models were not
always optimal for the original problem itself. Later researchers
developed the stochastic linear programming approach, but this too
has its limitations. Recently, interest has been given to linear
programming problems with data given as intervals, convex sets
and/or fuzzy sets. literature has not presented a unified theory.
Linear Optimization Problems with Inexact Data attempts to present
a comprehensive treatment of linear optimization with inexact data,
summarizing existing results and presenting new ones within a
unifying framework.
This revised and corrected second edition of a classic book on
special matrices provides researchers in numerical linear algebra
and students of general computational mathematics with an essential
reference.
Author Miroslav Fiedler, a Professor at the Institute of Computer
Science of the Academy of Sciences of the Czech Republic, Prague,
begins with definitions of basic concepts of the theory of matrices
and fundamental theorems. In subsequent chapters, he explores
symmetric and Hermitian matrices, the mutual connections between
graphs and matrices, and the theory of entrywise nonnegative
matrices. After introducing "M"-matrices, or matrices of class "K,
" Professor Fiedler discusses important properties of tensor
products of matrices and compound matrices and describes the
matricial representation of polynomials. He further defines band
matrices and norms of vectors and matrices. The final five chapters
treat selected numerical methods for solving problems from the
field of linear algebra, using the concepts and results explained
in the preceding chapters.
Simplex geometry is a topic generalizing geometry of the triangle
and tetrahedron. The appropriate tool for its study is matrix
theory, but applications usually involve solving huge systems of
linear equations or eigenvalue problems, and geometry can help in
visualizing the behaviour of the problem. In many cases, solving
such systems may depend more on the distribution of non-zero
coefficients than on their values, so graph theory is also useful.
The author has discovered a method that in many (symmetric) cases
helps to split huge systems into smaller parts. Many readers will
welcome this book, from undergraduates to specialists in
mathematics, as well as non-specialists who only use mathematics
occasionally, and anyone who enjoys geometric theorems. It
acquaints the reader with basic matrix theory, graph theory and
elementary Euclidean geometry so that they too can appreciate the
underlying connections between these various areas of mathematics
and computer science.
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