Computational Complexity and Feasibility of Data Processing and Interval Computations

460 Pages·1998·35.198 MB·English

by  Vladik Kreinovich, Anatoly Lakeyev, Jiří Rohn, Patrick Kahl (auth.)

Computational Complexity and Feasibility of Data
Processing and Interval Computations
Applied Optimization
Volume 10
Series Editors:
Panos M. Pardalos
University oiF lorida, U.S.A.
Donald Hearn
University oiF lorida, U.SA.
The titles published in this series are listed at the end 0/ this volurne.
Computational
Complexityand
Feasibility of Data
Processing and Interval
Computations
by
Vladik Kreinovich
University ofTexas at EI Paso,
Texas, U.S.A.
Anatoly Lakeyev
Computing Center, Russian Academy of Sciences,
Irkutsk, Russia
JiiiRohn
Charles University and Academy of Sciences,
Prague, Czech Republic
Patrick Kahl
IBM, Tueson, Arizona, U.S.A.
Springer-Science+Business Media, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4419-4785-7 ISBN 978-1-4757-2793-7 (eBook)
DOI 10.1007/978-1-4757-2793-7
Printed on acid-free paper
All Rights Reserved
© 1998 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1998.
Softcover reprint ofthe hardcover 1st edition 1998
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
inc1uding photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
CONTENTS
PREFACE
IX
1 INFORMAL INTRODUCTION:
DATA PROCESSING, INTERVAL
COMPUTATIONS, AND COMPUTATIONAL
COMPLEXITY 1
2 THE NOTIONS OF FEASIBILITY AND
NP-HARDNESS: BRIEF INTRODUCTION 23
3 IN THE GENERAL CASE,
THE BASIC PROBLEM OF
INTERV AL COMPUTATIONS
IS INTRACTABLE 41
4 BASIC PROBLEM OF INTERV AL
COMPUTATIONS FOR POLYNOMIALS OF
A FIXED NUMBER OF VARIABLES 53
5 BASIC PROBLEM OF INTERVAL
COMPUTATIONS FOR POLYNOMIALS OF
FIXEDORDER 71
6 BASIC PROBLEM OF INTERV AL
COMPUTATIONS FOR POLYNOMIALS
WITH BOUNDED COEFFICIENTS 79
v
VI COMPLEXITY OF INTERVAL COMPUTATIONS
7 FIXED DATA PROCESSING ALGORITHMS,
VARYING DATA: STILL NP-HARD 83
8 FIXED DATA, VARYING DATA
PROCESSING ALGORITHMS: STILL
INTRACTABLE 85
9 WHA T IF WE ONLY ALLOW SOME
ARITHMETIC OPERATIONS IN DATA
PROCESSING? 87
10 FOR FRACTIONALLY-LINEAR
FUNCTIONS,
A FEASIBLE ALGORITHM SOLVES
THE BASIC PROBLEM OF INTERV AL
COMPUTATIONS 91
11 SOLVING INTERV AL LINEAR SYSTEMS IS
NP-HARD 99
12 INTERV AL LINEAR. SYSTEMS: SEARCH
FOR FEASIBLE CLASSES 111
13 PHYSICAL COROLLARY: PREDICTION
IS NOT ALWAYS POSSIBLE, EVEN
FOR LINEAR SYSTEMS WITH KNOWN
DYNAMICS 143
14 ENGINEERING COROLLARY: SIGNAL
PROCESSING IS NP-HARD 153
15 BRIGHT SIDES OF NP-HARDNESS
OF INTERV AL COMPUTATIONS I:
NP-HARD MEANS THAT GOOD INTERV AL
HEURISTICS CAN SOLVE OTHER HARD
PROBLEMS 159
Contents VII
16 IF INPUT INTERVALS ARE NARROW
ENOUGH,THENINTERVAL
COMPUTATIONS ARE ALMOST ALWAYS
EASY 161
17 OPTIMIZATION - A FIRST EXAMPLE OF
A NUMERICAL PROBLEM IN WHICH
INTERV AL METHODS ARE USED:
COMPUTATIONAL COMPLEXITY AND
FEASIBILITY 173
18 SOLVING SYSTEMS OF EQUATIONS 197
19 APPROXIMATION OF INTERVAL
FUNCTIONS 207
20 SOLVING DIFFERENTIAL EQUATIONS 219
21 PROPERTIES OF INTERVAL MATRICES I:
MAIN RESULTS 225
22 PROPERTIES OF INTERVAL MATRICES 11:
PROOFS AND AUXILIARY RESULTS 257
23 NON-INTERVAL UNCERTAINTY I:
ELLIPSOID UNCERTAINTY AND ITS
GENERALIZATIONS 289
24 NON-INTERV AL UNCERTAINTY
11: MULTI-INTERVALS AND THEIR
GENERALIZATIONS 309
25 WHA T IF QUANTITIES ARE DISCRETE? 325
26 ERROR ESTIMATION FOR INDIRECT
MEASUREMENTS: INTERV AL
COMPUTATION PROBLEM IS
Vlll COMPLEXITY OF INTERVAL COMPUTATIONS
(SLIGHTLY) HARDER THAN A SIMILAR
PROBABILISTIC COMPUTATIONAL
PROBLEM 331
A IN CASE OF INTERVAL (OR MORE
GENERAL) UNCERTAINTY, NO
ALGORITHM CAN CHOOSE THE
SIMPLEST REPRESENTATIVE 347
B ERROR ESTIMATION FOR INDIRECT
MEASUREMENTS: CASE OF
APPROXIMATELY KNOWN FUNCTIONS 365
C FROM INTERV AL COMPUTATIONS TO
MODAL MATHEMATICS 381
D BEYOND NP: TWO ROOTS GOOD, ONE
ROOTBETTER 395
E DOES "NP-HARD" REALLY MEAN
"INTRACTABLE" ? 401
F BRIGHT SIDES OF NP-HARDNESS OF
INTERV AL COMPUTATIONS 11: FREEDOM
OF WILL? 405
G THE WORSE, THE BETTER:
PARADOXICAL COMPUTATIONAL
COMPLEXITY OF INTERV AL
COMPUTATIONS AND DATA PROCESSING 409
REFERENCES 413
INDEX 451
PREFACE
Targeted audience
• Specialists in numerical computations, especially in numerical optimiza
tion, who are interested in designing algorithms with automatie result ver
ification, and who would therefore be interested in knowing how general
their algorithms caIi in principle be.
• Mathematicians and computer scientists who are interested in the theory 0/
computing and computational complexity, especially computational com
plexity of numerical computations.
• Students in applied mathematics and computer science who are interested
in computational complexity of different numerical methods and in learning
general techniques for estimating this computational complexity. The book
is written with all explanations and definitions added, so that it can be
used as a graduate level textbook.
What this book .is about
Data processing. In many real-life situations, we are interested in the value
of a physical quantity y that is diflicult (or even impossible) to measure directly.
For example, it is impossible to directly measure the amount of oil in an oil
field or a distance to a star. Since we cannot measure such quantities directly,
we measure them indirectly, by measuring some other quantities Xi and using
the known relation between y and Xi'S to reconstruct y. The algorithm that
transforms the results Xi of measuring Xi into an estimate fj for y is called data
processing.
Error estimation for data processing: interval computations. The
input data for data processing algorithms come from measurements and are,
therefore, not precise: the measured value Xi is, in general, different from the
IX