Table Of ContentIN F O R M A T IO N L IN K A G E
B E T W E E N
A P P L IE D M A T H E M A T IC S
A N D IN D U S T R Y
I I
edited by
A R T H U R L. S C H O E N S T A D T
F R A N K D . F A U L K N E R
R IC H A R D F R A N K E
I. B E R T R U S S A K
Department of Mathematics
Naval Postgraduate School
Monterey, California
ACADEMIC PRESS
A Subsidiary of Harcourt Brace Jovanovich, Publishers
New York London Sydney Toronto San Francisco 1980
Copyright © 1980, by Academic Press, Inc.
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ACADEMIC PRESS, INC.
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ACADEMIC PRESS, INC. (LONDON) LTD.
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Library of Congress Cataloging in Publication Data
Workshop on the Information Linkage between Applied
Mathematics and Industry, 2d, Naval Postgraduate
School, Monterey, Calif., 1979.
Information linkage between applied mathematics
and industry II.
Bibliography: p.
1. Engineering mathematics—Congresses.
I. Schoenstadt, Arthur L. II. Title.
TA329.W67 1979 510'.2462 80-17975
ISBN 0-12-628750-3
PRINTED IN THE UNITED STATES OF AMERICA
80 81 82 83 9 8 7 6 5 4 3 2 1
N A V A L P O S T G R A D U A T E S C H O O L
SESSION C H A I R M E N
Theory and Computation of Linear Systems
Uno R. Kodres, Computer Sdence
Least Squares, Regression and Systems Identification
Louis V. Schmidt, Aeronautical Engineering
Applications in Controls
Thomas M. Houlihan, Mechanical Engineering
General Applications
Gilbert Aguirre-Ramirez, Mathematics
W O R K S H O P COMMITTEE
Arthur L. Schoenstadt, Coordinator
Frank D. Faulkner
Richard H. Franke
I. Bert Russak
Carroll O. Wilde
ADMINISTRATIVE A N D EDITORIAL C O N S U L T A N T
T O THE EDITOR
Ms. Rosemarie Stampfe l
CONTRIBUTORS
Numbers in parentheses indicate pages on which authors*contributions begin.
Leonard R. Anderson (197), Aerospace and Ocean Engineering Department, Vir
ginia Polytechnic Institute and State University, Blacksburg, Virginia 24061
Peter M. Bentler (195), Department of Psychology, University of California, Los
Angeles, California 90024
S. F. Borg (249), Department of Civil Engineering, Stevens Institute of Technol
ogy, Hoboken, New Jersy 07030
W. R. Burrus (1), Computer Sciences Division, Oak Ridge National Laboratory,
Union Carbide Corporation, Nuclear Division, Oak Ridge, Tennessee 37830
Chung-Yen Chiu (251), 1309 Ramsay Circle, Walnut Creek, California 94596
G. S. Christensen (223), Department of Electrical Engineering, University of Al
berta, Edmonton, Alberta, Canada TGG2E1
J. E. Cope (1), Computer Sciences Division, Oak Ridge National Laboratory,
Union Carbide Corporation, Nuclear Division, Oak Ridge, Tennessee 37830
John B. Dash (119), Bureau of Labor Statistics, 441 G Street, N.W., Washington,
DC 20212
M. E. El-Hawary (223), Faculty of Engineering, Memorial University, St. John’s,
Newfoundland, Canada
John Jones, Jr. (115), Department of Mathematics, Air Force Institute of Technol
ogy, Wright-Patterson AFB, Ohio 45433
Horst W. Kriebel (253), 904 Sandpiper Place, La Jolla, California 92037
Alan J. Laub (57), Department of Electrical Engineering Systems, PHE-422, Uni
versity of Southern California, Los Angeles, California 90007
Barrie W. Leach (139), NRC Building U-61, Flight Research Section, Montreal
Road, Ottawa, Canada K1A OR6
Roy B. Leipnik (117), Mathematics Department, University of California, Santa
Barbara, California 93106
James W. Longley (119), Bureau of Labor Statistics, 441 G Street, N.W.,
Washington, DC 20212
Victor Lovass-Nagy (215), Department of Mathematics, Clarkson College of
Technology, Potsdam, New York 13676
ix
X CONTRIBUTORS
Uy-Loi Ly (163), The Boeing Company, P.O. Box 3707, MS: 3N-43, Seattle,
Washington 98124
David P. Maher (269), Department of Mathematics, Worcester Polytechnic Insti
tute, Worcester, Massachusetts 01609
Theodore Mairson (221), Sanders Associates, Inc., NGQ 4-0126, Daniel Webster
Highway South, Nashua, New Hampshire 03061
Joseph F. McGrath (125), KMS Fusion, Inc., P.O. Box 1567, Ann Arbor, Michi
gan 48106
P. H. Merz (279), Chevron Research Company, Standard Avenue, Richmond,
California 94802
Rangaswamy Mukundan (215), Department of Electrical and Computer Engineer
ing, Clarkson College of Technology, Potsdam, New York 13676
Morris Newman (135), Department of Mathematics, University of California,
Santa Barbara, California 93106
David L. Powers (215), Department of Mathematics, Clarkson College of
Technology, Potsdam, New York 13676
Luis A. Prieto-Portar (281), 4025 Irvington Avenue, Coconut Grove, Florida
33133
J. Gary Reid (179), Department of Electrical Engineering, Air Force Institute of
Technology, Wright-Patterson AFB, Ohio 45433
Horst P. Richter (193, 283, 285), 2204 Nelda Way, Alamo, California 94507
Thomas E. Rosmond (85), Numerical Modeling, Naval Environmental Prediction
Facility, Monterey, California 93940
B. W. Rust (1), Computer Sciences Division, Oak Ridge National Laboratory,
Union Carbide Corporation, Nuclear Division, Oak Ridge, Tennessee 37830
A. Shamaly (223), Department of Electrical Engineering, University of Alberta,
Edmonton, Alberta, Canada TGG2E1
G. Stein (39), Honeywell, SRC, MS 17-2367, 2600 Ridgeway Road, Minneapolis,
Minnesota 55413
Roland A. Sweet (105), National Center for Atmospheric Research, Box 3000,
Boulder, Colorado
J. Arthur Woodward (195), Department of Psychology, University of California,
Los Angeles, California 90024
Min-Yen Wu (233), Department of Electrical Engineering, University of Colorado,
Boulder, Colorado 80309
Richard W. Young (137), Department of Engineering Science (ML 112), Univer
sity of Cincinnati, Cincinnati, Ohio 45221
P R E FA C E
These proceedings are the record of a symposium held from February 22 to 24,
1979. This symposium was the second of a series conducted at the Naval Post
graduate School, Monterey, California. As Peter Wang noted in the preface to the
proceedings from the first symposium, these were envisioned as an “attempt to
bring applied mathematicians and engineers together to exchange ideas in order to
create a working format in which mathematicians could share their experiences in
dealing with problems arising in engineering contexts.” This workshop was quite
successful in meeting this goal, as it brought together approximately 80 scientists
and engineers from throughout the United States and Canada.
In contrast to the first symposium in this series, which was organized along four
rather diverse themes, the second workshop concentrated exclusively on the topic of
linear systems of equations in applications. This concentration was done for several
reasons. Foremost among these was the feeling that addressing a narrower topic,
while it might attract a smaller audience, would attract a more homogeneous audi
ence and therefore lead to a more intense and vital exchange between the practition
ers and the theoreticians. In addition, linear systems, of course, are fundamental to
nearly all numerical studies, since they almost invariably arise in the discretization
of a continuous process. Lastly, the theme of linear systems was considered to be
especially appropriate since the timing of this workshop coincided with the publica
tion by the Society for Industrial and Applied Mathematics of the LINPACK li
brary of linear systems software. In fact, immediately preceding the symposium,
SIAM presented a two-day workshop on the LINPACK library.
In this volume we have included all invited papers at the symposium, and selected
contributed papers, plus the abstracts of the remaining contributed papers. In the
latter cases, we generally were unable to include full text because of their length,
and of a lack of necessary time in order to edit them down to an acceptable length.
On behalf of the workshop committee, I wish to extend our deep appreciation to
the Office of Naval Research, especially Dr. Stuart Brodsky, and to the Naval
Postgraduate School Foundation Research program, directed by Dean of Research,
Dr. William Tolies, for their continuing support of our endeavors. We would also
recognize the contribution of former Dean of Research at the Naval Postgraduate
School, Dr. Robert Fossum, now head of the Defense Advanced Research Project
xi
xii PREFACE
Agency, whose support was especially valuable in establishing the first of these
series. The workshop committee would also like to express our appreciation to our
colleagues at the Naval Postgraduate School, especially Professors Uno Kodres,
Louis Schmidt, and Thomas Houlihan, who assisted us by chairing the contributed
papers sessions during the symposium. And lastly, we would like to thank Ms. T.
Carol St. Onge, who was invaluable in assisting us with the mechanics of running
this symposium, and Ms. Rosemarie Stampfel, whose diligence, patience, and
superb typing skills produced this final manuscript.
Finally, I would like to add my personal thanks and appreciation for the assis
tance given me by the other members of the workshop committee, both in organiz
ing the symposium program, and in the sometimes laborious, generally rewarding
task of editing these proceedings.
Arthur L. Schoenstadt
Workshop Coordinator
Monterey, California
Information Linkage between Applied Mathematics and Industry II
CONSTRAINED INTERVAL ESTIMATION FOR
LINEAR MODELS WITH ILL-CONDITIONED EQUATIONS
W. R. Burrus^
B. W. Rust
J. E. Cope
Computer Sciences Division
Oak Ridge National Laboratory
Union Carbide Corporation, Nuclear Division
Oak Ridge, Tennessee
This paper is concerned with solving ill-conditioned and un
der determined linear systems Ax = b where the vector h is
known only approximately within a specified uncertainty. In gen
eral, the problem is to estimate upper and lower bounds for a
linear function φ = wTx where w is a given vector. Interval
estimates for the elements x% are obtained by successively tak
ing w equal to a unit vector with a 1 in position k. The stan
dard linear regression techniques are augmented by auxiliary in
formation which stabilizes the problem against ill conditioning
and rank deficiency. Commonly used kinds of auxiliary informa
tion are x non-negative or the length of x bounded. Numerical
approximations are derived from two points of view. The primal
approach involves the intersection of the uncertainty and con
straint regions and the dual approach poses the problem in terms
of biased estimation in the presence of uncertainty. In each case
the numerical approximations produce conservative or suboptimal
interval estimates. The procedures described have previously been
incorporated into the FERDO and FERD radiation spectrum unfolding
codes developed at Oak Ridge National Laboratory and are illus
trated here by a mathematical example very similar in form to an
unfolding problem.
^Associated with Science Applications, Inc., Oak Ridge, Tenn.
2
Operated by Union Carbide Corporation under contract W-7405-
eng-26 with the U. S. Department of Energy.
Copyright © 1980 Academic Press, Inc.
1 All rights of reproduction in any form reserved.
ISBN 0-12-628750-3
2 W . R. BURRUS et al
I . INTRODUCTION
Underdetermined systems of linear algebraic equations are
frequently encountered in solving physical problems in which
cause-effect relations are modeled by Fredholm integral equations
of the first kind. In this paper we assume that the problems
have already been converted from continuous integral equations
to discrete algebraic equations and consider the resulting ill-
conditioned numerical problems. The guiding philosophy of the
methods to be described is to use axuiliary information about the
solution to stabilize the problem against ill-conditioning and/or
rank deficiency.
We will take as our starting point the equation Ax = b
where the right-hand side b is uncertain, but is known (with
certainty) to be contained in a ball of radius σ about b. we
will consider three different forms of auxiliary information.
The first is that the solution vector x is limited in length.
The second is that, in addition, x is known to be nonnegative.
The third is that the matrix A has nonnegative elements, and x
is nonnegative.
The desired solution is an uncertainty interval (<l>l0,<f>Uf>)
T
which can be guaranteed to contain the value of φ = w x where
w is a given, arbitrary weight vector. In summary:
Given:
Ax = b (1)
\\b - b\\ <_ a (2)
IWI £ 3 (3a)
\\x\\ < 3, x >_ 0 (3b)
A. . > 0, x > 0 . (3c)
CONSTRAINED INTERVAL ESTIM ATIO N 3
Wanted: Guaranteed interval which contains
T
all possible values of φ = w x which satisfy
constraints (1), (2), and one of the constraints
(3a) , (3b) , or (3c) .
The simple constraints (2) and (3a) may be easily generalized
as will be shown, but the above statement of the problem is ade
quate to develop all the ideas of the methods.
The methods described in this paper were originally developed
at Oak Ridge National Laboratory to solve radiation spectrum un
folding problems. Computer codes employing these methods have
enjoyed great success throughout the world in spite of the lack
of coherent documentation of either the methods or the codes. The
guiding philosophy and underlying mathematics have been exten
sively discussed in [1], but that work did not describe the spe
cific methods to be treated here. The earliest descriptions of
these methods were given, at the time they were being developed,
in various brief papers and presentations by Burrus [2,3,4,5],
but these publications are not part of the readily available
mathematics literature. In 1961 Burrus wrote a we11-organized
overview in the form of a draft Ph.D. thesis which was rejected
by the Ohio State University Physics Department. In spite of this
setback, he and his coworkers continued to develop the methods
with striking successes [6,7,8]. In 1965 he wrote a new draft
Ph.D. thesis which was not nearly so well organized and coherent
as the first but which was accepted [9]. The main result of all
this work was a computer code called FERDOR which is based on the
mathematical techniques described in Section V of the present
paper. The mathematical technique and the code itself have pre
viously been described in References [8,10,11,12,13,14,15]. We
strive here to give a succinct mathematical description of the
method.
Sections VI and VII describe a mathematical technique that is
the basis of another, more recent, unfolding code called FERD.
