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Mapped continuation methods for computing all solutions to general systems of nonlinear equations
Compufersthem. Engng,Vol. 16, No. IO/II, pp. iii-v, 1992 Pcrgamon Press Ltd. F’rintedin Great Britain
BEST PAPER OF 1990 MAPPED CONTINUATION METHODS FOR COMPUTING ALL SOLUTIONS TO GENERAL SYSTEMS OF NONLINEAR EQUATIONS J. D. SEADER, M. KUNO, W.-J. LIN, S. A. JOHNSON, K. UNSWORTH and J. W. WISKIN
Computers & Chemical Engineering, Vol. 14, No. 1, pp. 71-85
J. D. !kder
_
.__~
S. A. Jobmon
M. Kuno
K. Uaswortb
W.-J. Lin
J. W. W&kin
ABOUT THE AUTHORS J. D. Seader-Is Professor of Chemical Engineering at the University of Utah. He obtained his M.S. from the University of California at Berkeley and his Ph.D. from the University of Wisconsin at Madison. After spending 13 years in industry with Chevron Research and the Rocketdyne Division of North American Aviation, he joined the University of Utah in 1966. His research interests include separation processes, restrictive diffusion effects in porous catalyst pellets, extraction of bitumen from tar sands and computing all solution for systems of nonlinear equations. Minoru Kmo-Is Director of Daitec Co., Ltd in Japan. He obtained his B.E. from Tokyo Institute Technology in 1962. During the period from 198687, he was a visiting researcher in the Department Chemical Engineering at the University of Utah.
of of
Weedhg Lb-Is currently a Senior Technology Engineer with Litwin Process Automation. He obtained an MS. from National Central University, Taiwan, following which he held a teaching position at that university.
iV
Best
paperof
1990
At the University of Utah, he obtained an M.S. in mathematics and a Ph.D. in chemical engineering. Prior to joining Litwin, he worked with the ChemShare Corporation. His professional experience includes apphed mathematics, advanced process control, simulation, design and on-line optimization. Steveo A. Jdumen- Is Research Professor of Bioengineering, Adjunct Associate Professor of Electrical Engineering, and Head of the Center for Inverse Problems, Imaging and Tomography (CIPIT) at the University of Utah. He obtained a Ph.D. in physics from Stanford University, following which he conducted research at the Mayo Foundation in Rochester, Minnesota under an NIH Postdoctoral Fellowship and Career Development Award. His research interests include inverse problems and imaging with medical, NDT, oceanographic, optical and geophysical data. Keith UtswobIs Lecturer in Computer Science at the University of Dundee, Scotland. He obtained his Ph.D. from the University of London. After a post-doctoral position at Imperial College of Science and Technology, London, he spent three years as a lecturer in The School of Mathematical Sciences, Science University of Malaysia, before joining the Department of Mathematics and Computer Science at Dun&e in 1979. His research interests are mainly in computer graphics and computer-aided geometric design Current research projects inchide 3-D object reconstruction from planar cross-sections, and shape-preservmg curve and surface interpolation. James W. Wiskim-Is Research Associate in Bioengineering at the University of Utah. He obtained an M.Sc. from Dalhousie University in Canada and his Ph.D. in mathematics from the University of Utah. His research interests involve computational inverse scattering theory and the application, via Maple, Fortran and C, of developed mathematical technology within dithxential geometry, algebraic geometry, algebmic topology, Lie group theory and knot and graph theory to biochemical, biological and physiological systems.
Abstract (see Vol. 14, pp. 71-86
for the full paper)
In many tlelds of engineering and science, use of mathematical models leads to systems of linear algebraic and nonlinear algebraic and transcendental equations. When all equations are linear, software packages such as LINPACK and YSMP can be applied to obtain the single solution. When the system contains nonlinear equations, more than one solution may exist, but until recently, software packages were designed to obtain at best just one solution from a specified starting guess. If the nonlinear equations are all of polynomial form, recent software packages such as HOMPACK and CONSOL can systematically locate alI solutions. The study reported here addresses the general case where the system may contain nonlinear equations with transcendental terms. By forming a 6xedpoint global homotopy and applying di&rential arclength continuation in finite mapped space, the two methods described (toroidal mapping and boomerang mapping) have located all solutions from a single starting guess for all cases studied. The two methods are illustrated for the case of an adiabatic continuous stirred-tank reactor operating in a steady-state mode with two consecutive reactions taking place, one of which is catalytic and irreversible with the other noncatalytic and reversible. All five steady-state solutions are found by each method.
EDITORIAL
NOTE
The Editorial Advisory Board of this Journal has assessed the papers published in Volume 14 by means of a three-stage process: nomination, tirst round elimination, and 6nal round balloting. The Seader et al. paper was the clear-cut result of that process. Professor Seader and his associates have pioneered in the development of continuation methods and their application to a variety of chemical engineering systems. This paper gives a well-written exposition of the issues arising in attempts to determine all the roots of systems of nonlinear equations. Using toroidal and boomerang mapping, general systems of nonlinear equations are convincingly solved, locating all of the roots from a single starting estimate. The work was judged to be a signigcant and lasting comibution to the chemical ag computing field in general and to the important family of continuation methods in particular.
RestPaper of 1990 The additional
”
highly ranked papers were:
J. T.-Y. Chetmg and G. Stephanopoulos, Representation of process trends--Part I. A formal representation framework; and Part II. The problem of scale and qualitatative scaling, No. 4/5, pp. 495-540. T. F. Yee, I. E. Grossmann and Z. Kravanja, Simultaneous optimization models for heat integration-I. Area and energy targeting and modeling of multi-stream exchanges; and Part II-Heat exchanger network synthesis, No. 10, pp. 1151-l 164. Part III-Process and heat exchanger network optimization, No. 11, pp. 1185-1200. Co-Editor,
Computers di Chemical Engineering
G. V. REKLAIT~S
BEST PAPER OF 1990 MAPPED CONTINUATION METHODS FOR COMPUTING ALL SOLUTIONS TO GENERAL SYSTEMS OF NONLINEAR EQUATIONS J. D. SEADER, M. KUNO, W.-J. LIN, S. A. JOHNSON, K. UNSWORTH and J. W. WISKIN
Computers & Chemical Engineering, Vol. 14, No. 1, pp. 71-85
J. D. !kder
_
.__~
S. A. Jobmon
M. Kuno
K. Uaswortb
W.-J. Lin
J. W. W&kin
ABOUT THE AUTHORS J. D. Seader-Is Professor of Chemical Engineering at the University of Utah. He obtained his M.S. from the University of California at Berkeley and his Ph.D. from the University of Wisconsin at Madison. After spending 13 years in industry with Chevron Research and the Rocketdyne Division of North American Aviation, he joined the University of Utah in 1966. His research interests include separation processes, restrictive diffusion effects in porous catalyst pellets, extraction of bitumen from tar sands and computing all solution for systems of nonlinear equations. Minoru Kmo-Is Director of Daitec Co., Ltd in Japan. He obtained his B.E. from Tokyo Institute Technology in 1962. During the period from 198687, he was a visiting researcher in the Department Chemical Engineering at the University of Utah.
of of
Weedhg Lb-Is currently a Senior Technology Engineer with Litwin Process Automation. He obtained an MS. from National Central University, Taiwan, following which he held a teaching position at that university.
iV
Best
paperof
1990
At the University of Utah, he obtained an M.S. in mathematics and a Ph.D. in chemical engineering. Prior to joining Litwin, he worked with the ChemShare Corporation. His professional experience includes apphed mathematics, advanced process control, simulation, design and on-line optimization. Steveo A. Jdumen- Is Research Professor of Bioengineering, Adjunct Associate Professor of Electrical Engineering, and Head of the Center for Inverse Problems, Imaging and Tomography (CIPIT) at the University of Utah. He obtained a Ph.D. in physics from Stanford University, following which he conducted research at the Mayo Foundation in Rochester, Minnesota under an NIH Postdoctoral Fellowship and Career Development Award. His research interests include inverse problems and imaging with medical, NDT, oceanographic, optical and geophysical data. Keith UtswobIs Lecturer in Computer Science at the University of Dundee, Scotland. He obtained his Ph.D. from the University of London. After a post-doctoral position at Imperial College of Science and Technology, London, he spent three years as a lecturer in The School of Mathematical Sciences, Science University of Malaysia, before joining the Department of Mathematics and Computer Science at Dun&e in 1979. His research interests are mainly in computer graphics and computer-aided geometric design Current research projects inchide 3-D object reconstruction from planar cross-sections, and shape-preservmg curve and surface interpolation. James W. Wiskim-Is Research Associate in Bioengineering at the University of Utah. He obtained an M.Sc. from Dalhousie University in Canada and his Ph.D. in mathematics from the University of Utah. His research interests involve computational inverse scattering theory and the application, via Maple, Fortran and C, of developed mathematical technology within dithxential geometry, algebraic geometry, algebmic topology, Lie group theory and knot and graph theory to biochemical, biological and physiological systems.
Abstract (see Vol. 14, pp. 71-86
for the full paper)
In many tlelds of engineering and science, use of mathematical models leads to systems of linear algebraic and nonlinear algebraic and transcendental equations. When all equations are linear, software packages such as LINPACK and YSMP can be applied to obtain the single solution. When the system contains nonlinear equations, more than one solution may exist, but until recently, software packages were designed to obtain at best just one solution from a specified starting guess. If the nonlinear equations are all of polynomial form, recent software packages such as HOMPACK and CONSOL can systematically locate alI solutions. The study reported here addresses the general case where the system may contain nonlinear equations with transcendental terms. By forming a 6xedpoint global homotopy and applying di&rential arclength continuation in finite mapped space, the two methods described (toroidal mapping and boomerang mapping) have located all solutions from a single starting guess for all cases studied. The two methods are illustrated for the case of an adiabatic continuous stirred-tank reactor operating in a steady-state mode with two consecutive reactions taking place, one of which is catalytic and irreversible with the other noncatalytic and reversible. All five steady-state solutions are found by each method.
EDITORIAL
NOTE
The Editorial Advisory Board of this Journal has assessed the papers published in Volume 14 by means of a three-stage process: nomination, tirst round elimination, and 6nal round balloting. The Seader et al. paper was the clear-cut result of that process. Professor Seader and his associates have pioneered in the development of continuation methods and their application to a variety of chemical engineering systems. This paper gives a well-written exposition of the issues arising in attempts to determine all the roots of systems of nonlinear equations. Using toroidal and boomerang mapping, general systems of nonlinear equations are convincingly solved, locating all of the roots from a single starting estimate. The work was judged to be a signigcant and lasting comibution to the chemical ag computing field in general and to the important family of continuation methods in particular.
RestPaper of 1990 The additional
”
highly ranked papers were:
J. T.-Y. Chetmg and G. Stephanopoulos, Representation of process trends--Part I. A formal representation framework; and Part II. The problem of scale and qualitatative scaling, No. 4/5, pp. 495-540. T. F. Yee, I. E. Grossmann and Z. Kravanja, Simultaneous optimization models for heat integration-I. Area and energy targeting and modeling of multi-stream exchanges; and Part II-Heat exchanger network synthesis, No. 10, pp. 1151-l 164. Part III-Process and heat exchanger network optimization, No. 11, pp. 1185-1200. Co-Editor,
Computers di Chemical Engineering
G. V. REKLAIT~S