LIBRARY OF THE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 5IQ64 The person charging this material is re- sponsible for its return to the library from which it was withdrawn on or before the Latest Date stamped below. Theft, mutilation, and underlining of books are reasons for disciplinary action and may result in dismissal from the University. UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN MAR 2 2 1978 OEC 1 4 1976 DEC 12 RECD L161 — O-1096 Digitized by the Internet Archive in 2013 http://archive.org/details/noteonnonexisten569gear 5/*'*T uiucDcs-R-73-569 jmut< coo-ii+69-0219 A NOTE ON THE NON-EXISTENCE OF MULTIVALUE A-STABLE METHODS OF ORDER GREATER THAN TWO by C. W. Gear March 1973 «frj '&■: DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN URBANA, ILLINOIS UIUCDCS-R-73-569 A NOTE ON THE NON-EXISTENCE OF MULTIVALUE A-STABLE METHODS OF ORDER GREATER THAN TWO* by C. W. Gear March 1973 DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF ILLINOIS AT URBAN A- CHAMPAIGN URBANA, ILLINOIS 6l801 Supported in part by contract US AEC AT(ll-l)lU69. -1- ABS TRACT It is shown that the Dahlquist result limiting the order of A-stable multistep methods also applies to multivalue methods. INTRODUCTION In 1956, Dahlquist [l] showed that the maximum order of a strongly stable k-step linear multistep method is k+1. In 1963, he showed [2] that the maximum order of an A-stable multistep method is two. In 1966, the author [3] showed that an extension of multistep methods, called multivalue methods , led to a larger class of methods , called modified k-step methods , such that the order of a stable method could be increased to 2k. Since then, the question of whether a multi- value method of order greater than two could be A-stable has remained. The answer is no and is shown by reducing the problem to the problem studied by Dahlquist. PROOF OF RESULT Consider the test equation y' - Ay = 0. Let a = [a ,a , . . . a ] = [y ,hy ' , . . h y /k! ] be the data saved in the normal T T form of a multivalue method, and let F(a.) = 5.-, a, - H ^ a = a - ya where y = hA. Then, from equation (11.21) in reference [U], the multivalue method is -2- 9F -1 a = Aa _ - I [— £ ] F(Aa . ) -n -n-1 —3a— -n-1 = a^.! - Llii i - y «2 - ] ( ^-i " y ^S )A ^h-i i - L *i"^o Aa _ -n-1 = Sa „ -n-1 where l_ = [&,£,. .. ] T . If the method is A-s table, then a = Sa__-*Oasn->- ° for fixed h whenever -n —0 Re(y) < 0. Hence, A-s t ability implies that all eigenvalues of S are inside the unit circle when Re(y) < 0. The eigenvalues of S are the same as the eigenvalues of I - I Let A - EI = [c^,c ,... ,c, ] where c. is the i-th column of A - £1 , and let A£ = v. Then we want to study the solutions of = det = det[c_ + v = dettcQ.^.Cg,... ,c k ] + det[ ^r"^7 ' %' C 2"-"% ] 1 M " ^Uo'^- 1 ^ • % S*} -3- 1 ^0 v ^{detU-Sl) - det[c , — , Cg,...,^]} 1 *1 " ^ £ o - v ydet(A-£l) - det[j^ , c.^ ,. . . .cj} [p(5) - ya(C)] (1) where p(£) and a(0 are the weighted sums of some determinants which are polynomials in £ and independent of y. Now this is the same as the polynomial that determines the roots of the constant coefficient difference equation arising when a linear multistep method is applied to y' = Ay. If the method has order p, then one eigenvalue of S, namely one zero of (1) must approximate e to order p, and this is the necessary and sufficient condition for p and a to determine a p-th order multistep method. If the eigenvalues of S are less than one for all Re(u) < 0, then the corresponding multistep method is A-stable , but this is not possible if p > 2. LIST OF REFERENCES [l] Dahlquist, G. , "Numerical Integration of Ordinary Differential Equations," Math. Scandinavica , k,(l956) pp. 33-50. [2] Dahlquist, G. , "A Special Stability Problem for Linear Multistep Methods," BIT , 3 (1963) pp. 27-^3. [3] Gear, C. W. , "The Numerical Integration of Ordinary Differential Equations of Various Orders," Argonne National Lab Report #ANL 7126 , Argonne , Illinois (1966). [k] Gear, C. W. , NUMERICAL INITIAL VALUE PROBLEMS IN ORDINARY DIFFERENTIAL EQUATIONS, Prentice-Hall (l97l). Form AEC-427 (6/68) AECM 3201 U.S. ATOMIC ENERGY COMMISSION UNIVERSITY-TYPE CONTRACTOR'S RECOMMENDATION FOR DISPOSITION OF SCIENTIFIC AND TECHNICAL DOCUMENT ( See Instructions on Reverse Side ) 1. AEC REPORT NO. COO-1^69-0219 2. TITLE A NOTE ON THE NON-EXISTENCE OF MULTIVALUE A-STABLE METHODS OF ORDER GREATER THAN TWO 3. TYPE OF DOCUMENT (Check one): a. Scientific and technical report l~~l b. Conference paper not to be published in a journal: Title of conference Date of conference Exact location of conference. Sponsoring organization □ c. Other (Specify) 4. RECOMMENDED ANNOUNCEMENT AND DISTRIBUTION (Check one): Q a. AEC's normal announcement and distribution procedures may be followed. ] b. Make available only within AEC and to AEC contractors and other U.S. Government agencies and their contractors. "2 c. Make no announcement or distrubution. 5. REASON FOR RECOMMENDED RESTRICTIONS: 6. SUBMITTED BY: NAME AND POSITION (Please print or type) C. W. Gear, Professor and Principal Investigator Organization Department of Computer Science university of Illinois Urbana, Illinois 61801 Signature ^hcMi^o ^-. Date March 1973 FOR AEC USE ONLY 7. AEC CONTRACT ADMINISTRATOR'S COMMENTS, IF ANY, ON ABOVE ANNOUNCEMENT AND DISTRIBUTION RECOMMENDATION: 8. PATENT CLEARANCE: I I a. AEC patent clearance has been granted by responsible AEC patent group. I I b. Report has been sent to responsible AEC patent group for clearance. I I c. Patent clearance not required. 4. Title and Subtitle BIBLIOGRAPHIC DATA SHEET 1. Report No. UIUCDCS-R- 73-569 3. Recipient's Accession No. A NOTE ON THE NON-EXISTENCE OF MULTIVALUE A-S TABLE METHODS OF ORDER GREATER THAN TWO 5. Report Date March 1973 7. Author(s) C. ¥. Gear 8. Performing Organization Rept. No. 9. Performing Organization Name and Address Department of Computer Science University of Illinois Urbana, Illinois 6l801 10. Project/Task/Work Unit No. 11. Contract/Grant No. US AEC AT(ll-l)lU69 12. Sponsoring Organization Name and Address US AEC Chicago Operations Office 9800 South Cass Avenue Argonne , Illinois 60*+39 13. Type of Report & Period Covered 14. 15. Supplementary Notes 16. Abstracts It is shown that the Dahlquist result limiting the order of A-staole multistep methods also applies to multi value methods. 17. Key Words and Document Analysis. 17a. Descriptors A-Stability Multi value Methods 17b. Identifiers/Open-Ended Terms 17e. COSATI Field/Group 18. Availability Statement unlimited distribution 19. Security Class (This Report) UNCLASSIFIED 20. Security Class (This Page UNCLASSIFIED 21. No. of Pages 7 22. Price FORM NTIS-3B ( 10-70) USCOMM-DC 40329-P71 OCT 2 9 1973