Algorithm 454: the complex method for constrained optimization [E4]

TLDR: An efficient solut ion has been found to depend on a number of ad hoc strategies, which are descr ibed in detail in the paper .

Abstract: s o f the papers by Daily and Lynning A Sel f -Modi fy ing Ex t rapo la t ion M e t h o d for Solving O r d i n a r y Differential Equa t ions Dar D. Dai ly Kansas State University, Manhattan, Kansas Author's address: 8133 Dearborn Drive, Prairie Village, KS 66208 This paper outl ines a p rogram that searches for the p r e d o m i n a n t terms of the a sympto t i c er ror expans ion of initial value problems in o rd ina ry differential equa t ions and uses this i n fo rma t ion in a se l f -modifying ex t rapo la t ion process. Dur ing the in tegra t ion process, using a ra t io that Carl de Boor (1971) used in an integra t ion program, the me thod seeks to recognize t rends of change in the error expans ion of the differential equat ion and to adjus t the me thod of ex t rapola t ion . A basic a lgo r i thm used in the modi fy ing process is presented a long with a br ief exp lana t ion . Also, a compa r i son made with the wel l -known ra t iona l ex t rapo la t ion m e t h o d shows ra t ional ex t rapo la t ion to be general ly less efficient in terms of funct ion eva lua t ions but also demons t ra t e s tha t the se l f -modifying me thod is general ly not able to reduce its e r ror to the level of ra t iona l ex t rapo la t ion . A note, though, shows the se l f -modifying m e t h o d to be super ior to the regular R o m b e r g ex t rapo la t ion . Key W o r d s and Phrases : se l f -modifying ex t rapo la t ion, ra t iona l ex t rapo la t ion , modif ied m i d p o i n t me thod , R o m b e r g in tegrat ion, a sympto t i c e r ror expans ion , predominan t , s ingulari ty, ini t ial value p rob lems in o rd ina ry differential equa t ions ; C R Categor ies : 5.10, 5.17 A C o m p u t e r Solut ion of Po lygona l Jigsaw Puzzles Ejv ind Lynning University of Arhus, Arhus, Denmark Author's address: Brandeis University, Waltham, M A. 02154 A p rogram to solve any j igsaw puzzle involving pieces of polygonal shape is described. An efficient solut ion has been found to depend on a number of ad hoc strategies, which are descr ibed in detail in the paper . The puzzles are solved by successively placing individual pieces in the region to be covered using a depth-f i rs t tree search a lgor i thm. A formal representa t ion of regions, pieces, and placings of pieces is defined. The main idea behind the chosen represen ta t ion is to or ient clockwise the po lygons mak ing up a region, and to or ient counter c lockwise the pieces to be placed. Placing a piece means compu t ing a valid new region, i.e. one or more c lockwise or ien ted polygons , cons t ruc ted f rom the previous one by removing the par t co r re spond ing to the piece which is placed. The da ta s t ructure and the p rocedures requi red to examine where pieces can be placed and how to pe r fo rm the placing of the pieces are also descr ibed. Al l puzzles so far presented to the p r o g r a m have been successfully solved in a r easonab le t ime. K e y W o r d s and Phrases : art if icial intelligence, p roblem solving, pa t te rn recogni t ion , puzzles, po lygona l puzzles, j igsaw puzzles, back t r ack p rog ramming , tree search a lgor i thms ; C R Categor ies : 3.6, 3.63, 3.64 482 Algor i t hms L.D. F o s d i c k and A.K. Cline, Ed i to r s Submittal of an algorithm for consideration for publication in Communications of the A C M implies unrestricted use of the algorithm within a computer is permissible. Copyright © 1973, Association for Computing Machinery, Inc. General permission to republish, but not for profit, all or part of this material is granted provided that ACM's copyright notice is given and that reference is made to the publication, to its date of issue, and to the fact that reprinting privileges were granted by permission of the Association for Computing Machinery.