[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-326847-105":3,"doc-detail-326847-en":81,"detail-sidebar-cat-0-en-105":99},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","modular-algorithms-in-symbolic-summation-and-symbolic-integration-foreword","Modular Algorithms in Symbolic Summation and Symbolic Integration - Foreword","","Front matter and foreword for the lecture notes/published volume “Modular Algorithms in Symbolic Summation and Symbolic Integration”, authored by Jürgen Gerhard. The foreword positions the work at the intersection of computer algebra—symbolic integration and summation—and fast algorithmics, explaining how algorithm analysis, modular techniques, and nearly optimal algorithms address key limitations in traditional approaches to symbolic computation.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & Report",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/modular-algorithms-in-symbolic-summation-and-symbolic-integration-foreword/326847/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/modular-algorithms-in-symbolic-summation-and-symbolic-integration-foreword/326847.png","ImageObject",300,407,{"name":42,"@type":43},"Riley","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-24","2026-09-21",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",5,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What two main research streams does the foreword highlight?","Question",{"text":63,"@type":64},"It highlights symbolic integration and summation on one hand, and fast algorithmics on the other.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"Why is algorithm analysis considered a clear success criterion?",{"text":68,"@type":64},"Because designing an algorithm that is faster asymptotically in the worst case leads to immediate recognition of the result’s value.",{"name":70,"@type":61,"acceptedAnswer":71},"What is a fundamental requirement for modular algorithms described in the foreword?",{"text":72,"@type":64},"That the solution modulo p equals the solution of the problem modulo p, typically ensured for primes not dividing a nonzero resultant r.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},326847,1790240915,{"code":4,"msg":82,"data":83},"success",{"doc_id":79,"user_id":84,"nickname":42,"user_avatar":85,"doc_module":4,"category_id":86,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":87,"file_id":88,"file_url":89,"file_type":90,"file_size":91,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":92,"language":93,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":94,"faqs":95,"seo_title":96,"seo_description":12,"update_tm":97,"read_time":98,"preview_page_count":4},1374391975076,"https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051",8,"Lecture Notes in Computer Science 3218  \nCommenced Publication in 1973 Founding and Former Series Editors:  \nGerhard Goos, Juris Hartmanis, and Jan van Leeuwen  \nEditorial Board  \nDavid Hutchison  \nLancaster University, UK Takeo Kanade  \nCarnegie Mellon University, Pittsburgh, PA, USA Josef Kittler  \nUniversity of Surrey, Guildford, UK Jon M. Kleinberg  \nCornell University, Ithaca, NY, USA Friedemann Mattern  \nETH Zurich, Switzerland John C. Mitchell  \nStanford University, CA, USA Moni Naor  \nWeizmann Institute of Science, Rehovot, Israel Oscar Nierstrasz  \nUniversity of Bern, Switzerland  \nC. Pandu Rangan  \nIndian Institute of Technology, Madras, India Bernhard Steffen  \nUniversity of Dortmund, Germany Madhu Sudan  \nMassachusetts Institute of Technology, MA, USA Demetri Terzopoulos  \nNew York University, NY, USA Doug Tygar  \nUniversity of California, Berkeley, CA, USAMosheY. Vardi  \nRice University, Houston, TX, USA Gerhard Weikum  \nMax-Planck Institute of Computer Science, Saarbruecken, Germany  \nJürgen Gerhard  \nModular Algorithms in Symbolic Summation and Symbolic Integration  \n1 3  \nAuthor  \nJürgen Gerhard Maplesoft  \n615 Kumpf Drive, Waterloo, ON, N2V 1K8, Canada [E-mail: Gerhard.Juergen@web.de](E-mail: Gerhard.Juergen@web.de)  \nThis work was accepted as PhD thesis on July 13, 2001, at  \nFachbereich Mathematik und Informatik Universität Paderborn  \n33095 Paderborn, Germany  \nLibrary of Congress Control Number: 2004115730  \nCR Subject Classification (1998): F.2.1, G.1, I.1  \nISSN 0302-9743  \nISBN 3-540-24061-6 Springer Berlin Heidelberg New York  \nThis work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law.  \nSpringer is a part of Springer Science+Business Media [springeronline.com](springeronline.com)  \n© Springer-Verlag Berlin Heidelberg 2004 Printed in Germany  \nTypesetting: Camera-ready by author, data conversion by Boller Mediendesign  \nPrinted on acid-free paper SPIN: 11362159 06/3142 5 4 3 2 1 0  \nTo Herbert Gerhard (1921–1999)  \nForeword  \nThis work brings together two streams in computer algebra: symbolic integration and summation on the one hand, and fast algorithmics on the other hand.  \nIn many algorithmically oriented areas of computer science, the analysis of algorithms – placed into the limelight by Don Knuth’s talk at the 1970 ICM – provides a crystal-clear criterion for success. The researcher who designs an algorithm that is faster (asymptotically, in the worst case) than any previous method receives instant gratification: her result will be recognized as valuable. Alas, the downside is that such results come along quite infrequently, despite our best efforts.  \nAn alternative evaluation method is to run a new algorithm on examples; this has its obvious problems, but is sometimes the best we can do. George Collins, oneof the fathers of computer algebra and a great experimenter, wrote in 1969: “I think this demonstrates again that a simple analysis is often more revealing than a ream of empirical data (although both are important) .”  \nWithin computer algebra, some areas have traditionally followed the former methodology, notably some parts of polynomial algebra and linear algebra. Other areas, such as polynomial system solving, have not yet been amenable to this approach. The usual “input size” parameters of computer science seem inadequate, and although some natural “geometric” parameters have been identified (solution dimension, regularity), not all (potential) major progress can be expresse","cbCaicRPHQqAvhNR","https://ap.wps.com/l/cbCaicRPHQqAvhNR","pdf",1442961,231,"English","# Foreword\n## Algorithm analysis and evaluation methods\n## Modular algorithms: solution modulo p and resultant r\n## Symbolic integration as a symbolic problem","[{\"question\":\"What two main research streams does the foreword highlight?\",\"answer\":\"It highlights symbolic integration and summation on one hand, and fast algorithmics on the other.\"},{\"question\":\"Why is algorithm analysis considered a clear success criterion?\",\"answer\":\"Because designing an algorithm that is faster asymptotically in the worst case leads to immediate recognition of the result’s value.\"},{\"question\":\"What is a fundamental requirement for modular algorithms described in the foreword?\",\"answer\":\"That the solution modulo p equals the solution of the problem modulo p, typically ensured for primes not dividing a nonzero resultant r.\"}]","Modular Algorithms in Symbolic Summation and Symbolic Integration - Foreword | PDF",1789975536,582,{"code":4,"msg":82,"data":100},[101,105,109,113,117,122,127,130,135,138,142],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":102,"show_sort_weight":103,"slug":104},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":106,"show_sort_weight":107,"slug":108},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":110,"show_sort_weight":111,"slug":112},"Exam",70,"exam",{"id":55,"doc_module":4,"doc_module_name":25,"category_name":114,"show_sort_weight":115,"slug":116},"Comic",60,"comic",{"id":118,"doc_module":4,"doc_module_name":25,"category_name":119,"show_sort_weight":120,"slug":121},6,"Technology",50,"technology",{"id":123,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":125,"slug":126},7,"Healthcare",40,"healthcare",{"id":86,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":128,"slug":129},30,"research-report",{"id":131,"doc_module":4,"doc_module_name":25,"category_name":132,"show_sort_weight":133,"slug":134},9,"Religion & Spirituality",20,"religion-spirituality",{"id":133,"doc_module":4,"doc_module_name":25,"category_name":136,"show_sort_weight":133,"slug":137},"World Cup","world-cup",{"id":139,"doc_module":4,"doc_module_name":25,"category_name":140,"show_sort_weight":139,"slug":141},10,"Lifestyle","lifestyle",{"id":143,"doc_module":4,"doc_module_name":25,"category_name":144,"show_sort_weight":55,"slug":145},19,"General","general"]