This paper presents a brief historical survey of iterative methods for solving linear systems of equations. The journey begins with Gauss who developed the first known method that can be termed iterative. The early 20th century saw good progress of these methods which were initially used to solve least-squares systems, and then linear systems arising from the discretization of partial different equations. Then iterative methods received a big impetus in the 1950s - partly because of the development of computers. The survey does not attempt to be exhaustive. Rather, the aim is to underline the way of thinking at a specific time and to highlight the major ideas that steered the field.
{"title":"Iterative methods for linear systems of equations: A brief historical journey","authors":"Y. Saad","doi":"10.1090/conm/754/15141","DOIUrl":"https://doi.org/10.1090/conm/754/15141","url":null,"abstract":"This paper presents a brief historical survey of iterative methods for solving linear systems of equations. The journey begins with Gauss who developed the first known method that can be termed iterative. The early 20th century saw good progress of these methods which were initially used to solve least-squares systems, and then linear systems arising from the discretization of partial different equations. Then iterative methods received a big impetus in the 1950s - partly because of the development of computers. The survey does not attempt to be exhaustive. Rather, the aim is to underline the way of thinking at a specific time and to highlight the major ideas that steered the field.","PeriodicalId":369218,"journal":{"name":"75 Years of Mathematics of\n Computation","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122072496","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article explains why the classical Murphy-E ranking function might fail to correctly rank polynomial pairs in the Number Field Sieve, and proposes a new ranking function.
本文解释了经典Murphy-E排序函数在Number Field Sieve中不能正确排序多项式对的原因，并提出了一种新的排序函数。
{"title":"A new ranking function for polynomial selection in the number field sieve","authors":"N. David, P. Zimmermann","doi":"10.1090/CONM/754/15139","DOIUrl":"https://doi.org/10.1090/CONM/754/15139","url":null,"abstract":"This article explains why the classical Murphy-E ranking function might fail to correctly rank polynomial pairs in the Number Field Sieve, and proposes a new ranking function.","PeriodicalId":369218,"journal":{"name":"75 Years of Mathematics of\n Computation","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127298567","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Summary: We present algorithms for computing j -invariants, modular polynomials and explicit isogenies for ordinary rank 2 Drinfeld modules over finite fields and describe how Drinfeld modular polynomials can be used to compute isogeny graphs and endomorphism rings of ordinary rank 2 Drinfeld modules. Our technique for computing Drinfeld modular polynomials is based on the traditional analytic approach for obtaining classical modular polynomials. Our ideas for generating isogeny graphs and finding endomorphism rings for rank 2 Drinfeld modules closely follows the work of Kohel and Fouquet. All our algorithms were implemented in SAGE and numerical examples are included. For the entire collection see [Zbl 1461.11002].
{"title":"Computing modular polynomials and isogenies of rank two Drinfeld modules over finite fields","authors":"Perlas Caranay, Matthew Greenberg, R. Scheidler","doi":"10.1090/conm/754/15148","DOIUrl":"https://doi.org/10.1090/conm/754/15148","url":null,"abstract":"Summary: We present algorithms for computing j -invariants, modular polynomials and explicit isogenies for ordinary rank 2 Drinfeld modules over finite fields and describe how Drinfeld modular polynomials can be used to compute isogeny graphs and endomorphism rings of ordinary rank 2 Drinfeld modules. Our technique for computing Drinfeld modular polynomials is based on the traditional analytic approach for obtaining classical modular polynomials. Our ideas for generating isogeny graphs and finding endomorphism rings for rank 2 Drinfeld modules closely follows the work of Kohel and Fouquet. All our algorithms were implemented in SAGE and numerical examples are included. For the entire collection see [Zbl 1461.11002].","PeriodicalId":369218,"journal":{"name":"75 Years of Mathematics of\n Computation","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121462096","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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