Multiplication of n-digit integers by long multiplication requires O(n 2 ) operations and can be time-consuming. A. Schonhage and V. Strassen published an algorithm in 1970 that is capable of performing the task with only O(nlog(n)) arithmetic operations over C; naturally, finite-precision approximations to C are used and rounding errors need to be accounted for. Overall, using variable-precision fixed-point numbers, this results in an O(n(log(n)) 2+" )-time algorithm. However, to make this algorithm more efficient and practical we need to make use of hardware-based floating- point numbers. How do we deal with rounding errors? and how do we determine the limits of the fixed-precision hardware? Our solution is to use interval arithmetic to guarantee the correctness of results and deter- mine the hardware's limits. We examine the feasibility of this approach and are able to report that 75,000-digit base-256 integers can be han- dled using double-precision containment sets. This demonstrates that our approach has practical potential; however, at this stage, our implemen- tation does not yet compete with commercial ones, but we are able to demonstrate the feasibility of this technique.
{"title":"A Rigorous Extension of the Schönhage-Strassen Integer Multiplication Algorithm Using Complex Interval Arithmetic","authors":"R. Sainudiin, T. Steinke","doi":"10.4204/EPTCS.24.19","DOIUrl":"https://doi.org/10.4204/EPTCS.24.19","url":null,"abstract":"Multiplication of n-digit integers by long multiplication requires O(n 2 ) operations and can be time-consuming. A. Schonhage and V. Strassen published an algorithm in 1970 that is capable of performing the task with only O(nlog(n)) arithmetic operations over C; naturally, finite-precision approximations to C are used and rounding errors need to be accounted for. Overall, using variable-precision fixed-point numbers, this results in an O(n(log(n)) 2+\" )-time algorithm. However, to make this algorithm more efficient and practical we need to make use of hardware-based floating- point numbers. How do we deal with rounding errors? and how do we determine the limits of the fixed-precision hardware? Our solution is to use interval arithmetic to guarantee the correctness of results and deter- mine the hardware's limits. We examine the feasibility of this approach and are able to report that 75,000-digit base-256 integers can be han- dled using double-precision containment sets. This demonstrates that our approach has practical potential; however, at this stage, our implemen- tation does not yet compete with commercial ones, but we are able to demonstrate the feasibility of this technique.","PeriodicalId":54499,"journal":{"name":"Reliable Computing","volume":"41 1","pages":"151-159"},"PeriodicalIF":0.0,"publicationDate":"2010-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82675247","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}
Pub Date : 2007-12-01DOI: 10.1007/s11155-007-9045-6
G. Xiang, M. Ceberio, V. Kreinovich
{"title":"Computing Population Variance and Entropy under Interval Uncertainty: Linear-Time Algorithms","authors":"G. Xiang, M. Ceberio, V. Kreinovich","doi":"10.1007/s11155-007-9045-6","DOIUrl":"https://doi.org/10.1007/s11155-007-9045-6","url":null,"abstract":"","PeriodicalId":54499,"journal":{"name":"Reliable Computing","volume":"13 1","pages":"467-488"},"PeriodicalIF":0.0,"publicationDate":"2007-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s11155-007-9045-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"53079624","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}
Pub Date : 2007-12-01DOI: 10.1007/s11155-007-9044-7
G. Chabert, L. Jaulin
{"title":"Computing the Pessimism of Inclusion Functions","authors":"G. Chabert, L. Jaulin","doi":"10.1007/s11155-007-9044-7","DOIUrl":"https://doi.org/10.1007/s11155-007-9044-7","url":null,"abstract":"","PeriodicalId":54499,"journal":{"name":"Reliable Computing","volume":"13 1","pages":"489-504"},"PeriodicalIF":0.0,"publicationDate":"2007-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s11155-007-9044-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"53079584","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}
Pub Date : 2007-09-19DOI: 10.1007/s11155-007-9043-8
N. Delanoue, L. Jaulin, Bertrand Cottenceau
{"title":"Guaranteeing the Homotopy Type of a Set Defined by Non-Linear Inequalities","authors":"N. Delanoue, L. Jaulin, Bertrand Cottenceau","doi":"10.1007/s11155-007-9043-8","DOIUrl":"https://doi.org/10.1007/s11155-007-9043-8","url":null,"abstract":"","PeriodicalId":54499,"journal":{"name":"Reliable Computing","volume":"13 1","pages":"381-398"},"PeriodicalIF":0.0,"publicationDate":"2007-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s11155-007-9043-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"53079526","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}
Pub Date : 2007-09-19DOI: 10.1007/S11155-007-9042-9
S. Lagrange, N. Delanoue, L. Jaulin
{"title":"On Sufficient Conditions of the Injectivity: Development of a Numerical Test Algorithm via Interval Analysis","authors":"S. Lagrange, N. Delanoue, L. Jaulin","doi":"10.1007/S11155-007-9042-9","DOIUrl":"https://doi.org/10.1007/S11155-007-9042-9","url":null,"abstract":"","PeriodicalId":54499,"journal":{"name":"Reliable Computing","volume":"13 1","pages":"409-421"},"PeriodicalIF":0.0,"publicationDate":"2007-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/S11155-007-9042-9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"53079477","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}
Pub Date : 2007-09-18DOI: 10.1007/s11155-007-9038-5
O. Castillo, V. Kreinovich
{"title":"Interval-Related Talks at the North American Fuzzy Information Processing Society Annual Conference NAFIPS’07","authors":"O. Castillo, V. Kreinovich","doi":"10.1007/s11155-007-9038-5","DOIUrl":"https://doi.org/10.1007/s11155-007-9038-5","url":null,"abstract":"","PeriodicalId":54499,"journal":{"name":"Reliable Computing","volume":"13 1","pages":"441-443"},"PeriodicalIF":0.0,"publicationDate":"2007-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s11155-007-9038-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"53079832","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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