This partitioned system, [EQUATION] is to be solved many times with the same n by n matrix, A, but differing n-vectors, b, c, and f, and differing scalars d and g. By using a bordering algorithm, A need be inverted once only. The remaining computation, done often, uses 2n2+3n+constant long operations (multiplications + divisions). Govaerts and Pryce (1987) consider the situation when A is nearly singular although (1) itself is well conditioned, reporting that using a bordering computation followed by iterative refinement of the initial poor solution works well, being both simple and practical.
{"title":"Another alternative for bordered systems","authors":"W. Knight","doi":"10.1145/37523.37526","DOIUrl":"https://doi.org/10.1145/37523.37526","url":null,"abstract":"This partitioned system, [EQUATION] is to be solved many times with the same n by n matrix, A, but differing n-vectors, b, c, and f, and differing scalars d and g. By using a bordering algorithm, A need be inverted once only. The remaining computation, done often, uses 2n2+3n+constant long operations (multiplications + divisions). Govaerts and Pryce (1987) consider the situation when A is nearly singular although (1) itself is well conditioned, reporting that using a bordering computation followed by iterative refinement of the initial poor solution works well, being both simple and practical.","PeriodicalId":177516,"journal":{"name":"ACM Signum Newsletter","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1987-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123383095","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 reports on a panel discussion with the same title at the "Workshop on Problems Relating Computer Science to Numerical Analysis" held at the Mathematical Sciences Research Institute at Berkeley, California on January 17, 1986. Speeches by the four panelists (Chorin, Collela, Kahan and Karp) are summarized, as well as discussions initiated by other participants. Additional comments by four leading scientists in this interdisciplinary area (Denning, Gear, Perlis and Rice) are included in an appendix.
{"title":"Obstacles which split computer science and numerical analysis","authors":"T. Chan","doi":"10.1145/36318.36321","DOIUrl":"https://doi.org/10.1145/36318.36321","url":null,"abstract":"This article reports on a panel discussion with the same title at the \"Workshop on Problems Relating Computer Science to Numerical Analysis\" held at the Mathematical Sciences Research Institute at Berkeley, California on January 17, 1986. Speeches by the four panelists (Chorin, Collela, Kahan and Karp) are summarized, as well as discussions initiated by other participants. Additional comments by four leading scientists in this interdisciplinary area (Denning, Gear, Perlis and Rice) are included in an appendix.","PeriodicalId":177516,"journal":{"name":"ACM Signum Newsletter","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1987-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132740265","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}
New non-linear Runge-Kutta methods for solving initial value problems are shown to be obtained by the strategic use of geometric mean (GM) rather than arithmetic mean averaging of the functional values in the standard integration formula.
{"title":"A nonlinear Runge-Kutta formula for initial value problems","authors":"D. J. Evans, B. Sanugi","doi":"10.1145/36318.36322","DOIUrl":"https://doi.org/10.1145/36318.36322","url":null,"abstract":"New non-linear Runge-Kutta methods for solving initial value problems are shown to be obtained by the strategic use of geometric mean (GM) rather than arithmetic mean averaging of the functional values in the standard integration formula.","PeriodicalId":177516,"journal":{"name":"ACM Signum Newsletter","volume":"118 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1987-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116347439","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 paper describes the author's experience of implementing a comprehensive library of elementary functions in the INMOS proprietry language occam, to support scientific and engineering applications of transputers.
{"title":"Implementing an elementary function library","authors":"P. Thompson","doi":"10.1145/24936.24937","DOIUrl":"https://doi.org/10.1145/24936.24937","url":null,"abstract":"This paper describes the author's experience of implementing a comprehensive library of elementary functions in the INMOS proprietry language occam, to support scientific and engineering applications of transputers.","PeriodicalId":177516,"journal":{"name":"ACM Signum Newsletter","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1987-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123665051","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}
In the preceding note (Dunham, 1987) were given forms of approximation that were provably monotone for suitable arithmetic, but left open the forms(1) c + x * r (x),where r could be a rational (the Kuki form on middle p. 9) or the next to last stage in Horner's method (top p. 10). If r is positive and non-decreasing on a positive interval I, (1) is obviously monotone on I for monotone arithmetic. This leaves two cases, (a)r positive and (slowly) decreasing on I, and (b) r negative on I. We consider them separately.
在前面的注释中(Dunham, 1987)给出了对于合适的算法来说可以证明是单调的近似形式，但保留了(1)c + x * r (x)的形式，其中r可以是有理性的(中间第9页上的Kuki形式)或Horner方法的下一个最后阶段(第10页上)。如果r在正区间I上为正且不递减，则(1)对于单调算法在I上明显是单调的。这就留下了两种情况，(a)r在I上为正且(缓慢)减小，(b) r在I上为负，我们分别考虑它们。
{"title":"Provably monotone approximations","authors":"C. Dunham","doi":"10.1145/24936.24938","DOIUrl":"https://doi.org/10.1145/24936.24938","url":null,"abstract":"In the preceding note (Dunham, 1987) were given forms of approximation that were provably monotone for suitable arithmetic, but left open the forms(1) c + x * r (x),where r could be a rational (the Kuki form on middle p. 9) or the next to last stage in Horner's method (top p. 10). If r is positive and non-decreasing on a positive interval I, (1) is obviously monotone on I for monotone arithmetic. This leaves two cases, (a)r positive and (slowly) decreasing on I, and (b) r negative on I. We consider them separately.","PeriodicalId":177516,"journal":{"name":"ACM Signum Newsletter","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1987-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128289218","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}
The Gaussian elimination method has been widely used for solutions of sets of linear equations for a very long time. This paper describes another use for the Gaussian elimination techniques. An algorithm will be developed for computing the coefficients for the central difference expressions approximating the first or second derivative of any order error by expressing the problem in matrix notation and applying Gaussian techniques for the solution.
{"title":"Using Gaussian elimination for computation of the central difference equation coefficients","authors":"S. Cynar","doi":"10.1145/24936.24939","DOIUrl":"https://doi.org/10.1145/24936.24939","url":null,"abstract":"The Gaussian elimination method has been widely used for solutions of sets of linear equations for a very long time. This paper describes another use for the Gaussian elimination techniques. An algorithm will be developed for computing the coefficients for the central difference expressions approximating the first or second derivative of any order error by expressing the problem in matrix notation and applying Gaussian techniques for the solution.","PeriodicalId":177516,"journal":{"name":"ACM Signum Newsletter","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1987-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134427861","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}
A topic that does not seem to have been considered is how to do function evaluations in parallel, that is, how to evaluate a mathematical function, eg. exp, for several different arguments simultaneously on a machine with parallel processors eg. ILLIAC IV: that machine had 64 processors, whose reaction to a (parallel) command from the master computer was either to execute or (on a condition) to decline to execute the instruction.
{"title":"The challenge of parallel function evaluation","authors":"C. Dunham","doi":"10.1145/12492.12495","DOIUrl":"https://doi.org/10.1145/12492.12495","url":null,"abstract":"A topic that does not seem to have been considered is how to do function evaluations in parallel, that is, how to evaluate a mathematical function, eg. exp, for several different arguments simultaneously on a machine with parallel processors eg. ILLIAC IV: that machine had 64 processors, whose reaction to a (parallel) command from the master computer was either to execute or (on a condition) to decline to execute the instruction.","PeriodicalId":177516,"journal":{"name":"ACM Signum Newsletter","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1986-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124592351","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}
I regret the copy of "Numerical Methods Disguised" (SIGNUM Newsletter, July 1986, 21: 31-32) that I forwarded to you had the last paragraph omitted. It reads:An analytical solution of the same problem should also be presented. Thus for a seller's market, the compromise price will be B + (2/3)(S - B). The figure 2/3 is obtained by observing that the bargaining procedure uses successive halving. Hence the series 1 - 1/2+1/4 - 1/8+1/16 - 1/32+1/64 - ... which converges to 2/3. How close to 2/3 the compromise price will be depends on the tolerance, i.e., on how many terms in the series will be summed. Identical reasoning gives the compromise price in a buyer's market as B+(1/3)(S - B). Note that these expressions are algebraically identical to (2/3)S+(1/3)B and (1/3)S+(2/3)B respectively, which, in that form, nicely covey the idea behind a seller's or buyer's market.
{"title":"Addendum to \"Numerical Methods Disguised\" (SIGNUM Newsletter, July 1986, 21: 31-32)","authors":"J. S. Fulda","doi":"10.1145/12492.1057971","DOIUrl":"https://doi.org/10.1145/12492.1057971","url":null,"abstract":"I regret the copy of \"Numerical Methods Disguised\" (SIGNUM Newsletter, July 1986, 21: 31-32) that I forwarded to you had the last paragraph omitted. It reads:An analytical solution of the same problem should also be presented. Thus for a seller's market, the compromise price will be B + (2/3)(S - B). The figure 2/3 is obtained by observing that the bargaining procedure uses successive halving. Hence the series 1 - 1/2+1/4 - 1/8+1/16 - 1/32+1/64 - ... which converges to 2/3. How close to 2/3 the compromise price will be depends on the tolerance, i.e., on how many terms in the series will be summed. Identical reasoning gives the compromise price in a buyer's market as B+(1/3)(S - B). Note that these expressions are algebraically identical to (2/3)S+(1/3)B and (1/3)S+(2/3)B respectively, which, in that form, nicely covey the idea behind a seller's or buyer's market.","PeriodicalId":177516,"journal":{"name":"ACM Signum Newsletter","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1986-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114623265","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}
Mathematical switches have been used to choose the suitable numerical scheme automatically at every computing step while solving differential equation problems with a multischeme approach. The concept of using mathematical switches in a numerical computation with a multiprocessor computer is feasible. The use of the switches in a multischeme computation yields a more accurate solution than the use of only a single scheme. We examined the use of two mathematical switches, namely, the one of exact error and of truncation error. The numerical results from the use of the former switch confirmed our conjecture that a combined solution of several schemes is indeed more accurate than one of an individual scheme. However, a switch of exact error is not practical and the one of truncation error is not satisfactory. Hence, further study is needed to design more practical and more satisfactory mathematical switches, and to establish the practical value on their use.
{"title":"Use of mathematical switches to solve differential equation problems","authors":"Yi-ling F. Chiang","doi":"10.1145/12492.12494","DOIUrl":"https://doi.org/10.1145/12492.12494","url":null,"abstract":"Mathematical switches have been used to choose the suitable numerical scheme automatically at every computing step while solving differential equation problems with a multischeme approach. The concept of using mathematical switches in a numerical computation with a multiprocessor computer is feasible. The use of the switches in a multischeme computation yields a more accurate solution than the use of only a single scheme. We examined the use of two mathematical switches, namely, the one of exact error and of truncation error. The numerical results from the use of the former switch confirmed our conjecture that a combined solution of several schemes is indeed more accurate than one of an individual scheme. However, a switch of exact error is not practical and the one of truncation error is not satisfactory. Hence, further study is needed to design more practical and more satisfactory mathematical switches, and to establish the practical value on their use.","PeriodicalId":177516,"journal":{"name":"ACM Signum Newsletter","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1986-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128826917","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}
DPUP is a library of utilities that support distributed concurrent computing on a local area network of computers. The library is built upon the interprocess communication facilities in Berkeley Unix 4.2bsd. Thus it will run on any network, connected by an Ethernet, where each computer runs a version of the Unix operating system that supports the Berkeley Unix interprocess communication facilities. DPUP supports two models of distributed concurrent computation, a master-slave model based upon stream sockets, and a broadcast model based upon datagram sockets. With each model, facilities for creating and terminating remote processes, establishing communications between them, and sending and receiving data between these processes are provided. This paper describes the facilities provided in DPUP and gives examples of their use.
{"title":"DPUP: a distributed processing utilities package","authors":"Timothy J Gardner","doi":"10.1145/12492.12493","DOIUrl":"https://doi.org/10.1145/12492.12493","url":null,"abstract":"DPUP is a library of utilities that support distributed concurrent computing on a local area network of computers. The library is built upon the interprocess communication facilities in Berkeley Unix 4.2bsd. Thus it will run on any network, connected by an Ethernet, where each computer runs a version of the Unix operating system that supports the Berkeley Unix interprocess communication facilities. DPUP supports two models of distributed concurrent computation, a master-slave model based upon stream sockets, and a broadcast model based upon datagram sockets. With each model, facilities for creating and terminating remote processes, establishing communications between them, and sending and receiving data between these processes are provided. This paper describes the facilities provided in DPUP and gives examples of their use.","PeriodicalId":177516,"journal":{"name":"ACM Signum Newsletter","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1986-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127023940","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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