It is shown that use of double precision in complex multiplication and division can significantly reduce rounding errors in these operations. Use of double precision accumulation of inner products in complex multiplication gives very close to the best possible error bound. Use of the same in complex division reduces error bounds, but a double precision divide is necessary to get close to the best possible bound.
{"title":"Improvement of complex arithmetic by use of double elements","authors":"C. Dunham","doi":"10.1145/74654.74655","DOIUrl":"https://doi.org/10.1145/74654.74655","url":null,"abstract":"It is shown that use of double precision in complex multiplication and division can significantly reduce rounding errors in these operations. Use of double precision accumulation of inner products in complex multiplication gives very close to the best possible error bound. Use of the same in complex division reduces error bounds, but a double precision divide is necessary to get close to the best possible bound.","PeriodicalId":177516,"journal":{"name":"ACM Signum Newsletter","volume":"241 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1989-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133714408","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}
There has recently been an upsurge of work on the (by now) classical problem of the stability of Gaussian elimination e.g., the papers of Trefethen & Schreiber [5], and Higham and Higham [2]. This short note describes some experimental results in this area derived by the author. They are described more fully in a technical report [4].
最近对高斯消去的稳定性(到目前为止)这一经典问题的研究兴起，例如Trefethen &Schreiber [5]， Higham and Higham[2]。这篇短文描述了作者在这一领域得出的一些实验结果。它们在技术报告[4]中有更全面的描述。
{"title":"Some statistics on Gaussian elimination with partial pivoting","authors":"A. MacLeod","doi":"10.1145/74650.74653","DOIUrl":"https://doi.org/10.1145/74650.74653","url":null,"abstract":"There has recently been an upsurge of work on the (by now) classical problem of the stability of Gaussian elimination e.g., the papers of Trefethen & Schreiber [5], and Higham and Higham [2]. This short note describes some experimental results in this area derived by the author. They are described more fully in a technical report [4].","PeriodicalId":177516,"journal":{"name":"ACM Signum Newsletter","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1989-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121801804","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}
We argue that subroutine libraries are not enough but need to be supplemented by common file formats and directly executable commands. A concrete example, tensor spline primitives, illustrates this style of programming.
{"title":"A philosophy for scientific computing tools","authors":"W. M. Coughran, E. Grosse","doi":"10.1145/74650.74651","DOIUrl":"https://doi.org/10.1145/74650.74651","url":null,"abstract":"We argue that subroutine libraries are not enough but need to be supplemented by common file formats and directly executable commands. A concrete example, tensor spline primitives, illustrates this style of programming.","PeriodicalId":177516,"journal":{"name":"ACM Signum Newsletter","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1989-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123191109","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}
Under favourable conditions, evaluation of polynomials by Homer's rule has an error not exceeding a few units in the last place.It has been observed that for moderate sized x and coefficients decreasing rapidly in magnitude that floating point evaluation of polynomials by Homer's method gives results accurate to around one or two units in the last place. A hand calculated example together with an informal justification is given by Fike (1968, pp. 52--53). In this note the observation is justified by a Wilkinson-type backward error analysis. It would be helpful for the reader to be familiar with the approach of Wilkinson (1963, Chap. 1), in which fl (expr) denotes the effect of evaluation of expression expr in floating point.We will assume that floating point is done using guard digits (Johnston, 1982, p. 11: Sterbenz, 1974) in which case we have by analysis similar to those of Wilkinson (1963, Chap. 1).
在有利的条件下，用荷马法则计算多项式的误差在最后一个地方不会超过几个单位。已经观察到，对于中等大小的x和大小迅速下降的系数，用荷马的方法对多项式进行浮点计算，结果精确到最后一个或两个单位左右。Fike(1968，第52—53页)给出了一个手工计算的例子和非正式的论证。在这篇笔记中，观察结果是通过威尔金森式的反向误差分析来证明的。熟悉Wilkinson(1963，第1章)的方法会对读者有所帮助，其中fl (expr)表示浮点表达式expr求值的效果。我们将假设浮点数是使用保护数字完成的(Johnston, 1982, p. 11; Sterbenz, 1974)，在这种情况下，我们通过分析与Wilkinson(1963，第1章)相似。
{"title":"Perturbation analysis of Horner's method for nice cases","authors":"C. Dunham","doi":"10.1145/74650.74652","DOIUrl":"https://doi.org/10.1145/74650.74652","url":null,"abstract":"Under favourable conditions, evaluation of polynomials by Homer's rule has an error not exceeding a few units in the last place.It has been observed that for moderate sized x and coefficients decreasing rapidly in magnitude that floating point evaluation of polynomials by Homer's method gives results accurate to around one or two units in the last place. A hand calculated example together with an informal justification is given by Fike (1968, pp. 52--53). In this note the observation is justified by a Wilkinson-type backward error analysis. It would be helpful for the reader to be familiar with the approach of Wilkinson (1963, Chap. 1), in which fl (expr) denotes the effect of evaluation of expression expr in floating point.We will assume that floating point is done using guard digits (Johnston, 1982, p. 11: Sterbenz, 1974) in which case we have by analysis similar to those of Wilkinson (1963, Chap. 1).","PeriodicalId":177516,"journal":{"name":"ACM Signum Newsletter","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1989-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116810498","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}
Equations linear in n out of n + 1 variables can be solved by reduction to a linear subproblem and variation of the nonlinear parameter.
在n + 1个变量中有n个变量的线性方程可以通过简化为线性子问题和非线性参数的变化来求解。
{"title":"Solving equations nonlinear in only one of N + 1 variables","authors":"C. Dunham, C. Zhu","doi":"10.1145/58859.58860","DOIUrl":"https://doi.org/10.1145/58859.58860","url":null,"abstract":"Equations linear in n out of n + 1 variables can be solved by reduction to a linear subproblem and variation of the nonlinear parameter.","PeriodicalId":177516,"journal":{"name":"ACM Signum Newsletter","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1988-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122191352","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 most widely used is GAMS, which is specifically designed for Further details on GAMS can be found in the GAMS User's Guide. The GAMS User's Guide. Mathematical Programming, 87:153–176, 2000. (4). A. Brooke, D. Kendrick, and A. Meeraus. GAMS: A User's Guide. The Scientific Press, South San Francisco. JuMP is an open-source modeling language that allows users to express a wide D. Kendrick, A. Meeraus, and R. Raman, GAMS: A User's Guide, Scientific.
使用最广泛的是GAMS，它是专门为GAMS设计的，更多关于GAMS的详细信息可以在GAMS用户指南中找到。GAMS用户指南。数学规划，87:153-176,2000。(4). A.布鲁克，D.肯德里克，A.密劳斯。GAMS:用户指南。科学出版社，南旧金山。JuMP是一种开源建模语言，允许用户广泛表达D. Kendrick, a . Meeraus和R. Raman, GAMS: a User's Guide, Scientific。
{"title":"GAMS, a user's guide","authors":"A. Brook, David Kendrick, A. Meeraus","doi":"10.1145/58859.58863","DOIUrl":"https://doi.org/10.1145/58859.58863","url":null,"abstract":"The most widely used is GAMS, which is specifically designed for Further details on GAMS can be found in the GAMS User's Guide. The GAMS User's Guide. Mathematical Programming, 87:153–176, 2000. (4). A. Brooke, D. Kendrick, and A. Meeraus. GAMS: A User's Guide. The Scientific Press, South San Francisco. JuMP is an open-source modeling language that allows users to express a wide D. Kendrick, A. Meeraus, and R. Raman, GAMS: A User's Guide, Scientific.","PeriodicalId":177516,"journal":{"name":"ACM Signum Newsletter","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1988-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133653999","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 entertainment industry graphical images are often produced without adherence to physically accurate models. This is usually the case because there is a need to reduce production time and costs. The physical scientist, on the other hand, must produce graphical images that are accurate simulations of the physical phenomena they are attempting to visualize. Frequently the data that an accurate simulation produces can be difficult to use for rendering the animation. This paper describes the use of numerical methods to minimize the number of keyframes for an accurate animation.
{"title":"The use of forward difference equations in animation of physical phenomena","authors":"S. Cynar","doi":"10.1145/58859.58861","DOIUrl":"https://doi.org/10.1145/58859.58861","url":null,"abstract":"In the entertainment industry graphical images are often produced without adherence to physically accurate models. This is usually the case because there is a need to reduce production time and costs. The physical scientist, on the other hand, must produce graphical images that are accurate simulations of the physical phenomena they are attempting to visualize. Frequently the data that an accurate simulation produces can be difficult to use for rendering the animation. This paper describes the use of numerical methods to minimize the number of keyframes for an accurate animation.","PeriodicalId":177516,"journal":{"name":"ACM Signum Newsletter","volume":"308 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1988-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131969609","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}
{"title":"New version of MACHAR available","authors":"W. Cody","doi":"10.1145/58859.58862","DOIUrl":"https://doi.org/10.1145/58859.58862","url":null,"abstract":"","PeriodicalId":177516,"journal":{"name":"ACM Signum Newsletter","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1988-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116088090","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}
Under stated favorable conditions, the Lagrange formula for polynomial interpolation is computationally exact at the nodes. This is applied to approximation with Lagrange-type interpolation.
在规定的有利条件下，多项式插值的拉格朗日公式在节点处计算精确。这被应用于拉格朗日型插值的近似。
{"title":"Approximation with exact Lagrange-type interpolation","authors":"C. Dunham, Z. Zhu","doi":"10.1145/47917.47920","DOIUrl":"https://doi.org/10.1145/47917.47920","url":null,"abstract":"Under stated favorable conditions, the Lagrange formula for polynomial interpolation is computationally exact at the nodes. This is applied to approximation with Lagrange-type interpolation.","PeriodicalId":177516,"journal":{"name":"ACM Signum Newsletter","volume":"788 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1988-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"113995422","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}
We wish to generate all unique permutations of a list of elements with possibly repeated items. We develop four algorithms to generate permutations, two from Knuth's Art of Computer Programming and two of our own devising. The last one developed also gives a numbering scheme for permutations that can be inverted to regenerate original permutations.
{"title":"Generations of permutations with non-unique elements","authors":"T. Rolfe","doi":"10.1145/47917.47922","DOIUrl":"https://doi.org/10.1145/47917.47922","url":null,"abstract":"We wish to generate all unique permutations of a list of elements with possibly repeated items. We develop four algorithms to generate permutations, two from Knuth's Art of Computer Programming and two of our own devising. The last one developed also gives a numbering scheme for permutations that can be inverted to regenerate original permutations.","PeriodicalId":177516,"journal":{"name":"ACM Signum Newsletter","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1988-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132054977","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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