In this paper we study the ladder operators for little q-Jacobi polynomials and for polynomials orthogonal with respect to a generalized q-Jacobi weight. We also briefly discuss the computational aspects in the computer algebra system Mathematica (www.wolfram.com).
{"title":"On Ladder Operators for Little q-Jacobi Polynomials and Their Generalizations","authors":"G. Filipuk, Maciej Haneczok","doi":"10.1109/SYNASC.2013.20","DOIUrl":"https://doi.org/10.1109/SYNASC.2013.20","url":null,"abstract":"In this paper we study the ladder operators for little q-Jacobi polynomials and for polynomials orthogonal with respect to a generalized q-Jacobi weight. We also briefly discuss the computational aspects in the computer algebra system Mathematica (www.wolfram.com).","PeriodicalId":293085,"journal":{"name":"2013 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125164837","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 present generalized multisets in the Zermelo-Fraenkel framework, in Reverse Mathematics, and in the Fraenkel-Mostowski framework. In the Zermelo-Fraenkel framework, we prove that the set of all generalized multisets over a certain finite set is a finitely-generated, lattice-ordered, free abelian group. Similar properties are then discussed in Reverse Mathematics. Finally, we study the generalized multisets in the Fraenkel-Mostowski framework, and present their nominal properties. Several Zermelo-Fraenkel algebraic properties of generalized multisets are translated into the Fraenkel-Mostowski framework by using the finite support axiom of the Fraenkel-Mostowski set theory.
{"title":"Algebraic Properties of Generalized Multisets","authors":"A. Alexandru, Gabriel Ciobanu","doi":"10.1109/SYNASC.2013.55","DOIUrl":"https://doi.org/10.1109/SYNASC.2013.55","url":null,"abstract":"We present generalized multisets in the Zermelo-Fraenkel framework, in Reverse Mathematics, and in the Fraenkel-Mostowski framework. In the Zermelo-Fraenkel framework, we prove that the set of all generalized multisets over a certain finite set is a finitely-generated, lattice-ordered, free abelian group. Similar properties are then discussed in Reverse Mathematics. Finally, we study the generalized multisets in the Fraenkel-Mostowski framework, and present their nominal properties. Several Zermelo-Fraenkel algebraic properties of generalized multisets are translated into the Fraenkel-Mostowski framework by using the finite support axiom of the Fraenkel-Mostowski set theory.","PeriodicalId":293085,"journal":{"name":"2013 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123591915","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 LHCb experiment at CERN processes its datasets over hundred different grid sites within the Worldwide LHC Computing Grid (WLCG). All those grid sites consist of multicore CPUs nowadays. However, the number of cores per worker node will increase in the near future. Using such worker nodes more efficiently requires parallelization of software as well as modifications at the level of scheduling. This paper will evaluate a moldable job model for LHCb grid jobs where the main challenge is the definition of the best degree of parallelism. Choosing an appropriate degree of parallelism depends on the parameters, on which optimization shall be applied. Commonly used features are for example scalability, workload and turnaround time. Prediction of run time is another major problem and it will be discussed how it can be handled using historical information. Furthermore, the advantages and disadvantages of a moldable job model will be discussed as well on how it must be extended to meet the requirements of LHCb jobs.
{"title":"Evaluating Moldability of LHCb Jobs for Multicore Job Submission","authors":"N. Rauschmayr, A. Streit","doi":"10.1109/SYNASC.2013.74","DOIUrl":"https://doi.org/10.1109/SYNASC.2013.74","url":null,"abstract":"The LHCb experiment at CERN processes its datasets over hundred different grid sites within the Worldwide LHC Computing Grid (WLCG). All those grid sites consist of multicore CPUs nowadays. However, the number of cores per worker node will increase in the near future. Using such worker nodes more efficiently requires parallelization of software as well as modifications at the level of scheduling. This paper will evaluate a moldable job model for LHCb grid jobs where the main challenge is the definition of the best degree of parallelism. Choosing an appropriate degree of parallelism depends on the parameters, on which optimization shall be applied. Commonly used features are for example scalability, workload and turnaround time. Prediction of run time is another major problem and it will be discussed how it can be handled using historical information. Furthermore, the advantages and disadvantages of a moldable job model will be discussed as well on how it must be extended to meet the requirements of LHCb jobs.","PeriodicalId":293085,"journal":{"name":"2013 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129568063","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}
K is a rewrite-based executable semantic framework for defining languages. The K framework is designed to allow implementing a variety of generic tools that can be used with any language defined in K, such as parsers, interpreters, symbolic execution engines, semantic debuggers, test-case generators, state-space explorers, model checkers, and even deductive program verifiers. The latter are based on matching logic for expressing static properties, and on reachability logic for expressing dynamic properties. Several large languages have been already defined or are being defined in K, including C, Java, Python, Javascript, and LLVM.
{"title":"Specifying Languages and Verifying Programs with K","authors":"Grigore Roşu","doi":"10.1109/SYNASC.2013.81","DOIUrl":"https://doi.org/10.1109/SYNASC.2013.81","url":null,"abstract":"K is a rewrite-based executable semantic framework for defining languages. The K framework is designed to allow implementing a variety of generic tools that can be used with any language defined in K, such as parsers, interpreters, symbolic execution engines, semantic debuggers, test-case generators, state-space explorers, model checkers, and even deductive program verifiers. The latter are based on matching logic for expressing static properties, and on reachability logic for expressing dynamic properties. Several large languages have been already defined or are being defined in K, including C, Java, Python, Javascript, and LLVM.","PeriodicalId":293085,"journal":{"name":"2013 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128989342","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}
Lang's "universal molecule" algorithm solves a variant of the origami design problem. It takes as input a metric tree and a convex polygonal region (the "paper") having a certain metric relationship with the tree. It computes a crease- pattern which allows for the paper to "fold" to a uniaxial base, which is a 3-dimensional shape projecting onto the given tree. Lang's universal molecule algorithm runs in cubic time and quadratic space. We investigate two implementations which improve the running time to sub-cubic time. The first uses a cyclic tournament forest, a new data structure which extends kinetic tournament trees to allow for cycle splitting operations, and the second uses a priority queue to store events.
{"title":"Computing Origami Universal Molecules with Cyclic Tournament Forests","authors":"J. Bowers, I. Streinu","doi":"10.1109/SYNASC.2013.13","DOIUrl":"https://doi.org/10.1109/SYNASC.2013.13","url":null,"abstract":"Lang's \"universal molecule\" algorithm solves a variant of the origami design problem. It takes as input a metric tree and a convex polygonal region (the \"paper\") having a certain metric relationship with the tree. It computes a crease- pattern which allows for the paper to \"fold\" to a uniaxial base, which is a 3-dimensional shape projecting onto the given tree. Lang's universal molecule algorithm runs in cubic time and quadratic space. We investigate two implementations which improve the running time to sub-cubic time. The first uses a cyclic tournament forest, a new data structure which extends kinetic tournament trees to allow for cycle splitting operations, and the second uses a priority queue to store events.","PeriodicalId":293085,"journal":{"name":"2013 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129929479","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 study P-critical edge-bipartite graphs (bigraphs for short) Δ with n ≥ 2 vertices, by means of the nonsymmetric Gram matrix ǦΔ ∈ Mn(Z), the Coxeter matrix CoxΔ := -ǦΔ · ǦΔ-tr ∈ Mn(Z), its Coxeter polynomial coxΔ(t) = det(t · E + ǦΔ · ǦΔ-tr), and its Coxeter spectrum sρeccΔ. We recall that Δ is positive if the symmetric matrix ǦΔ := ǦΔ + ǦΔtr is positive definite; and Δ is P-critical if it is not positive and each of its proper full subbigraphs is positive. It is easy to see that if two bigraphs Δ, Δ' are Z-bilinear equivalent Δ ≈Z Δ' (i.e., there exists a matrix B ∈ Gl(n, Z) such that ǦΔ = Btr · ǦΔ' · B) then their Coxeter spectra speccΔ and speccΔ' coincide; but the converse implication does not hold in general. One of the main questions of the Coxeter spectral analysis of connected P-critical bigraphs Δ, Δ' is whether the congruence Δ ≈Z Δ' holds if and only if speccΔ = speccΔ'. In this note we discuss the problem in case when n ≤ 10 and Δ and Δ' are P-critical looρ-free bigraphs such that their Euclidean types DΔ, DΔ' ∈ {An, n > 1, D̃m, m ≥ 4, Ẽ6,Ẽ7, Ẽ8} coincide. In particular, we get an affirmative answer to the stated question, for a large class of P-critical bigraphs and Tits P-critical finite posets. By applying symbolic and numerical algorithms in Maple and C# we compute the set of Coxeter polynomials coxΔ(t) for P-critical loop-free bigraphs Δ, with at most 10 vertices.
我们研究P-critical edge-bipartite图(简称bigraphs)Δn≥2顶点,通过非对称格拉姆矩阵ǦΔ∈Mn (Z),考克斯Coxeter矩阵Δ:= -ǦΔ·ǦΔtr∈Mn (Z),其Coxeter多项式考克斯Δ(t) =检波器(t·E +ǦΔ·ǦΔtr),及其Coxeter频谱sρeccΔ。我们记得，如果对称矩阵ǦΔ:= ǦΔ + ǦΔtr是正定的，Δ是正的;而Δ是p临界的，如果它不是正的，并且它的每一个适当的满子图都是正的。很容易看出，如果两个图Δ， Δ'是Z-双线性等价的Δ≈Z Δ'(即存在一个矩阵B∈Gl(n, Z)使得ǦΔ = Btr·ǦΔ'·B)，则它们的Coxeter谱speccΔ和speccΔ'重合;但相反的含义并不普遍成立。连通p临界图Δ， Δ'的Coxeter谱分析的一个主要问题是当且仅当speccΔ = speccΔ'时，同余Δ≈Z Δ'是否成立。在本文中，我们讨论了当n≤10且Δ和Δ'是p临界无循环图且它们的欧几里得类型DΔ， DΔ'∈{An, n > 1, D / m, m≥4，Ẽ6，Ẽ7， Ẽ8}重合时的问题。特别地，对于一大类p临界图和Tits p临界有限偏集，我们得到了上述问题的肯定答案。通过在Maple和c#中应用符号和数值算法，我们计算了p临界无循环图形Δ的Coxeter多项式集coxΔ(t)，最多有10个顶点。
{"title":"Algorithmic Experiences in Coxeter Spectral Study of P-critical Edge-Bipartite Graphs and Posets","authors":"A. Polak, D. Simson","doi":"10.1109/SYNASC.2013.56","DOIUrl":"https://doi.org/10.1109/SYNASC.2013.56","url":null,"abstract":"We study P-critical edge-bipartite graphs (bigraphs for short) Δ with n ≥ 2 vertices, by means of the nonsymmetric Gram matrix Ǧ<sub>Δ</sub> ∈ M<sub>n</sub>(Z), the Coxeter matrix Cox<sub>Δ</sub> := -Ǧ<sub>Δ</sub> · Ǧ<sub>Δ</sub><sup>-tr</sup> ∈ M<sub>n</sub>(Z), its Coxeter polynomial cox<sub>Δ</sub>(t) = det(t · E + Ǧ<sub>Δ</sub> · Ǧ<sub>Δ</sub><sup>-tr</sup>), and its Coxeter spectrum sρecc<sub>Δ</sub>. We recall that Δ is positive if the symmetric matrix Ǧ<sub>Δ</sub> := Ǧ<sub>Δ</sub> + Ǧ<sub>Δ</sub><sup>tr</sup> is positive definite; and Δ is P-critical if it is not positive and each of its proper full subbigraphs is positive. It is easy to see that if two bigraphs Δ, Δ' are Z-bilinear equivalent Δ ≈<sub>Z</sub> Δ' (i.e., there exists a matrix B ∈ Gl(n, Z) such that Ǧ<sub>Δ</sub> = B<sup>tr</sup> · Ǧ<sub>Δ'</sub> · B) then their Coxeter spectra specc<sub>Δ</sub> and specc<sub>Δ'</sub> coincide; but the converse implication does not hold in general. One of the main questions of the Coxeter spectral analysis of connected P-critical bigraphs Δ, Δ' is whether the congruence Δ ≈<sub>Z</sub> Δ' holds if and only if specc<sub>Δ</sub> = specc<sub>Δ'</sub>. In this note we discuss the problem in case when n ≤ 10 and Δ and Δ' are P-critical looρ-free bigraphs such that their Euclidean types DΔ, DΔ' ∈ {A<sub>n</sub>, n > 1, D̃<sub>m</sub>, m ≥ 4, Ẽ<sub>6</sub>,Ẽ<sub>7</sub>, Ẽ<sub>8</sub>} coincide. In particular, we get an affirmative answer to the stated question, for a large class of P-critical bigraphs and Tits P-critical finite posets. By applying symbolic and numerical algorithms in Maple and C# we compute the set of Coxeter polynomials cox<sub>Δ</sub>(t) for P-critical loop-free bigraphs Δ, with at most 10 vertices.","PeriodicalId":293085,"journal":{"name":"2013 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing","volume":"106 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128079569","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 presents a study of techniques used to speedup a scientific simulation code. The techniques include sequential optimizations as well as the parallelization with OpenMP. This work is carried out on two different multicore shared memory architectures, namely a cutting edge 8×8 core CPU and a more common 2×6 core board. Our target application is representative of many memory bound codes, and the techniques we present show how to overcome the burden of the memory bandwidth limit, which is quickly reached on multi-core or many-core with shared memory architectures. To achieve efficient speedups, strategies are applied to lower the computation costs, and to maximize the use of processors caches. Optimizations are: minimizing memory accesses, simplifying and reordering computations, and tiling loops. On 12 cores processor Intel X5675, aggregation of these optimizations results in an execution time 21.6 faster, compared to the original version on one core.
{"title":"Optimization and Parallelization of Emedge3D on Shared Memory Architecture","authors":"M. Kuhn, G. Latu, S. Genaud, N. Crouseilles","doi":"10.1109/SYNASC.2013.72","DOIUrl":"https://doi.org/10.1109/SYNASC.2013.72","url":null,"abstract":"This paper presents a study of techniques used to speedup a scientific simulation code. The techniques include sequential optimizations as well as the parallelization with OpenMP. This work is carried out on two different multicore shared memory architectures, namely a cutting edge 8×8 core CPU and a more common 2×6 core board. Our target application is representative of many memory bound codes, and the techniques we present show how to overcome the burden of the memory bandwidth limit, which is quickly reached on multi-core or many-core with shared memory architectures. To achieve efficient speedups, strategies are applied to lower the computation costs, and to maximize the use of processors caches. Optimizations are: minimizing memory accesses, simplifying and reordering computations, and tiling loops. On 12 cores processor Intel X5675, aggregation of these optimizations results in an execution time 21.6 faster, compared to the original version on one core.","PeriodicalId":293085,"journal":{"name":"2013 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130455281","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 Interpreter design pattern provides an elegant and natural way of implementing systems based on term-rewriting in a OO fashion. The idea is simply associating each term of a language, either terminal or non-terminal, with a corresponding class provided with a suitable simplify() method.Reducing a term to a normal form is thus performed through a series of recursive calls to such a method.The main weakness of this approach is that it does not take into account similarities existing among different domains, thus enforcing programmers to pollute generic and domain-specific rules. The resulting code if often wordy, hard to maintain, non-reusable. In this paper we adapt the Interpreter pattern so that a clean separation between generic (common to different domains) and domain-specific rules is possible. The new pattern significantly helps design even complex rewriting systems. A running example which refers to a generic Logical domain is used throughout the paper. An application to High Level Petri nets analysis is sketched. Without any loss of generality we refer to Java as representative of a large class of languages.
{"title":"An Extension of the Interpreter Pattern to Define Domain-Parametric Rewriting Systems","authors":"L. Capra, Vincenzo Stile","doi":"10.1109/SYNASC.2013.32","DOIUrl":"https://doi.org/10.1109/SYNASC.2013.32","url":null,"abstract":"The Interpreter design pattern provides an elegant and natural way of implementing systems based on term-rewriting in a OO fashion. The idea is simply associating each term of a language, either terminal or non-terminal, with a corresponding class provided with a suitable simplify() method.Reducing a term to a normal form is thus performed through a series of recursive calls to such a method.The main weakness of this approach is that it does not take into account similarities existing among different domains, thus enforcing programmers to pollute generic and domain-specific rules. The resulting code if often wordy, hard to maintain, non-reusable. In this paper we adapt the Interpreter pattern so that a clean separation between generic (common to different domains) and domain-specific rules is possible. The new pattern significantly helps design even complex rewriting systems. A running example which refers to a generic Logical domain is used throughout the paper. An application to High Level Petri nets analysis is sketched. Without any loss of generality we refer to Java as representative of a large class of languages.","PeriodicalId":293085,"journal":{"name":"2013 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133918859","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}
For scalable distributed database systems, weak consistency models are essential. Distributed databases, such as Google Spanner, scale to millions of nodes that replicate data across datacentres possibly located on different continents. At this scale, it is infeasible to maintain serialisability, which assumes that a global total order over committed transactions can be established. Instead, weaker consistency models, such as eventual consistency, causal consistency, sequential consistency and external consistency, are assumed. The problem is that operational models, such as labelled transition systems, tend to assume an interleaving semantics, which serialises transactions. To address this limitation, we provide an operational model that allows a weaker notion of consistency for a geographically distributed database inspired by Spanner. We reduce the timing guarantees provided by Spanner's TrueTime protocol to causal dependencies that are specified in a formal calculus.
{"title":"Non-interleaving Operational Semantics for Geographically Replicated Databases","authors":"Gabriel Ciobanu, Ross Horne","doi":"10.1109/SYNASC.2013.64","DOIUrl":"https://doi.org/10.1109/SYNASC.2013.64","url":null,"abstract":"For scalable distributed database systems, weak consistency models are essential. Distributed databases, such as Google Spanner, scale to millions of nodes that replicate data across datacentres possibly located on different continents. At this scale, it is infeasible to maintain serialisability, which assumes that a global total order over committed transactions can be established. Instead, weaker consistency models, such as eventual consistency, causal consistency, sequential consistency and external consistency, are assumed. The problem is that operational models, such as labelled transition systems, tend to assume an interleaving semantics, which serialises transactions. To address this limitation, we provide an operational model that allows a weaker notion of consistency for a geographically distributed database inspired by Spanner. We reduce the timing guarantees provided by Spanner's TrueTime protocol to causal dependencies that are specified in a formal calculus.","PeriodicalId":293085,"journal":{"name":"2013 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing","volume":"104 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115676573","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 investigates the cancellation errors which may occur in algebraic preprocessing with floating-point numbers, of numerical model simulation. We consider two operations, the substitution of a polynomial for terms of other polynomials and solving system of parametric linear equations. For the first operation, we clarify that the "gsystematic term-cancellation" may cause large errors and propose a simple method to avoid large errors. The idea is to introduce system parameters of value 1 to avoid mixing of terms which may cancel one another. For the second operation, we propose a combined method of numerical Gauss-Jordan elimination with pivoting and the symbolic minor expansion of the parametric determinants. There may occur the case that the pivoting is unsatisfactory or restricted severely, hence we propose two error-avoiding methods, they can be used together with pivoting. We convince ourselves of the effectiveness of proposed methods by experiments.
{"title":"On Algebraic Preprocessing of Floating-Point DAEs for Numerical Model Simulation","authors":"Tateaki Sasaki, Tetsu Yamaguchi","doi":"10.1109/SYNASC.2013.18","DOIUrl":"https://doi.org/10.1109/SYNASC.2013.18","url":null,"abstract":"This paper investigates the cancellation errors which may occur in algebraic preprocessing with floating-point numbers, of numerical model simulation. We consider two operations, the substitution of a polynomial for terms of other polynomials and solving system of parametric linear equations. For the first operation, we clarify that the \"gsystematic term-cancellation\" may cause large errors and propose a simple method to avoid large errors. The idea is to introduce system parameters of value 1 to avoid mixing of terms which may cancel one another. For the second operation, we propose a combined method of numerical Gauss-Jordan elimination with pivoting and the symbolic minor expansion of the parametric determinants. There may occur the case that the pivoting is unsatisfactory or restricted severely, hence we propose two error-avoiding methods, they can be used together with pivoting. We convince ourselves of the effectiveness of proposed methods by experiments.","PeriodicalId":293085,"journal":{"name":"2013 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing","volume":"76 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128821599","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：