We consider the problem of finding closed form solutions of a linear homogeneous ordinary differential equation having coefficients which are elliptic functions. In particular, the input coefficients are assumed to be represented as elements of C(p,p'), where C is the complex number field, and p(x) and p' (x) are the Weierstrass p function and its first derivative, respectively. The specific closed form solutions y(x) which we seek are hyperexponential over C(p,p'), i.e., solutions y(x) such that y' (x)/y(x) is in C(p,p'). Such solutions correspond to first order right-hand factors of the associated linear differential operator, and are analogous to hyperexponential solutions over C(x), in the more well-known case where the coefficients of the ode are in C(x). A previous paper [4] gave an algorithm for equations of second order. The algorithm presented here works for equations of arbitrary order, and will find all such hyperexponential solutions that may exist. It relies on determining the structure of such first order factors to construct an ansatz of a solution, which can then be completely determined by solving a system of multivariate polynomial equations. The algorithm works well for solutions having few singularities and hidden poles, but can slow as the number of such points increases.
{"title":"Solving higher order linear differential equations having elliptic function coefficients","authors":"Reinhold Burger","doi":"10.1145/2608628.2608675","DOIUrl":"https://doi.org/10.1145/2608628.2608675","url":null,"abstract":"We consider the problem of finding closed form solutions of a linear homogeneous ordinary differential equation having coefficients which are elliptic functions. In particular, the input coefficients are assumed to be represented as elements of C(p,p'), where C is the complex number field, and p(x) and p' (x) are the Weierstrass p function and its first derivative, respectively. The specific closed form solutions y(x) which we seek are hyperexponential over C(p,p'), i.e., solutions y(x) such that y' (x)/y(x) is in C(p,p'). Such solutions correspond to first order right-hand factors of the associated linear differential operator, and are analogous to hyperexponential solutions over C(x), in the more well-known case where the coefficients of the ode are in C(x). A previous paper [4] gave an algorithm for equations of second order. The algorithm presented here works for equations of arbitrary order, and will find all such hyperexponential solutions that may exist. It relies on determining the structure of such first order factors to construct an ansatz of a solution, which can then be completely determined by solving a system of multivariate polynomial equations. The algorithm works well for solutions having few singularities and hidden poles, but can slow as the number of such points increases.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128844737","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 this paper, we propose an SL(n)-invariant division of SL(n)-invariant polynomials by establishing an admissible order among the invariant polynomials in normal form. The invariant division leads to an invariant Gröbner basis theory. The invariant division is closely related to multivariate coordinate polynomial division. This feature leads to a proof of the result that if f1,..., fk are SL(n,K)-invariant where K is an arbitrary field, possibly of positive characteristic, then the invariant ideal generated by them is the intersection of the ideal generated by the fi in the polynomial ring of coordinates with the algebra of invariants.
{"title":"Reduction among bracket polynomials","authors":"Hongbo Li, Changpeng Shao, Lei Huang, Yue Liu","doi":"10.1145/2608628.2608630","DOIUrl":"https://doi.org/10.1145/2608628.2608630","url":null,"abstract":"In this paper, we propose an <i>SL</i>(<i>n</i>)-invariant division of <i>SL</i>(<i>n</i>)-invariant polynomials by establishing an admissible order among the invariant polynomials in normal form. The invariant division leads to an invariant Gröbner basis theory.\u0000 The invariant division is closely related to multivariate coordinate polynomial division. This feature leads to a proof of the result that if <i>f</i><sub>1</sub>,..., <i>f</i><sub><i>k</i></sub> are <i>SL</i>(<i>n</i>,K)-invariant where K is an arbitrary field, possibly of positive characteristic, then the invariant ideal generated by them is the intersection of the ideal generated by the <i>f</i><sub><i>i</i></sub> in the polynomial ring of coordinates with the algebra of invariants.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114429320","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}
Order bases are a fundamental tool for linear algebra with polynomial coefficients. In particular, block Wiedemann methods are nowadays able to tackle large sparse matrix problems because they benefit from fast order basis algorithms. However, such fast algorithms suffer from two practical drawbacks: they are not designed for early termination and often require more knowledge on the input than necessary. In this paper, we propose an online algorithm for order basis which allows for both early termination and minimal input requirement while keeping quasi-optimal complexity in the order. Using this algorithm inside block Wiedemann methods leads to an improvement of their practical performance by a constant factor.
{"title":"Online order basis algorithm and its impact on the block Wiedemann algorithm","authors":"Pascal Giorgi, R. Lebreton","doi":"10.1145/2608628.2608647","DOIUrl":"https://doi.org/10.1145/2608628.2608647","url":null,"abstract":"Order bases are a fundamental tool for linear algebra with polynomial coefficients. In particular, block Wiedemann methods are nowadays able to tackle large sparse matrix problems because they benefit from fast order basis algorithms. However, such fast algorithms suffer from two practical drawbacks: they are not designed for early termination and often require more knowledge on the input than necessary. In this paper, we propose an online algorithm for order basis which allows for both early termination and minimal input requirement while keeping quasi-optimal complexity in the order. Using this algorithm inside block Wiedemann methods leads to an improvement of their practical performance by a constant factor.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132743765","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 prove that every affine rational surface, parametrized by means of an affine rational parametrization without projective base points, can be covered by at most three parametrizations. Moreover, we give explicit formulas for computing the coverings. We provide two different approaches: either covering the surface with a surface parametrization plus a curve parametrization plus a point, or with the original parametrization plus two surface reparametrizations of it.
{"title":"Covering of surfaces parametrized without projective base points","authors":"J. Sendra, David Sevilla, Carlos Villarino","doi":"10.1145/2608628.2608635","DOIUrl":"https://doi.org/10.1145/2608628.2608635","url":null,"abstract":"We prove that every affine rational surface, parametrized by means of an affine rational parametrization without projective base points, can be covered by at most three parametrizations. Moreover, we give explicit formulas for computing the coverings. We provide two different approaches: either covering the surface with a surface parametrization plus a curve parametrization plus a point, or with the original parametrization plus two surface reparametrizations of it.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116116312","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}
Randomized algorithms are given for linear algebra problems on an input matrix A ∈ Knxm over a field K. We give an algorithm that simultaneously computes the row and column rank profiles of A in 2r3 + (r2 + n + m + |A|)1+o(1) field operations from K, where r is the rank of A and |A| denotes the number of nonzero entries of A. Here, the +o(1) in cost estimates captures some missing log n and log m factors. The rank profiles algorithm is randomized of the Monte Carlo type: the correct answer will be returned with probability at least 1/2. Given a b ∈ Knx1, we give an algorithm that either computes a particular solution vector x ∈ Kmx1 to the system Ax = b, or produces an inconsistency certificate vector u ∈ K1xn such that uA = 0 and ub ≠ 0. The linear solver examines at most r + 1 rows and r columns of A and has running time 2r3 + (r2 + n + m + |R| + |C|)1+o(1) field operations from K, where |R| and |C| are the number of nonzero entries in the rows and columns, respectively, that are examined. The solver is randomized of the Las Vegas type: an incorrect result is never returned but the algorithm may report FAIL with probability at most 1/2. These cost estimates are achieved by making use of a novel randomized online data structure for the detection of linearly independent rows and columns.
随机算法给出了线性代数问题的输入矩阵∈Knxm字段K .我们给出一个算法,同时计算的行和列等级资料在2 r3 + (r2 + n + m + | |) 1 + o(1)操作从K, r是A的秩和| |表示数量的非零项A, + o(1)成本估计捕捉一些失踪的o (log n)和日志m因素。排名概要算法是蒙特卡罗类型的随机化:正确答案将以至少1/2的概率返回。给定b∈Knx1，我们给出一种算法，该算法可以计算系统Ax = b的特解向量x∈Kmx1，或者产生一个不一致证书向量u∈K1xn，使得uA = 0且ub≠0。线性解算器最多检查A的r + 1行和r列，运行时间为2r3 + (r2 + n + m + | r | + |C|)1+o(1)次字段操作，其中| r |和|C|分别是检查的行和列中的非零条目的数量。求解器是拉斯维加斯类型的随机化:永远不会返回不正确的结果，但算法可能以最多1/2的概率报告FAIL。这些成本估计是通过使用一种新的随机在线数据结构来检测线性无关的行和列来实现的。
{"title":"Linear independence oracles and applications to rectangular and low rank linear systems","authors":"A. Storjohann, Shiyun Yang","doi":"10.1145/2608628.2608673","DOIUrl":"https://doi.org/10.1145/2608628.2608673","url":null,"abstract":"Randomized algorithms are given for linear algebra problems on an input matrix <i>A</i> ∈ K<sup><i>n</i>x<i>m</i></sup> over a field K. We give an algorithm that simultaneously computes the row and column rank profiles of <i>A</i> in 2<i>r</i><sup>3</sup> + (<i>r</i><sup>2</sup> + <i>n</i> + <i>m</i> + |<i>A</i>|)<sup>1+<i>o</i>(1)</sup> field operations from K, where <i>r</i> is the rank of <i>A</i> and |<i>A</i>| denotes the number of nonzero entries of <i>A</i>. Here, the +<i>o</i>(1) in cost estimates captures some missing log <i>n</i> and log <i>m</i> factors. The rank profiles algorithm is randomized of the Monte Carlo type: the correct answer will be returned with probability at least 1/2. Given a <i>b</i> ∈ K<sup><i>n</i>x1</sup>, we give an algorithm that either computes a particular solution vector <i>x</i> ∈ K<sup><i>m</i>x1</sup> to the system <i>Ax</i> = <i>b</i>, or produces an inconsistency certificate vector <i>u</i> ∈ K<sup>1x<i>n</i></sup> such that <i>uA</i> = 0 and <i>ub</i> ≠ 0. The linear solver examines at most <i>r</i> + 1 rows and <i>r</i> columns of <i>A</i> and has running time 2<i>r</i><sup>3</sup> + (<i>r</i><sup>2</sup> + <i>n</i> + <i>m</i> + |<i>R</i>| + |<i>C</i>|)<sup>1+<i>o</i>(1)</sup> field operations from K, where |<i>R</i>| and |<i>C</i>| are the number of nonzero entries in the rows and columns, respectively, that are examined. The solver is randomized of the Las Vegas type: an incorrect result is never returned but the algorithm may report FAIL with probability at most 1/2. These cost estimates are achieved by making use of a novel randomized online data structure for the detection of linearly independent rows and columns.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125560719","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}
Let B be a basis of a Euclidean lattice, and B an approximation thereof. We give a sufficient condition on the closeness between B and B so that an LLL-reducing transformation U for B remains valid for B. Further, we analyse an efficient reduction algorithm when B is itself a small deformation of an LLL-reduced basis. Applications include speeding-up reduction by keeping only the most significant bits of B, reducing a basis that is only approximately known, and efficiently batching LLL reductions for closely related inputs.
{"title":"LLL reducing with the most significant bits","authors":"Goel Sarushi, I. Morel, D. Stehlé, G. Villard","doi":"10.1145/2608628.2608645","DOIUrl":"https://doi.org/10.1145/2608628.2608645","url":null,"abstract":"Let B be a basis of a Euclidean lattice, and B an approximation thereof. We give a sufficient condition on the closeness between B and B so that an LLL-reducing transformation U for B remains valid for B. Further, we analyse an efficient reduction algorithm when B is itself a small deformation of an LLL-reduced basis. Applications include speeding-up reduction by keeping only the most significant bits of B, reducing a basis that is only approximately known, and efficiently batching LLL reductions for closely related inputs.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133534715","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 algorithms to construct and do arithmetic operations in the algebraic closure of the finite field Fp. Our approach is inspired by algorithms for constructing irreducible polynomials, which first reduce to prime power degrees, then use composita techniques. We use similar ideas to give efficient algorithms for embeddings and isomorphisms.
{"title":"Fast arithmetic for the algebraic closure of finite fields","authors":"L. Feo, Javad Doliskani, É. Schost","doi":"10.1145/2608628.2608672","DOIUrl":"https://doi.org/10.1145/2608628.2608672","url":null,"abstract":"We present algorithms to construct and do arithmetic operations in the algebraic closure of the finite field Fp. Our approach is inspired by algorithms for constructing irreducible polynomials, which first reduce to prime power degrees, then use composita techniques. We use similar ideas to give efficient algorithms for embeddings and isomorphisms.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"79 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128254885","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 discrete dynamical systems, or recurrence relations, of the general form [EQUATION] with explicitly known series coefficients αk and α1 ≠ 0. We associate with the discrete system an interpolating continuous system Y (t), such that Y (n) = yn. The asymptotic behaviour of yn can then be investigated through Y (t). The corresponding continuous system is [EQUATION] where G is called the generator (following Labelle's terminology), and is given by an explicit formula in terms of the recurrence relation. This continuous system may fail to be smooth everywhere but nonetheless may be useful. Analytic solution is only rarely possible. We analyze the equation for Y under assumptions of an asymptotic limit, and show that the asymptotic behaviour can be obtained by reverting a series containing logarithms and powers. We introduce a novel reversion based on the Wright ω function. An application of the theory is made to functional iteration of the Lambert W function and the asymptotic behaviour of the iteration is obtained. The iteration of functions is a central topic in the theory of complex dynamical system, and a sophisticated use of conjugation is only one key tool used there. We show here that Labelle's theory and generator can be used to compute the conjugated mapping of functional iterations to simple non-iterative functions in general. We use the Lambert W function again as an example to illustrate this. We also discuss the curious asymptotic series ln z ~ Σk ≥ 1W(z). This study uses the truncated generalized series tools available in Maple, particularly the logarithmic-and-power series that is usual in Maple. We also use Levin's u-transform as a key piece in interpolating the discrete dynamical system.
{"title":"The asymptotic analysis of some interpolated nonlinear recurrence relations","authors":"Robert M Corless, D. J. Jeffrey, Fei Wang","doi":"10.1145/2608628.2608677","DOIUrl":"https://doi.org/10.1145/2608628.2608677","url":null,"abstract":"We study discrete dynamical systems, or recurrence relations, of the general form\u0000 [EQUATION]\u0000 with explicitly known series coefficients <i>α</i><sub><i>k</i></sub> and <i>α</i><sub>1</sub> ≠ 0. We associate with the discrete system an interpolating continuous system <i>Y</i> (<i>t</i>), such that <i>Y</i> (<i>n</i>) = <i>y</i><sub><i>n</i></sub>. The asymptotic behaviour of <i>y</i><sub><i>n</i></sub> can then be investigated through <i>Y</i> (<i>t</i>). The corresponding continuous system is\u0000 [EQUATION]\u0000 where <i>G</i> is called the generator (following Labelle's terminology), and is given by an explicit formula in terms of the recurrence relation. This continuous system may fail to be smooth everywhere but nonetheless may be useful. Analytic solution is only rarely possible.\u0000 We analyze the equation for <i>Y</i> under assumptions of an asymptotic limit, and show that the asymptotic behaviour can be obtained by reverting a series containing logarithms and powers. We introduce a novel reversion based on the Wright ω function.\u0000 An application of the theory is made to functional iteration of the Lambert <i>W</i> function and the asymptotic behaviour of the iteration is obtained.\u0000 The iteration of functions is a central topic in the theory of complex dynamical system, and a sophisticated use of conjugation is only one key tool used there. We show here that Labelle's theory and generator can be used to compute the conjugated mapping of functional iterations to simple non-iterative functions in general. We use the Lambert <i>W</i> function again as an example to illustrate this. We also discuss the curious asymptotic series ln <i>z</i> ~ Σ<sub><i>k</i> ≥ 1</sub> <i>W</i><sup><i><k></i></sup>(<i>z</i>).\u0000 This study uses the truncated generalized series tools available in Maple, particularly the logarithmic-and-power series that is usual in Maple. We also use Levin's u-transform as a key piece in interpolating the discrete dynamical system.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129020670","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}
Historically, a lot of computer algebra systems were designed to have a command line interface, then GUI was added if required. AsirPad -- a computer algebra system with a handwriting interface on PDA developed by the author is one of them. Risa/Asir -- a CAS with a command line interface is the CAS engine of AsirPad, and AsirPad is created by covering it with GUI. This method is suitable to develop an application based on an existing CAS for mobile devices such as tablets and smartphones. In this tutorial, I would like to explain the details of this method.
{"title":"How to develop a mobile computer algebra system","authors":"M. Fujimoto","doi":"10.1145/2608628.2627492","DOIUrl":"https://doi.org/10.1145/2608628.2627492","url":null,"abstract":"Historically, a lot of computer algebra systems were designed to have a command line interface, then GUI was added if required. AsirPad -- a computer algebra system with a handwriting interface on PDA developed by the author is one of them. Risa/Asir -- a CAS with a command line interface is the CAS engine of AsirPad, and AsirPad is created by covering it with GUI. This method is suitable to develop an application based on an existing CAS for mobile devices such as tablets and smartphones. In this tutorial, I would like to explain the details of this method.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"76 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123129684","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 report on on-going efforts to apply real quantifier elimination to the synthesis of optimal numerical algorithms. In particular, we describe a case study on the square root problem: given a real number x and an error bound ε, find a real interval such that it contains [EQUATION] and its width is less than or equal to ε. A typical numerical algorithm starts with an initial interval and repeatedly updates it by applying a "refinement map" on it until it becomes narrow enough. Thus the synthesis amounts to finding a refinement map that ensures the correctness and optimality of the resulting algorithm. This problem can be formulated as a real quantifier elimination. Hence, in principle, the synthesis can be carried out automatically. However, the computational requirement is huge, making the automatic synthesis practically impossible with the current general real quantifier elimination software. We overcame the difficulty by (1) carefully reducing a complicated quantified formula into several simpler ones and (2) automatically eliminating the quantifiers from the resulting ones using the state of the art quantifier elimination software. As the result, we were able to synthesize semi-automatically, under mild assumptions, a class of optimal maps, which are significantly better than the well known hand-crafted Secant-Newton map. Interestingly, the optimal synthesized maps are not contracting as one would naturally expect.
{"title":"Synthesis of optimal numerical algorithms using real quantifier elimination (case study: square root computation)","authors":"Madalina Erascu, H. Hong","doi":"10.1145/2608628.2608654","DOIUrl":"https://doi.org/10.1145/2608628.2608654","url":null,"abstract":"We report on on-going efforts to apply real quantifier elimination to the synthesis of optimal numerical algorithms. In particular, we describe a case study on the square root problem: given a real number x and an error bound ε, find a real interval such that it contains [EQUATION] and its width is less than or equal to ε.\u0000 A typical numerical algorithm starts with an initial interval and repeatedly updates it by applying a \"refinement map\" on it until it becomes narrow enough. Thus the synthesis amounts to finding a refinement map that ensures the correctness and optimality of the resulting algorithm.\u0000 This problem can be formulated as a real quantifier elimination. Hence, in principle, the synthesis can be carried out automatically. However, the computational requirement is huge, making the automatic synthesis practically impossible with the current general real quantifier elimination software.\u0000 We overcame the difficulty by (1) carefully reducing a complicated quantified formula into several simpler ones and (2) automatically eliminating the quantifiers from the resulting ones using the state of the art quantifier elimination software.\u0000 As the result, we were able to synthesize semi-automatically, under mild assumptions, a class of optimal maps, which are significantly better than the well known hand-crafted Secant-Newton map. Interestingly, the optimal synthesized maps are not contracting as one would naturally expect.","PeriodicalId":243282,"journal":{"name":"International Symposium on Symbolic and Algebraic Computation","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116770206","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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