The Experts below are selected from a list of 7302 Experts worldwide ranked by ideXlab platform
Lijun Zhang - One of the best experts on this subject based on the ideXlab platform.
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counterexample guided polynomial loop invariant generation by Lagrange Interpolation
Computer Aided Verification, 2015Co-Authors: Yufang Chen, Chihduo Hong, Bowyaw Wang, Lijun ZhangAbstract:We apply multivariate Lagrange Interpolation to synthesizing polynomial quantitative loop invariants for probabilistic programs. We reduce the computation of a quantitative loop invariant to solving constraints over program variables and unknown coefficients. Lagrange Interpolation allows us to find constraints with less unknown coefficients. Counterexample-guided refinement furthermore generates linear constraints that pinpoint the desired quantitative invariants. We evaluate our technique by several case studies with polynomial quantitative loop invariants in the experiments.
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counterexample guided polynomial loop invariant generation by Lagrange Interpolation
arXiv: Software Engineering, 2015Co-Authors: Yufang Chen, Chihduo Hong, Bowyaw Wang, Lijun ZhangAbstract:We apply multivariate Lagrange Interpolation to synthesize polynomial quantitative loop invariants for probabilistic programs. We reduce the computation of an quantitative loop invariant to solving constraints over program variables and unknown coefficients. Lagrange Interpolation allows us to find constraints with less unknown coefficients. Counterexample-guided refinement furthermore generates linear constraints that pinpoint the desired quantitative invariants. We evaluate our technique by several case studies with polynomial quantitative loop invariants in the experiments.
Chinchen Chang - One of the best experts on this subject based on the ideXlab platform.
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grouped secret sharing schemes based on Lagrange Interpolation polynomials and chinese remainder theorem
Security and Communication Networks, 2021Co-Authors: Fuyou Miao, Keju Meng, Yan Xiong, Chinchen ChangAbstract:In a threshold secret sharing (SS) scheme, whether or not a shareholder set is an authorized set totally depends on the number of shareholders in the set. When the access structure is not threshold, (t,n) threshold SS is not suitable. This paper proposes a new kind of SS named grouped secret sharing (GSS), which is specific multipartite SS. Moreover, in order to implement GSS, we utilize both Lagrange Interpolation polynomials and Chinese remainder theorem to design two GSS schemes, respectively. Detailed analysis shows that both GSS schemes are correct and perfect, which means any authorized set can recover the secret while an unauthorized set cannot get any information about the secret.
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lossless and unlimited multi image sharing based on chinese remainder theorem and Lagrange Interpolation
Signal Processing, 2014Co-Authors: Chinchen Chang, Ngoctu HuynhAbstract:This study proposes a novel multi-image threshold sharing scheme based on Chinese remainder theorem and Lagrange Interpolation. The exceptional property of the scheme is its ability to retrieve any secret image without recovering all the other images. Therefore, it works efficiently and reduces computation cost in case it needs to recover only one image from shares. In term of capacity, the scheme has no limitation on number of input secret images, output shares and the recovery threshold. Another advantage of the scheme is that it can be used for many image formats whether it is binary or grayscale or color. Moreover, the scheme can recover the secret images without any distortion.
Takuya Tsuchiya - One of the best experts on this subject based on the ideXlab platform.
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error analysis of Lagrange Interpolation on tetrahedrons
Journal of Approximation Theory, 2020Co-Authors: Kenta Kobayashi, Takuya TsuchiyaAbstract:Abstract This paper describes the analysis of Lagrange Interpolation errors on tetrahedrons. In many textbooks, the error analysis of Lagrange Interpolation is conducted under geometric assumptions such as shape regularity or the (generalized) maximum angle condition. In this paper, we present a new estimation in which the error is bounded in terms of the diameter and projected circumradius of the tetrahedron. Because we do not impose any geometric restrictions on the tetrahedron itself, our error estimation may be applied to any tetrahedralizations of domains including very thin tetrahedrons.
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extending babuska aziz s theorem to higher order Lagrange Interpolation
Applications of Mathematics, 2016Co-Authors: Kenta Kobayashi, Takuya TsuchiyaAbstract:We consider the error analysis of Lagrange Interpolation on triangles and tetrahedrons. For Lagrange Interpolation of order one, Babuska and Aziz showed that squeezing a right isosceles triangle perpendicularly does not deteriorate the optimal approximation order. We extend their technique and result to higher-order Lagrange Interpolation on both triangles and tetrahedrons. To this end, we make use of difference quotients of functions with two or three variables. Then, the error estimates on squeezed triangles and tetrahedrons are proved by a method that is a straightforward extension of the original one given by Babuska-Aziz.
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a priori error estimates for Lagrange Interpolation on triangles
Applications of Mathematics, 2015Co-Authors: Kenta Kobayashi, Takuya TsuchiyaAbstract:We present the error analysis of Lagrange Interpolation on triangles. A new a priori error estimate is derived in which the bound is expressed in terms of the diameter and circumradius of a triangle. No geometric conditions on triangles are imposed in order to get this type of error estimates. To derive the new error estimate, we make use of the two key observations. The first is that squeezing a right isosceles triangle perpendicularly does not reduce the approximation property of Lagrange Interpolation. An arbitrary triangle is obtained from a squeezed right triangle by a linear transformation. The second key observation is that the ratio of the singular values of the linear transformation is bounded by the circumradius of the target triangle.
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extenting of babu v s ka aziz s theorem to higher order Lagrange Interpolation
arXiv: Numerical Analysis, 2015Co-Authors: Kenta Kobayashi, Takuya TsuchiyaAbstract:We consider the error analysis of Lagrange Interpolation on triangles and tetrahedrons. For Lagrange Interpolation of order one, Babuska and Aziz showed that squeezing a right isosceles triangle perpendicularly does not deteriorate the optimal approximation order. We extend their technique and result to higher-order Lagrange Interpolation on both triangles and tetrahedrons. To this end, we make use of difference quotients of functions with two or three variables. Then, the error estimates on squeezed triangles and tetrahedrons are proved by a method that is a straightforward extension of the original given by Babuska-Aziz.
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a priori error estimates for Lagrange Interpolation on triangles
arXiv: Numerical Analysis, 2014Co-Authors: Kenta Kobayashi, Takuya TsuchiyaAbstract:We present the error analysis of Lagrange Interpolation on triangles. A new \textit{a priori} error estimate is derived in which the bound is expressed in terms of the diameter and circumradius of a triangle. No geometric conditions on triangles are imposed in order to get this type of error estimates.
Yufang Chen - One of the best experts on this subject based on the ideXlab platform.
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counterexample guided polynomial loop invariant generation by Lagrange Interpolation
Computer Aided Verification, 2015Co-Authors: Yufang Chen, Chihduo Hong, Bowyaw Wang, Lijun ZhangAbstract:We apply multivariate Lagrange Interpolation to synthesizing polynomial quantitative loop invariants for probabilistic programs. We reduce the computation of a quantitative loop invariant to solving constraints over program variables and unknown coefficients. Lagrange Interpolation allows us to find constraints with less unknown coefficients. Counterexample-guided refinement furthermore generates linear constraints that pinpoint the desired quantitative invariants. We evaluate our technique by several case studies with polynomial quantitative loop invariants in the experiments.
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counterexample guided polynomial loop invariant generation by Lagrange Interpolation
arXiv: Software Engineering, 2015Co-Authors: Yufang Chen, Chihduo Hong, Bowyaw Wang, Lijun ZhangAbstract:We apply multivariate Lagrange Interpolation to synthesize polynomial quantitative loop invariants for probabilistic programs. We reduce the computation of an quantitative loop invariant to solving constraints over program variables and unknown coefficients. Lagrange Interpolation allows us to find constraints with less unknown coefficients. Counterexample-guided refinement furthermore generates linear constraints that pinpoint the desired quantitative invariants. We evaluate our technique by several case studies with polynomial quantitative loop invariants in the experiments.
Xuezhang Liang - One of the best experts on this subject based on the ideXlab platform.
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some researches on trivariate Lagrange Interpolation
Journal of Computational and Applied Mathematics, 2006Co-Authors: Xuezhang Liang, Jielin Zhang, Renhong Wang, Lihong Cui, Ming ZhangAbstract:In this paper, in order to go a step further research on the problem of trivariate Lagrange Interpolation, we pose the concepts of sufficient intersection of algebraic surfaces and Lagrange Interpolation along a space algebraic curve, and extend Cayley-Bacharach theorem in algebraic geometry from R2 to R3. By using the conclusion of the extended theorem, we deduce a general method of constructing properly posed set of nodes for Lagrange Interpolation along a space algebraic curve, and give a series of corollaries for the practical applications. Moreover, we give a new method of constructing properly posed set of nodes for Lagrange Interpolation along an algebraic surface, and as a result we make clear the geometrical structure of it.
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multivariate Lagrange Interpolation and an application of cayley bacharach theorem for it
arXiv: Numerical Analysis, 2006Co-Authors: Xuezhang Liang, Jielin Zhang, Ming Zhang, Lihong CuiAbstract:In this paper,we deeply research Lagrange Interpolation of n-variables and give an application of Cayley-Bacharach theorem for it. We pose the concept of sufficient intersection about s algebraic hypersurfaces in n-dimensional complex Euclidean space and discuss the Lagrange Interpolation along the algebraic manifold of sufficient intersection. By means of some theorems (such as Bezout theorem, Macaulay theorem and so on) we prove the dimension for the polynomial space P(n) m along the algebraic manifold S of sufficient intersection and give a convenient expression for dimension calculation by using the backward difference operator. According to Mysovskikh theorem, we give a proof of the existence and a characterizing condition of properly posed set of nodes of arbitrary degree for Interpolation along an algebraic manifold of sufficient intersection. Further we point out that for s algebraic hypersurfaces of sufficient intersection, the set of polynomials must constitute the H-base of ideal. As a main result of this paper, we deduce a general method of constructing properly posed set of nodes for Lagrange Interpolation along an algebraic manifold, namely the superposition Interpolation process. At the end of the paper, we use the extended Cayley-Bacharach theorem to resolve some problems of Lagrange Interpolation along the 0-dimensional and 1-dimensional algebraic manifold. Just the application of Cayley-Bacharach theorem constitutes the start point of constructing properly posed set of nodes along the high dimensional algebraic manifold by using the superposition Interpolation process.
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the application of cayley bacharach theorem to bivariate Lagrange Interpolation
Journal of Computational and Applied Mathematics, 2004Co-Authors: Xuezhang Liang, Lihong Cui, Jielin ZhangAbstract:In this paper, we give a new proof of the famous Cayley-Bacharach theorem by means of Interpolation, and deduce a general method of constructing properly posed set of nodes for bivariate Lagrange Interpolation. As a result, we generalize the main results in Liang (On the Interpolations and approximations in several variables, Jilin University, 1965), Liang and Lu (Approximation Theory IX, Vanderbilt University Press, 1988) and Liang et al. (Analysis, Combinatorics and Computing, Nova Science Publishers, Inc., New York, 2002) to the more extensive situations.
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properly posed set of nodes for bivariate Lagrange Interpolation along an algebraic curve
Analysis combinatorics and computing, 2002Co-Authors: Xuezhang Liang, Jielin ZhangAbstract:In this paper, in order to make a further research on the problem of properly posed set of nodes for bivariate Lagrange Interpolation along an algebraic curve which was posed in [1], [2] and [3], we introduce the concept of weak Grobner basis and give a new method of constructing properly posed set of nodes for bivariate Lagrange Interpolation along an algebraic curve, which generalizes the main result in [2].
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properly posed sets of nodes for multivariate Lagrange Interpolation in c s
SIAM Journal on Numerical Analysis, 2001Co-Authors: Xuezhang Liang, Renzhong FengAbstract:In this paper, we apply techniques from the theory of ideals and varieties in algebraic geometry to study the geometric structure of a properly posed set of nodes (or PPSN, for short) for multivariate Lagrange Interpolation along an algebraic hypersurface. We provide a hyperplane-superposition process to construct the PPSN for Interpolation along an algebraic hypersurface, and as a result, we offer a clear understanding of the geometric structure of the PPSN for multivariate Lagrange Interpolation in Cs.
