In this paper we prove some interesting extensions and generalizations of Enestrom- Kakeya Theorem concerning the location of the zeros of a polynomial in a complex plane. We also obtain some zero-free regions for a c...In this paper we prove some interesting extensions and generalizations of Enestrom- Kakeya Theorem concerning the location of the zeros of a polynomial in a complex plane. We also obtain some zero-free regions for a class of related analytic functions. Our results not only contain some known results as a special case but also a variety of interesting results can be deduced in a unified way by various choices of the parameters.展开更多
We raise and partly answer the question: whether there exists a Markov system with respect to which the zeros of the Chebyshev polynomials are dense, but the maximum length of a zero free interval of the nth Chebyshev...We raise and partly answer the question: whether there exists a Markov system with respect to which the zeros of the Chebyshev polynomials are dense, but the maximum length of a zero free interval of the nth Chebyshev polynomial does not tends to zero. We also draw the conclu- tion that a Markov system, under an additional assumption, is dense if and only if the maxi- mum length of a zero free interval of the nth associated Chebyshev polynomial tends to zero.展开更多
In this note, we study zeroes of Clifford algebra-valued polynomials. We prove that if such a polynomial has only real coefficients, then it has two types of zeroes： the real isolated zeroes and the spherical conjuga...In this note, we study zeroes of Clifford algebra-valued polynomials. We prove that if such a polynomial has only real coefficients, then it has two types of zeroes： the real isolated zeroes and the spherical conjugate ones. The total number of zeroes does not exceed the degree of the polynomial. We also present a technique for computing the zeroes.展开更多
Presents the first estimation conditions for Nourein iterations for simultaneous finding all zeros of a polynomial under which the iteration processes are guaranteed to converge. Computational formulas; Theorems and p...Presents the first estimation conditions for Nourein iterations for simultaneous finding all zeros of a polynomial under which the iteration processes are guaranteed to converge. Computational formulas; Theorems and proofs.展开更多
A PL homotopy algorithm is modified to yield a polynomial-time result on its computational complexity.We prove that the cost of locating all zeros of a polynomial of degree n to an accuracy of ε(measured by the numbe...A PL homotopy algorithm is modified to yield a polynomial-time result on its computational complexity.We prove that the cost of locating all zeros of a polynomial of degree n to an accuracy of ε(measured by the number of evaluations of the polynomial)grows no faster than O(max{n^4,n^3log_2(n/ε)}).This work is in response to a question raised in a paper by S.Smale as to the efficiency of piecewise linear methods in solving equations.In comparison with a few results reported,the algorithm under discussion is the only one providing correct multiplicities and the only one employing vector labelling.展开更多
In this paper, the problem of computing zeros of a general degree bivariate Bernstein polynomial is considered. An efficient and robust algorithm is presented that takes into full account particular properties of the ...In this paper, the problem of computing zeros of a general degree bivariate Bernstein polynomial is considered. An efficient and robust algorithm is presented that takes into full account particular properties of the function considered. The algorithm works for rectangular as well as triangular domains. The outlined procedure can also be applied for the computation of the intersection of a Bezier patch and a plane as well as in the determination of an algebraic curve restricted to a compact domain. In particular, singular points of the algebraic curve are reliably detected.展开更多
Presents a family of parallel iterations for finding all zeros of a polynomial without evaluation of derivatives. Construction of iterations; Convergence of the iterations; Details on the numerical examples.
Let F be a field,and let e,k be integers such that 1≤e≤|F\{0}|and k≥0.We show that for any subset{a1,……,ae}■F\{0},the curious identity∑(i1+……ie)∈Z^(e)≥0,i1+……+ie=k a_(1)^(i1)…a_(e)^(ie)=∑i=1 e a_(i)^(k+...Let F be a field,and let e,k be integers such that 1≤e≤|F\{0}|and k≥0.We show that for any subset{a1,……,ae}■F\{0},the curious identity∑(i1+……ie)∈Z^(e)≥0,i1+……+ie=k a_(1)^(i1)…a_(e)^(ie)=∑i=1 e a_(i)^(k+e-1)/∏i≠j=1 e(a_(i)-a_(j))holds with Z≥0 being the set of nonnegative integers.As an application,we prove that for any subset{a_(1)…,a_(e)}■F_(q)\{0}with F_(q)being the finite field of q elements and e,l being integers such that 2≤e≤q-1 and 0≤l≤e-2,∑(i_(1),…,i_(e))∈Z^(e)≥0,i_(1)+…i_(e)=q-e+l a_(1)^(i1)…a_(e)^(ie)=0 Using this identity and providing an extension of the principle of cross-classification that slightly generalizes the one obtained by Hong in 1996,we show that if r is an integer with 1≤r≤q-2,then for any subset{a_(1),…a_(r)}■F_(q)^(*)we have x^(q-1)-1/∏i=1 r(x-a_(i))-∑i=1 q-1-r(∑i_(1)+…+i_(r)=q-1-r-i^(a_(1)^(i1)…a_(r)^(ir)))x^(i).This implies#{x∈Fq*|∑i=0 q-1-4(∑_(i1)+…+ir=q-1-r-i^(a_(1)^(i1)…a_(r)^(ir)))x^(i)=0}=q-1-r.展开更多
文摘In this paper we prove some interesting extensions and generalizations of Enestrom- Kakeya Theorem concerning the location of the zeros of a polynomial in a complex plane. We also obtain some zero-free regions for a class of related analytic functions. Our results not only contain some known results as a special case but also a variety of interesting results can be deduced in a unified way by various choices of the parameters.
文摘We raise and partly answer the question: whether there exists a Markov system with respect to which the zeros of the Chebyshev polynomials are dense, but the maximum length of a zero free interval of the nth Chebyshev polynomial does not tends to zero. We also draw the conclu- tion that a Markov system, under an additional assumption, is dense if and only if the maxi- mum length of a zero free interval of the nth associated Chebyshev polynomial tends to zero.
基金sponsored by the National Natural ScienceFunds for Young Scholars (10901166)the Scientific Research Foundation for the Youth Scholars of Sun Yat-SenUniversitythe Research Grant of University of Macao on Applications of Hyper-Complex Analysis (cativo:7560)
文摘In this note, we study zeroes of Clifford algebra-valued polynomials. We prove that if such a polynomial has only real coefficients, then it has two types of zeroes： the real isolated zeroes and the spherical conjugate ones. The total number of zeroes does not exceed the degree of the polynomial. We also present a technique for computing the zeroes.
基金National Natural Science Foundation of China Natural ScienceFoundation of Zhejiang Province.
文摘Presents the first estimation conditions for Nourein iterations for simultaneous finding all zeros of a polynomial under which the iteration processes are guaranteed to converge. Computational formulas; Theorems and proofs.
基金This work is supported in part by the Foundation of Zhongshan University Advanced Research Centrein part by the National Natural Science Foundation of China
文摘A PL homotopy algorithm is modified to yield a polynomial-time result on its computational complexity.We prove that the cost of locating all zeros of a polynomial of degree n to an accuracy of ε(measured by the number of evaluations of the polynomial)grows no faster than O(max{n^4,n^3log_2(n/ε)}).This work is in response to a question raised in a paper by S.Smale as to the efficiency of piecewise linear methods in solving equations.In comparison with a few results reported,the algorithm under discussion is the only one providing correct multiplicities and the only one employing vector labelling.
文摘In this paper, the problem of computing zeros of a general degree bivariate Bernstein polynomial is considered. An efficient and robust algorithm is presented that takes into full account particular properties of the function considered. The algorithm works for rectangular as well as triangular domains. The outlined procedure can also be applied for the computation of the intersection of a Bezier patch and a plane as well as in the determination of an algebraic curve restricted to a compact domain. In particular, singular points of the algebraic curve are reliably detected.
基金National Natural Science Foundation of ChinaNatural Science Foundation of Zhejiang Province
文摘Presents a family of parallel iterations for finding all zeros of a polynomial without evaluation of derivatives. Construction of iterations; Convergence of the iterations; Details on the numerical examples.
基金supported partially by the National Science Foundation of China(Grant#11771304)the Fundamental Research Funds for the Central Universities.
文摘Let F be a field,and let e,k be integers such that 1≤e≤|F\{0}|and k≥0.We show that for any subset{a1,……,ae}■F\{0},the curious identity∑(i1+……ie)∈Z^(e)≥0,i1+……+ie=k a_(1)^(i1)…a_(e)^(ie)=∑i=1 e a_(i)^(k+e-1)/∏i≠j=1 e(a_(i)-a_(j))holds with Z≥0 being the set of nonnegative integers.As an application,we prove that for any subset{a_(1)…,a_(e)}■F_(q)\{0}with F_(q)being the finite field of q elements and e,l being integers such that 2≤e≤q-1 and 0≤l≤e-2,∑(i_(1),…,i_(e))∈Z^(e)≥0,i_(1)+…i_(e)=q-e+l a_(1)^(i1)…a_(e)^(ie)=0 Using this identity and providing an extension of the principle of cross-classification that slightly generalizes the one obtained by Hong in 1996,we show that if r is an integer with 1≤r≤q-2,then for any subset{a_(1),…a_(r)}■F_(q)^(*)we have x^(q-1)-1/∏i=1 r(x-a_(i))-∑i=1 q-1-r(∑i_(1)+…+i_(r)=q-1-r-i^(a_(1)^(i1)…a_(r)^(ir)))x^(i).This implies#{x∈Fq*|∑i=0 q-1-4(∑_(i1)+…+ir=q-1-r-i^(a_(1)^(i1)…a_(r)^(ir)))x^(i)=0}=q-1-r.
