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We consider Konig’s method for finding roots of polynomials. Let K f , n be the function defined in Konig’s method for polynomial f of order n ( n ≥ 2 ) . For any given complex number λ ∈ C and any given set Ω ≔ { z 1 , z 2 , … , z k } of k ( k ≥ 2 ) distinct complex numbers, we present a procedure of constructing a polynomial f n , λ for which Ω is a k -cycle of K f n , λ , n with multiplier λ .

In this paper, we first propose a new block splitting (NBS) iteration method for solving the large sparse complex symmetric linear systems. The NBS iteration method avoids complex arithmetic compared with the combination method of real part and imaginary part (CRI) one established by Wang et al. (2017). The unconditional convergence and the quasi-optimal parameter of the NBS iteration method are given. Moreover, by further accelerating the NBS one with another parameter, we construct the parameterized BS (PBS) iteration method and establish its convergence theory. Also, the spectral properties of the PBS-preconditioned matrix are analyzed and the parameter selection strategy of the PBS iteration method is given. Numerical experiments are reported to illustrate the feasibility and effectiveness of the proposed methods.

In this paper we consider an extension of analytic solutions of a class of constrained matrix minimization problems and propose analytic solutions of the following matrix minimization problem: min S k S k H = I n , L k L k H = I m | det ( c I m ± ∏ k = 1 s A k S k B k L k ) | min S k S k H = I n , L k L k H = I m | t r ( c I m ± ∏ k = 1 s A k S k B k L k ) | , where A k ∈ C m × n , B k ∈ C n × m , c is a complex number, I m denotes m -order identity matrix, and det ( ⋅ ) , t r ( ⋅ ) denote matrix determinant and trace function. The proposed results improve the corresponding existing ones in Xu et al. (2019). Numerical examples are provided to illustrate the efficiency of the proposed theoretical results.

Sums of the form Σv=1xf(v) are defined traditionally only when the number of terms x is a positive integer or ∞. We propose a natural way to extend this definition to the case when x is a (rather arbitrary) real or complex number ("fractional sums"). This generalizes known special cases like the interpolation of the factorial by the Γ function, or Euler's little-known formula Σv=1-1/21/v = -2 ln 2. After giving the fundamental definition, we generalize several algebraic identities (such as the geometric series) to the case with a non-integer number of terms.We use these ideas to derive a number of unusual infinite sums, products and limits, such as limn→∞((2n)-n2/2-n/4 e-n/8Πv=12nv(-1)vv2/4) = e7ζ(3)/16π2.

We devise an efficient algorithm that, given points z 1 , ? , z k in the open unit disk D and a set of complex numbers { f i , 0 , f i , 1 , ? , f i , n i - 1 } assigned to each z i , produces a rational function f with a single (multiple) pole in D , such that f is bounded on the unit circle by a predetermined positive number, and its Taylor expansion at z i has f i , 0 , f i , 1 , ? , f i , n i - 1 as its first n i coefficients.

A discrete complex-valued bidirectional associative memory (CVBAM) is considered. The input and output of CVBAM are bipolar complex numbers, of which the two components take the value of either −1 or 1. We prove that the stored patterns are all fixed points of CVBAM provided certain sufficient conditions are satisfied. It is shown that each fixed point belongs to a fixed point group which consists of four fixed points. Supporting numerical results are provided.

The nonnegative inverse eigenvalue problem is that given a family of complex numbers λ={λ1,…,λn}, find a nonnegative matrix of order n with spectrum λ. This problem is difficult and remains unsolved partially. In this paper, we focus on its generalization that the reconstructed nonnegative matrices should have some prescribed entries. It is easy to see that this new problem will come back to the common nonnegative inverse eigenvalue problem if there is no constraint of the locations of entries. A numerical isospectral flow method which is developed by hybridizing the optimization theory and steepest descent method is used to study the reconstruction. Moreover, an error estimate of the numerical iteration for ordinary differential equations on the matrix manifold is presented. After that, a numerical method for the nonnegative symmetric inverse eigenvalue problem with prescribed entries and its error estimate are considered. Finally, the approaches are verified by the numerical test results.

We study particular sequences of rational matrix functions with poles outside the unit circle. These Schur–Nevanlinna–Potapov sequences are recursively constructed based on some complex numbers with norm less than one and some strictly contractive matrices. The main theme of this paper is a thorough analysis of the matrix functions belonging to the sequences in question. Essentially, such sequences are closely related to the theory of orthogonal rational matrix functions on the unit circle. As a further crosslink, we explain that the functions belonging to Schur–Nevanlinna–Potapov sequences can be used to describe the solution set of an interpolation problem of Nevanlinna–Pick type for matricial Schur functions.

Let [email protected]?C^n^x^n be a nontrivial involution, i.e., R=R^-^1 +/-I"n. We say that [email protected]?C^n^x^n is R-symmetric if RGR=G. The set of all nxnR-symmetric matrices is denoted by GSC^n^x^n. In this paper, we first give the solvability condition for the following inverse eigenproblem (IEP): given a set of vectors {x"i}"i"="1^m in C^n and a set of complex numbers {@l"i}"i"="1^m, find a matrix [email protected]?GSC^n^x^n such that {x"i}"i"="1^m and {@l"i}"i"="1^m are, respectively, the eigenvalues and eigenvectors of A. We then consider the following approximation problem: Given an nxn matrix [email protected]?, find [email protected][email protected]?S"E such that @[email protected][email protected][email protected]?=min"A"@?"S"""[email protected][email protected][email protected]?, where S"E is the solution set of IEP and @[email protected][email protected]? is the Frobenius norm. We provide an explicit formula for the best approximation solution [email protected]? by means of the canonical correlation decomposition.

Let µ be a finite positive Borel measure supported on a compact set of the real line and introduce the discrete Sobolev-type inner product (f, g) = ∫ f(x)g(x) dµ(x) + Σk=1K Sigma;i=0Nk Mk,if(i)(ck) g(i)(ck), where the mass points ck belong to supp(µ) and Mk,i are complex numbers such that Mk,Nk ≠ 0. In this paper we investigate the asymptotics of the polynomials orthogonal with this product. When the mass points ck belong to C\supp(µ), the problem was solved in a paper by G. Lopez, et al. (Constr. Approx. 11 (1995) 107-137) and, for mass points in supp(µ) = [-1, 1], the solution was given by I.A. Rocha et al. (J. Approx. Theory, 121 (2003) 336-356) provided that µ'(x) > 0 a.e.x ∈ [- 1, 1] and Mk,i are nonnegative constants. If µ ∈ M(0, 1), the possibility ck ∈ supp(µ)\[-1, 1] must be considered. Here we solve this last case with complex constants Mk,i.

In (simple) Pal-type interpolation the problem consists of finding a polynomial of degree n + m-1 that has prescribed values at, say, n pairwise distinct points and prescribed values for the rth derivative r ≤ 1 at (another) set of m distinct points.The problem is named regular if it has a unique solution for any set of data; this is equivalent to the homogeneous problem (all prescribed values are 0) having the trivial solution only.In this paper the effect of interchanging the value-nodes and the derivative-nodes on the regularity of the problem is studied for several sets of nodes.

Stability of θ-methods for delay integro-differential equations (DIDEs) is studied on the basis of the linear equation du/dt= λu(t) + µu(t - τ) + k ∫t-τt u(σ) dσ, where λ,µ,k are complex numbers and τ is a constant delay. It is shown that every A-stable θ-method possesses a similar stability property to P-stability, i.e., the method preserves the delay-independent stability of the exact solution under the condition that k is real and τ/h is an integer, where h is a step-size. It is also shown that the method does not possess the same property if τ/h is not an integer. As a result, no θ-method can possess a similar stability property to GP-stability with respect to DIDEs.

Let Λ = ∗λ 0 , λ1,…∗ be a given sequence of complex numbers such that ¦λ i ¦ > 1 and Re (λ i λ j − 1) > 0 for each i and j. For Muntz polynomials P(x) = ∑i=0npixλi and Q(x) = ∑j=0mqjxλj we define an inner product [P, Q] by [P,Q]= ∫ 0 1 (P Q )(x) d x x 2 where (P⊙ Q )(x) = ∑ i=0 n ∑ j=0 m p i q j x λ i λ j . We obtain the Muntz polynomials (Qn(x)) orthogonal with respect to this inner product and connect them with the Malmquist rational functions W ν (z)= ∏ v=0 n−1 ( z−1 λ v ) ∏ v=0 n (z−λ v ) orthogonal on the unit circle with respect to the inner product (u,v)= 1 2τ i ∮ || =1 u(z) v(z) d z z Several interesting properties of polynomials Qn(x) are given.

A bisequence of complex numbers {μn}−∞∞ determines a strong moment functional L satisfying L[xn] = μn. If L is positive-definite on a bounded interval (a,b) ⊂ R{0}, then L has an integral representation L , n=0, ±1, ±2,…, and quadrature rules {wni,xni} exist such that μk = ∑i=innsnikwni. This paper is concerned with establishing certain extremal properties of the weights wni and using these properties to obtain maximal mass results satisfied by distributions ψ(x) representing L when only a finite bisequence of moments {μk}k=−nn−1 is given.

Let p(x) be a polynomial of degree n⩾2 with coefficients in a subfield K of the complex numbers. For each natural number m⩾2, let Lm(x) be the m×m lower triangular matrix whose diagonal entries are p(x) and for each j=1,…,m−1, its jth subdiagonal entries are p j (x) j! . For i=1,2, let Lmi)(x) be the matrix obtained from Lm(x) by deleting its first i rows and its last i columns. L1(1)(x)≡1. Then, the function Bm(x)=x−p(x) det (L m−1 (1) (x)) det (L m (1) (x)) defines a fixed-point iteration function having mth order convergence rate for simple roots of p(x). For m=2 and 3, Bm(x) coincides with Newton's and Halley's, respectively. The function Bm(x) is a member of S(m,m+n−2), where for any M⩾m, S(m,M) is the set of all rational iteration functions g(x) ∈ K(x) such that for all roots θ of p(x), then g(x)=θ+∑i=mMγi(x)(θ−x)i, with γi(x) ∈ K(x) and well-defined at any simple root θ. Given g ∈ S(m,M), and a simple root θ of p(x), gi(θ)=0, i=1, …, m−1 and the asymptotic constant of convergence of the corresponding fixed-point iteration is γ m (θ) = (−1)g m (θ) m! . For Bm(x) we obtain γ m (θ)= (−1) m det (L m+1 (2) (θ)) det (L m (1)(θ)) . If all roots of p(x) are simple, Bm(x) is the unique member of S(m,m + n − 2). By making use of the identity 0 = ∑ i=0 n [p (i) (x) i!] (θ − x) i , we arrive at two recursive formulas for constructing iteration functions within the S(m,M) family. In particular, the family of Bm(x) can be generated using one of these formulas. Moreover, the other formula gives a simple scheme for constructing a family of iteration functions credited to Euler as well as Schroder, whose mth order member belongs to S(m,mn), m>2. The iteration functions within S(m,M) can be extended to any arbitrary smooth function f, with the uniform replacement of p(j) with f(j) in g as well as in γm(θ).

In this paper, multipoint rational approximants to the Riesz-Herglotz transform of a Borel measure µ, supported on [-π,π] are considered. We give estimates for the rate of convergence of these approximants. These estimates are obtained from the asymptotic behaviour of a sequence of rational functions, orthogonal with respect to µ and having prescribed poles 1/α_k, where {α_k:k=1...∞} is a sequence of complex numbers compactly contained in the open unit disk. As an application we also give the rate of convergence of rational Szegö quadrature formulas for integrals with respect to µ. ispartof: Journal of computational and applied mathematics vol:77 issue:1 pages:77-101 ispartof: location:GERMANY, LEIPZIG status: published

Let α = zn,mm=1n with |zn,m| < 1, n = 1,2,…, be an arbitrary sequence of complex numbers. We generalize the orthogonal rational functions with poles at α. We study the weak convergence and the interpolation properties of the orthogonal rational functions.

Let α = {zn,m}nm = 1 with |zn,m| < 1, n = 1,2,…, be an arbitrary sequence of complex numbers. We study the convergence of the rational functionpn−1/qn which interpolates the Carathéodory function F(z) = 1/ (2π)∫2π0u/(u − z) dμ(t), u = eit, at {zn,m, 1/z̄n,m}nm = 1, where dμ is a finite positive measure on the unit circle. We also consider the relations between pn − 1/qn and the orthogonal polynomials with respect to varying measures.

Interval methods for the simultaneous inclusion of polynomial zeros produce the approximations that contain the exact zeros providing not only error bounds automatically but also take into account rounding errors without altering the fundamental structure of the interval formula. However, at present, the computational costs of most interval methods are still great, in general. In this paper several effective algorithms which preserve the inclusion property concerning the complex zeros and which have a high computational efficiency are constructed. These algorithms combine the efficiency of ordinary floating-point iterations with the accuracy control that may be obtained by the iterations in interval arithmetic. Several examples are included to illustrate the efficiency and some advantages of the proposed combined methods.

A polynomial Pythagorean-hodograph (PH) curve r(t)=(x"1(t),...,x"n(t)) in R^n is characterized by the property that its derivative components satisfy the Pythagorean condition x"1^'^2(t)+...+x"n^'^2(t)=@s^2(t) for some polynomial @s(t), ensuring that the arc length s(t)=@!@s(t)dt is simply a polynomial in the curve parameter t. PH curves have thus far been extensively studied in R^2 and R^3, by means of the complex-number and the quaternion or Hopf map representations, and the basic theory and algorithms for their practical construction and analysis are currently well-developed. However, the case of PH curves in R^n for n>3 remains largely unexplored, due to difficulties with the characterization of Pythagorean (n+1)-tuples when n>3. Invoking recent results from number theory, we characterize the structure of PH curves in dimensions n=5 and n=9, and investigate some of their properties.

In the family S of normalized, univalent functions, an omitted point in F ⊂S is a complex number w 0 , such that there is at least one function f∈ F , satisfying f(z)≠w 0 for all |z| Let a set of m distinct complex numbers w 1 , w 2 ,…, w m all ≠0, be given such that 0⩽ arg w 1 arg w 2 arg w m π . The tuple ( w 1 , w 2 ,…, w m ) shall be called an omitted tuple for F if there exists at least one f∈ F such that f ( z )≠ w i , ∀ i =1,2,…, m and all | z | tw 1 , tw 2 ,…, tw m ), t an arbitrary positive number, is an omitted tuple in S or not, more precisely the number of functions omitting the tuple for different values of t . An answer to this question in full generality will not be offered, but some partial results are given. Moreover, two subfamilies where complete solutions are obtained, are briefly mentioned.

We investigate the growth of the meromorphic function defined by 1 + a1zb + a2z1 + … , where ai are complex numbers, ai ≠ 0, ¦ai¦ ⩾ ¦ai+1¦ (i = 1, 2, …) and limi→∞ ai =0. b is a complex number.

Let {cn} and {c∗n} be two sequences of complex numbers converging to the finite limit c. We say that {c∗n} converges faster to c than {cn} if (c ∗ n −c) (c n −c) →0 . Convergence acceleration is based on this definition. In this paper we draw attention to some of its deficiencies. As an example we consider the case where {cn} is the sequence of approximants of a continued fraction K ( a n 1 ) where an → 0.

We show that the simple conditional convergence set P α = {z ∈ C |z|- Re (z e -i2α )⩽ 1 2 cos 2 α} fixed α ∈〈- 1 2 π, 1 2 π〉 is maximal in the sense that if b is an arbitrary point in C ⧹P α , then Pα∪{b} is not conditional convergence set.

Consider a system of n linear first-order differential equations (d/dx)y = A(x)y in which A(x) is an n × n matrix of rational functions over a subfield F of the field C of complex numbers and let Γ = {α1; .... ,αd} ⊆ C be a set of conjugate singularities of this system, i.e., poles of A(x) which are roots in C of some irreducible polynomial p(x) in F[x]. We propose an algorithm for transforming the given system into an equivalent system over F(x) which is super-irreducible in each element α ∈ Γ. This algorithm does not require working in the algebraic extension F(Γ) that appears when one applies Hilali-Wazner's algorithm (Numer. Math. 50 (1987) 429) successively with the individual singularities α1,...,αd. The transformation matrix as well as the resulting system have their coefficients in F(x) and all the computations are performed in F[x]/(p) instead of the splitting field of p.

We show how Padé approximations are used to get Diophantine approximations of real or complex numbers, and so to prove the irrationality. We present two kinds of examples. First, we study two types of series for which Padé approximations provide exactly Diophantine approximations. Then, we show how Padé approximants to the asymptotic expansion of the remainder term of a value of a series also leads to Diophantine approximation.

Starting from a suitable fixed point relation, a new one-parameter family of iterative methods for the simultaneous inclusion of complex zeros in circular complex arithmetic is constructed. It is proved that the order of convergence of this family is four. The convergence analysis is performed under computationally verifiable initial conditions. An approach for the construction of accelerated methods with negligible number of additional operations is discussed. To demonstrate convergence properties of the proposed family of methods, two numerical examples results are given.

We characterize the sequences {zn} of complex numbers which are sequences of approximants of continued fractions K(an/bn) with |an|+1⩽|bn|, and study some of their properties. In particular we give truncation error bounds for such continued fractions.

The purpose of the paper is to present some procedures for the construction of simultaneous methods for finding all zeros of a polynomial f using Weierstrass' corrections W i (z) = f(z) ∏ k≠i (z−z k ) . In this manner some well-known methods as well as new methods are obtained. The proposed iterative algorithms are realized in ordinary complex arithmetic and circular interval arithmetic. The convergence analysis of the presented methods and numerical examples are given.

Define for integer m⩾1 and α a complex number with α2,αm≠1, the polynomial of degree mwm(α;z)=(z+α)m-(1+αz)m,whose (simple) zeros can be seen as the Möbius transforms of the mth roots of unity.In this paper it will be shown that (0,m) Pál-type interpolation on the zeros of {wn+m(α;z),wn(α;z)} is regular for all integers n,m⩾1 and complex numbers α with α2,αm,αn,αn+m≠1.