The convergence ball of the Secant method under Hölder continuous divided differences

Under the hypothesis that nonlinear operators have Hölder continuous divided differences of order one, an estimate of the radius of the convergence ball of the Secant method is obtained, error estimate is also established.     

On the Secant method under generalized Lipschitz conditions for the divided difference operator

In this work we introduce for the first time the generalized Lipschitz conditions for the divided difference operator. A positive integrable function, partial case of which is usual Lipschitz constant, is suggested. Under the given conditions the convergence of the Secant method for solving the operator equations in Banach spaces is investigated, the uniqueness ball for the solution is obtained. As a partial case the known results for the Lipschitz constants are received. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Secant method

We present a new semilocal convergence analysis for the Secant method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis is based on the weaker center-Lipschitz concept instead of the stronger Lipschitz condition which has been ubiquitously employed in other studies such as Amat et al. (2004)  [2], Bosarge and Falb (1969)  [9], Dennis (1971)  [10], Ezquerro et al. (2010)  [11], Hernández et al. (2005, 2000)   and , Kantorovich and Akilov (1982)  [14], Laasonen (1969)  [15], Ortega and Rheinboldt (1970)  [16], Parida and Gupta (2007)  [17], Potra (1982, 1984–1985, 1985)  ,  and , Proinov (2009, 2010)   and , Schmidt (1978) [23], Wolfe (1978)  [24] and Yamamoto (1987)  [25] for computing the inverses of the linear operators. We also provide lower and upper bounds on the limit point of the majorizing sequences for the Secant method. Under the same computational cost, our error analysis is tighter than that proposed in earlier studies. Numerical examples illustrating the theoretical results are also given in this study.     

A continuous function whose divided differences at the Chebyshev extrema are all zero

A nontrivial function having the properties described in the title is given explicitly.     

Note on trigonometric divided differences

An alternative to Lyche's recently proposed scheme for trigonometric divided differences and a Newton-type interpolation formula, obtained through the development of an earlier suggestion of the writer, appears to be simpler, neater in format, more symmetrical and easier to compute.     

Root finding by divided differences

Summary A recursive method is presented for computing a simple zero of an analytic functionf from information contained in a table of divided differences of its reciprocalh=1/f. A good deal of flexibility is permitted in the choice of ordinate and derivative values, and in the choice of the number of previous points upon which to base the next estimate of the required zero.The method is shown to be equivalent to a process of fitting rational functions with linear numerators to data sampled fromf. Asymptotic and regional convergence properties of such a process have already been studied; in particular, asymptotically quadratic convergence is easily obtained, at the expense of only one function evaluation and a moderate amount of overhead computation per step. In these respects the method is comparable with the Newton form of iterated polynomial inverse interpolation, while its regional convergence characteristics may be superior in certain circum-stances.It is also shown that the method is not unduly sensitive to round-off errors.     

Mysovskii-type theorem for the Secant method under Hölder continuous Fréchet derivative

The Mysovskii-type condition is considered in this study for the Secant method in Banach spaces to solve a nonlinear operator equation. We suppose the inverse of divided difference of order one is bounded and the Fréchet derivative of the nonlinear operator is Hölder continuous. By use of Fibonacci generalized sequence, a semilocal convergence theorem is established which matches with the convergence order of the method. Finally, two simple examples are provided to show that our results apply, where earlier ones fail.     

A convergence criterion for Secant method with approximate zeros

We estimate the speed of convergence of Secant method in one variable and multivariable case with a constant from the coefficients of Taylor series. We present a criterion to confirm thatz is close enough to a zero for Secant method and compare with that of Newton method.     

ECT-B-splines defined by generalized divided differences

ECT-spline curves are generated from different local ECT-systems via connection matrices. If they are nonsingular, lower triangular and totally positive there is a basis of the space of ECT-splines consisting of functions having minimal compact supports, normalized either to form a nonnegative partition of unity or to have integral one. In this paper such ECT-B-splines are defined by generalized divided differences. This definition reduces to the classical one in case of a Schoenberg space. Under suitable assumptions it leads to a recursive method for computing the ECT-B-splines that reduces to the de Boor–Mansion–Cox recursion in case of ordinary polynomial splines and to Lyche's recursion in case of Tchebycheff splines [Mühlbach and Tang, Calculation of ECT-B-splines and of ECT-spline curves recursively, in preparation].There is an ECT-spline space naturally adjoint to every ECT-spline space. We also construct B-splines via generalized divided differences for this space and study relations between the two adjoint spaces.     

The Secant method for nondifferentiable operators

In this paper, we use the Secant method to find a solution of a nonlinear operator equation in Banach spaces. A semilocal convergence result is obtained. For that, we consider a condition for divided differences which generalizes the usual ones, i.e., Lipschitz continuous or Hölder continuous conditions. Besides, we apply our results to approximate the solution of a nonlinear equation.     

Hermite插值多项式的差商表示及其应用

差商展开是个非常重要的解析工具.有迹象表明其内在的思想和技巧似乎被人们所忽视或淡忘.论文的目的是对H erm ite插值多项式的重节点差商表示予以系统的表述,并利用重节点差商的展开技巧证明一些在应用上相当重要的结果.     

A functional equation related to divided differences

Lithuanian Mathematical Journal -     

Integer-valued polynomials over matrices and divided differences

Let $D$ be an integrally closed domain with quotient field $K$ and $n$ a positive integer. We give a characterization of the polynomials in $K[X]$ which are integer-valued over the set of matrices $M_n(D)$ in terms of their divided differences. A necessary and sufficient condition on $f\in K[X]$ to be integer-valued over $M_n(D)$ is that, for each $k$ less than $n$ , the $k$ th divided difference of $f$ is integral-valued on every subset of the roots of any monic polynomial over $D$ of degree $n$ . If in addition $D$ has zero Jacobson radical then it is sufficient to check the above conditions on subsets of the roots of monic irreducible polynomials of degree $n$ , that is, conjugate integral elements of degree $n$ over $D$ .     

The differential mean value of divided differences

The paper deals with the asymptotic behaviour of the differential mean value of divided differences as the interval shrinks to zero by presenting an asymptotic expansion. The coefficients are given by a recurrence formula. For a wide class of analytic functions the differential mean value can be represented by a convergent sum. Our results generalize two recent theorems by Powers, Riedel and Sahoo [R.C. Powers, T. Riedel, P.K. Sahoo, Limit properties of differential mean values, J. Math. Anal. Appl. 227 (1998) 216-226].     

A chain rule for multivariate divided differences

In this paper we derive a formula for divided differences of composite functions of several variables with respect to rectangular grids of points. Letting the points coalesce yields a chain rule for partial derivatives of multivariate functions.     

A recurrence relation for generalized divided differences with respect to ECT-systems

It is well known that ordinary divided differences can be computed recursively. This holds true also for generalized divided differences with respect to complete Chebyshev-systems. In this note for extended complete Chebyshev-systems and possibly repeated nodes for the recurrence relation a simple proof is given which also covers the case of complex valued functions. As an application, interpolation by linear combinations of certain complex exponential functions is considered. Moreover, it is shown that generalized divided differences are also continuous functions of their nodes. This revised version was published online in August 2006 with corrections to the Cover Date.