Stability radius of polynomials occurring in the numerical solution of initial value problems

This paper deals with polynomial approximations(x) to the exponential function exp(x) related to numerical procedures for solving initial value problems. Motivated by stability requirements, we present a numerical study of the largest diskD()={z C: |z+|} that is contained in the stability regionS()={z C: |(z)|1}. The radius of this largest disk is denoted byr(), the stability radius. On the basis of our numerical study, several conjectures are made concerningr _m,p=sup {r(): _m,p}. Here _{m, p} (1pm; p, m integers) is the class of all polynomials(x) with real coefficients and degree m for which(x)=exp(x)+O(x ^p+1) (forx 0).     

Asymptotic Behavior of Entire Functions with Exceptional Values in the Borel Relation

Let M _f(r) and _f (r) be, respectively, the maximum of the modulus and the maximum term of an entire function f and let l(r) be a continuously differentiable function convex with respect to ln r. We establish that, in order that ln M _f(r) ln _f (r), r +, for every entire function f such that _f (r) l(r), r +, it is necessary and sufficient that ln (rl(r)) = o(l(r)), r +.     

Complete stability of large order statistics

Fix an integerr1. For eachnr, letM _nr be the rth largest ofX ₁,...,X _n, where {X _n,n1} is a sequence of i.i.d. random variables. Necessary and sufficient conditions are given for the convergence of _n=r n P[|M _nr /a _n –1|<] for every >0, where {a _n} is a real sequence and –1. Moreover, it is shown that if this series converges for somer1 and some >–1, then it converges for everyr1 and every >–1.     

Local theorems for some statistics of integer-valued sequences

Let Js={j₁, ..., j_s} be a collection of nonnegative numbers, j₁+...+j_s=n, j_s1, R(Js) be the set of sequencesf=(f(1), ...,f(n)) in each of which the integer m occurs j_s times. Randomly and equiprobably one chooses a sequence f from R(Js). Let _n,r be the number of r-drops in f; _n be the r-principal index off. Local limit theorems are established in this paper for the random variables _n,r and _n as n .Translated from Teoriya Sluchainykh Protsessov, No. 16, pp. 61–66, 1988.     

Lattice point theory of irractional ellipsoids with an arbitrary center

We shall consider positive definite quadratic formsQ inr2 variables of the almost diagonal shape where 2, and for 1j,Q _j is a positive definite quadratic form with integral coefficients inr _j variables, _j is a positive real number,r _j1 andr ₁+...+r =r Letb ₁,...,b _r be a system of real numbers with 0b _j<1. For x>0 letA(x) be the number of lattice points in the ellipsoidQ(u+b)x, letV(x) be the volume of this ellipsoid and letP(x)=A(x)-V(x). Our purpose is to find the exact order ofP(x); i. e., the numberf for which for each >0P(x)=O(x^f+) andP(x)=(x ^f–).     

On the Growth of the Maximum of the Modulus of an Entire Function on a Sequence

Let M _f(r) and _f(r) be, respectively, the maximum of the modulus and the maximum term of an entire function f and let be a continuously differentiable function convex on (–, +) and such that x = o((x)) as x +. We establish that, in order that the equality be true for any entire function f, it is necessary and sufficient that ln (x) = o((x)) as x +.     

Acceleration of Superlinearly Convergent Sequences

A method analogous to the Aitken extrapolation is proposed to accelerate the convergence of sequences of real or complex numbers with asymptotic behavior where e _i is the error of the ith element of the sequence, d _j1 for all 1jr, and d_j > 1. The R-order of the resulting sequences is computed using methods which are also of independent interest.     

Characterization theorem of generalized polynomial of best approximation having bounded coefficients

Let the set of generalized polynomials having bounded coefficients beK={p= _jg_j. _{j j j},j=1, 2, ...,n}, whereg ₁,g ₂, ...,g _n are linearly independent continuous functions defined on the interval [a, b], _j, _j are extended real numbers satisfying _j<+, _j>-, and _{j j}. Assume thatf is a continuous function defined on a compact setX [a, b]. This paper gives the characterization theorem forp being the best uniform approximation tof fromK, and points out that the characterization theorem can be applied in calculating the approximate solution of best approximation tof fromK.     

Fast Summation of Power Series with Coefficients Analytic at Infinity

We propose a fast summation algorithm for slowly convergent power series of the form _{j=j 0} z ^j _j j _i=1 ^s (j+ _i )^– _i , where R, _i 0 and _i C, 1is, are known parameters, and _j =(j), being a given real or complex function, analytic at infinity. Such series embody many cases treated by specific methods in the recent literature on acceleration. Our approach rests on explicit asymptotic summation, started from the efficient numerical computation of the Laurent coefficients of . The effectiveness of the resulting method, termed ASM (Asymptotic Summation Method), is shown by several numerical tests.