For the binary floating point arithmetic with proper rounding of the sum, the cumulation of errors can be made practically independent of N

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria III: MATEMATYKA STOSOWANA XXIX (1987)

Summaries

Wa l e r ia n Du b n i c k i, Kr y s t ia n Zor y c h t a

Quasiconvex and pseudoconvex functions in nonlinear programming

The paper based on B. Martos’ ideas (B. Martos, Nonlinear programming theory and methods, Akademiai Kiado, Budapest 1975. Polish translation PWN 1983) presents theoretical results concerning quasiconvex and pseudoconvex functions.

J^{a n u s z} Ch o j n a c k i, An d r z e j Kie l ba s in s k i

Rounding Errors in Romberg Algorithm

It is shown that each quadrature computed by Romberg algorithm is exact for slightly perturbed data (computed values of the integrand). For the ordinary summation algorithm the cumulation of rounding errors is pro- portional to N, the number of quadrature modes. For more elaborate summation the cumulation is proportional to log N.

For the binary floating point arithmetic with proper rounding of the sum, the cumulation of errors can be made practically independent of N.

In each case the influence of Romberg extrapolation on the cumulation of rounding errors is bounded by a constant.

Maria Z. We s o l o w s k a

On time series.

Remarks about Lecture Notes edited by Wladyslaw Milo Comments on what is and what is not the time series analysis accompanied by a short historical review.

94 Summaries Wiesla w Pa se w ic z

Estimators of the Quadratic Discriminant Function in the Case of Special Covariance Matrices Structure Let us assume that the observed random vector Z, has a p-dimensional normal distribution with zero-mean vector and covariance matrix of the form

I i = a f [ i \ - Qi)I + QiE^ (i = 1, 2),

where of is the variance, is the correlation coefficient, Ipxp is the identity matrix and Epxp is the matrix with all elements equal to 1.

Further, let f(z) (i — 1, 2) denote a density function of the random vector

Z t -

In the present paper we discuss estimators of the quadratic discriminant function U (z) =

Zd z is l a w Po r o s i n s k i

Optimal stopping of the maximum of an observed sequence over an order statistic of an unobserved sequence

In this paper an optimal stopping problem is considered. Only one of two sequences of random variables which are independent copies of a known continuously distributed random variable is observed. It is necessary to stop the observation at the moment in which at most k values of the unobserved sequence are greater than the observed maximum, with maximal probability.

The optimal stopping rule for the finite length of the observation is obtained.