Tak Zwanej Aproksymacji Jednokrotnie Optymalnej i Niektórych Jej Zastosowaniach w Mechanice

The Tchebyshev approximation called also the best or optimum approximation shows, in addition to the principal advantage of minimizing the approximation error in a certain interval, a few draw-backs from the practical viewpoint. The approximation error becomes maximum at the ends of the interval, where the boundary values have often an important physical interpretation it being therefore desirable to preserve the accurate values at these points; The extrapolation error is, with this method as a rule higher than for other approximation types. Finally, the selection of the coefficients of the approximation polynomial is a difficult problem and requires considerable labour. These drawbacks can be avoided, for instance, if the Hermite boundary approximation is used in which the agreement of the function itself and a few of its derivatives at the ends of the interval is made use of; however, the error of this approximation inside the interval is, as a rule, relatively large. The «onefold optimum» approximation proposed in the present paper constitutes a certain intermediate type, n - 1 Hermite's conditions of onefold are used one parameter remaining to be determined from the condition of minimum deviation. The iteration procedure of obtaining the value of this parameter presents no major difficulties. In Sec. 2 a condition is established for this procedure while Sec. 3 is concerned with the application of falsi rule (3.2).
Sec. 4 is devoted to a numerical determination of the extremum values of the function, necessary for the application of the equations derived. The extremum value fm is determined first by means of the series (4.7) and then by means of finite differences the function being approximated by means of a polynomial of the second (4.11), third (4.18) and fourth order polynomial (4.12) and (4.13). As an example the equations obtained are used to calculate the extremum of the function f (x) = - x In x.
Section 5 brings an estimate of the error of the «onefold optimum» approximation by means of the classical method consisting of making use of the general formula (5.7). This estimate is quoted merely for comparison purposes, a more accurate estimate being concerned with the technique of obtaining the coefficients of the approximating polynomial by means of the condition (2.4)., Some examples are also given concerning the application of the approximation method proposed to the theory of stability and the theory of plasticity; the derivation of approximate equations for the critical force are discussed in detail for a clamped bar loaded as shown by Fig. 1. Figure 2 and Table 6 represent the results. The equations (6.24) and (6.25) are proposed as final formulae.