Engineering Transactions, 11, 3, pp. 463-490, 1963

### Tak Zwanej Aproksymacji Jednokrotnie Optymalnej i Niektórych Jej Zastosowaniach w Mechanice

M. Życzkowski
Politechnika Krakowska
Poland

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.

Full Text: PDF

#### References

N.I. ACHIEZER, Teoria aproksymacji (tlum. z ros.), PWN, Warszawa 1957.

[in Russian]

[in Russian]

[in Russian]

L. Fox, The numerical solution of two-point boundary problems, Clarendon Press, Oxford 1957.

[in Russian]

D.R. HARTREE, Numerical analysis, Clarendon Press, Oxford 1958.

D. JACKSON, The theory of approximation, Am. Math. Soc. Colloquium Publ., Vol. 11, 1930.

J. ŁUKASZEWICZ, M. WARMUS, Metody numeryczne i graficzne, t. 1, PWN, Warszawa 1956.

[in Russian]

[in Russian]

W. E. MILNE, Numerical Culculus, Princeton UP 1949 (thum. ros., Moskwa 1952).

ST. PIECHNIK, M. ŻYCZKOWSKI, On the plastic interaction-curve for bending and torsion of a circular bar, Arch. Mech. Stos. 5, 13 (1961), 669-692.

S. D. PONOMARIEW i inni, Współczesne metody obliczeń wytrzymałościowych w budowie maszyn, t. II, PWN, Warszawa 1958 (tłum. z ros.).

[in Russian]

[in Russian]

W. WALTER, Über Tschebyscheff-Approximation differenzierbarer Funktionen, GAMM-Tagung Würzburg, April 1961.

H. WERNER, Bemerkungen zur Tschebyscheffschen Approximation mit rationalen Funktionen, GAMM-Tagung Würzburg, April 1961.

W. WETTERLING, Zur Anwendung des Newtonschen Iterationsverfahrens bei der numerischen Behandlung des Tschebyscheff-Approximation, GAMM-Tagung Würzburg, April 1961.

M. WNUK, Stan graniczny pręta jednocześnie skręcanego i rozciąganego przy dowolnym kształcie przekroju, Rozpr. Inzyn., 3, 10 (1962).

M. ŻYCZKOWSKI, Obliczanie sił krytycznych dla sprężystych prętów niepryzmatycznych metoda interpolacji częściowej, Rozpr. Inzyn., 3, 4 (1956), 367-412.

M. ŻYCZKOWSKI, Potenzieren von verallgemeinerten Potenzreihen mit beliebigem Exponent, Z. angew. Math, Physik, 6, 12 (1961), 572-576.

M. ŻYCZKOWSKI, Krzywe graniczne dla belek jednocześnie rozciąganych i zginanych i o dowolnym przekroju (w druku).