By Samuel Daniel Conte
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INTEGER IFLAG,NTABLE, J,NEXT,NEXTL,NEXTR REAL F(NTABLE),TOL,X(NTABLE),XBAR, A(20),ERROR,PSIK,XK(20) C****** I N P U T ****** C XBAR POINT AT WHICH TO INTERPOLATE . ,NTABLE CONTAINS THE FUNCTION TABLE . C A S S U M P T I O N ... ) C NTABLE NUMBER OF ENTRIES IN FUNCTION TABLE. C TOL DESIRED ERROR BOUND . C****** O U T P U T ****** C TABLE THE INTERPOLATED FUNCTION VALUE . C IFLAG AN INTEGER, C =l , SUCCESSFUL EXECUTION , C =2 , UNABLE TO ACHIEVE DESIRED ERROR IN 20 STEPS, C =3 , XBAR LIES OUTSIDE OF TABLE RANGE.
Thus we can assume that the local round-off errors are either uniformly or normally distributed between their extreme values. Using statistical methods, we can then obtain the standard deviation, the variance of distribution, and estimates of the accumulated roundoff error. The statistical approach is considered in some detail by Hamming [1] and Henrici [2]. The method does involve substantial analysis and additional computer time, but in the experiments conducted to date it has obtained error estimates which are in remarkable agreement with experimentally available evidence.
Although we cannot where prove that a certain n is “large enough,” we can test the hypothesis that n is “large enough” by comparing with If for k near n, say for k = n - 2, n - 1, n, then we accept the hypothesis that n is “large enough” for to be true, and therefore accept Example Let p > 1. , within 1/10 of 1 for n = 3 and p = 2. 12005 · · · . is then a the error This notation carries over to functions of a real variable. If we say that the convergence is provided for some finite constant K and all small enough h.