or the first three steps, we obtain 9 = 1.721549 =1+0.721549 0.721549 =1+1.385907263401380 1.385907263401380 1 More precisely, the corresponding accuracy is E = 16 - = 3.69901547059293 X 10-7 representing in the binary system (to be actually used for the algorithm implementation), an accuracy of up to 10 bits, since 2-11 = Appendix R Table R1 "Split and divide" expansion up to 12 steps. Un = 1/ n an (Un. - an-1) qn Pn/qn Valid 1.721549000000000 1 1.00000000000000 7.215x10-1 5.00 x 10-1 -2.215x10-1 1.385907263401380 2.00000000000000 2.785 x 10-1 5.00 x 10-1 2.215x10-1 2.591296134687970 3 1.66666666666667 5.488 x 10-2 5.56x10-2 6.726 x 10-4 1.691199961128960 4 1.75000000000000 2.845 x 10-2 3.13x10-2 2.800 x 10-3 1.446759340620720 7 1.71428571428571 7.264 x 10-3 1.02 x 10-2 2.940 x 10-3 5 2 2.238341561276870 18 1.72222222222222 6.726x10-4 1.54 x 10-3 8.706x10-4 6 4 4.195659349727770 79 1.72151898734177 3.065x10-5 8.01 x 10-5 4.947x10-5 7 5 5.110923660900250 1.72154963680387 0.000 x 10° 2.93x10-6 2.931 x 10-6 8 9 9.015209125664090 3796 1.72154899894626 6.379 x 10-7 3.47x10-8 -6.032 x 10-7 No 9 65 65.749999183803000 425486 247153 1.72154900001214 6.368 x 10-7 8.19x10-12 -6.368x10-7 No 1.333334784351880 432021 250949 1.72154899999602 6.368 x 10-7 7.94 x 10-12 -6.368x10-7 No 2 2.999986940889930 1289528 749051 1.72154900000133 6.368 x 10-2 8.91 x 10-13 -6.368 x 10-7 No 0.385907263401380 0.385907263401380 2.591296134687970 The results of this "split and divide" expansion up to 12 steps are summarized in Table R1. For each step n, the table lists the expansion coefficient an = Lun], the ratio Un = 1/(un-1-an-1), - starting from uo = X, the integer numbers Pn, qn as defined by Eq. (R2), the ratio Pn/qn, the error A = /pn/qn - 41, the upper bound 1/2q in Eq. (R11), and the difference 1/2q2 - A. If this difference is negative, this means that the condition in Eq. (R11) is not valid, and hence that pn/qn is not convergent on Q. We observe from Table R1 that for n > 1 the condition is valid up to the order n : 7, which yields s