Half-Interval Bisection

For a case such as this, fundamental calculus says that there will be one real root in this interval. Note that we can determine the root graphically by plotting the function f(x) and estimating where it intersects the a -axis. 

The bisection or half-interval method is a systematic trial-and-error solution consisting of the following steps  

Choose points x — a and x = 6 so that the interval a x b encompasses one and only one real root. 

Determine the midpoint of the interval, point c. If as in this case, /(c) 0, then the root must lie between the midpoint of the original interval, c, and the lower end of the interval, a. If /(c) 0, then the root must be between c and 6. 

Repeat the previous step and calculate /(x) at the midpoint of the new interval at each step of the iteration and discard the half of the interval that does not contain the root. 

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The case KEY= 2 in the routine QXG in FLR [5] uses a family of four formulas in the RMS formulas, namely, 13-point (L), 19-point h), 27-point (A), 41-point (D rules. Including Ninomiya s stable 11-point rule (7o) yields a sequence of rules, L, A,..., A, with increasing accuracy, see Fig. 2 for the arrangement of sample points in the right-half interval with the sample points symmetrically arranged in the left-half interval. From Fig. 2 we find that the sample points for 4 are all reused in 4+i, 0 < /c < 3. Further, they can be also reused after the bisection of the interval. See also Sugiura and Sakurai [15]. 

In order to test the measurements of the 2S — 8S and 2S — 8D transitions, the frequencies of the 2S — 12D intervals have also been measured in Paris [49]. This transition yields complementary information, because the 12D levels are very sensitive to stray electric fields (the quadratic Stark shift varies as n7), and thus such a measurement provides a stringent test of Stark corrections to the Rydberg levels. The frequency difference between the 2S — Y2D transitions (A 750 nm, u 399.5 THz) and the LD/Rb standard laser is about 14.2 THz, i.e. half of the frequency of the CO2/OSO4 standard. This frequency difference is bisected with an optical divider [56] (see Fig. 5). The frequency chain (see Fig. 11) is split between the LPTF and the LKB the two optical fibers are used to transfer the CO2/OSO4 standard from the LPTF to the LKB, where the hydrogen transitions are observed. This chain includes an auxiliary source at 809 nm (u 370.5 THz) such that the laser frequencies satisfy the equations ... 

Tj+i should be closer to the better one of the points aj bj. That is the one with the smaller absolute value of Q. Tj+i is determined by bisection if the distance to the better point is larger than half of the interval length, because the bisection step is expected to result in a larger reduction of the interval length. 

The scan method may be a rather time-consuming procedure for polynomials whose roots lie in a large region of search. A variation of this search is the method of bisection that divides the interval of search by 2 and always retains that half of the search interval in which the change of sign has occurred. When the range of search has been narrowed down sufficiently, a more accurate search technique would then be applied within that step in order to refine the value of the root. 

Stability of the system is examined at the minimum, maximum, and midpoints of the range of search of K. . That half of the interval in which the change from negative to positive (stable to unstable system) occurs is retained by the bisection algorithm. This new interval is bisected again and the evaluation of the system stability is repeated, until the convergence criterion, which is < 0.001, is met. 

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