Dichotomy method

Dichotomy in the translation from Greek means "consecutive division in two" or "bifurcation". Dichotomous division is quite successfully used in mathematics and logic for the classification of elements, and in philosophy and linguistics for the formation of subsections of one term mutually exclusive.

The method of dichotomy must be distinguished from ordinary division. For example, the word "person" can be divided into the concepts of "men" and "women", and can be divided into "men" and "not men." So, in the first case, the two concepts do not contradict each other, so there is no dichotomy. In the second case, "male" and "not male" are two definitions that contradict each other and do not overlap, and this is the definition of a dichotomy.

The dichotomy method is attractive by its simplicity, since there are always only two classes that are exhausted by the volume of the divisible concept. In other words, there is always proportionality in the dichotomous division. The next main property is the exclusion of each other by the members of the division, because each dividend set can only fall into one of the "b" or "not b" classes, and the division is performed only on one basis associated with the presence or absence of a certain attribute.

With all its merits, the dichotomy method also has a disadvantage, consisting in the uncertainty of that part of it that has a particle "not." For example, if all scientists are divided into mathematicians and not mathematicians, then concerning the second group there is a certain ambiguity. In addition to this shortcoming, there is one more, consisting in the difficult establishment of a concept that contradicts the first meaning, in terms of the degree of distance from the first pair.

As mentioned above, the dichotomy is often used as an auxiliary technique for the classification of any concepts. The dichotomy method is actively used to find the values of functions determined by a certain criterion (for example, a comparison to a maximum or a minimum).

Quite often, the method of the dichotomy is used unconsciously, the algorithm of which can be described literally step-by-step. For example, in the "Guess a Number" game, one of the players guesses a number in the range from 1 to 100, and the other makes attempts to guess it based on the "less" or "greater" clues of the first. If you think logically, 50 is always called the first number, and in the case of the conceited smaller one it is 25, the largest is 75. Therefore, at each stage the uncertainty of the number is reduced by half, and even the unlucky person will guess this unknown in about 7 attempts.

When using the dichotomy method in solving various equations, finding the correct solution is only possible when it is reliably known to find a single root at a given interval. This does not mean that the application of this method is possible to find the roots of only linear equations. When solving equations of a higher order using the method of half-division, it is first of all necessary to divide the roots along the segments. The process of separating them is carried out by finding the first and second derivatives of the function and equating the resulting equations to zero (f '(x) = 0, f' '(x) = 0). The next step is to determine the values of f (x) at the boundary and critical points. The result of all the calculations performed is the interval | a, b | on which the sign of the function changes and where f (a) * f (b) <0.

When considering a graphical method for solving an equation using a dichotomy, the decision algorithm is fairly simple. For example, there is a segment | a, b | within which there is one root x.

The first step is to calculate the mean algebraic x = (a + b) / 2. Further, the value of the function at a given point is calculated. If f (x) <0, then [a, x], otherwise - [x, b]. Thus, the interval is narrowed, as a result of which a certain sequence x is formed. The calculation is terminated upon reaching a difference of lesser error.