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MATHEMATICAL EXPERIENCE Doshimova M.A.
Doshimova Minayim A'senbaevna - Teacher of Mathematics, SECONDARY SCHOOL № 11, KEGEYLI, REPUBLIC OF UZBEKISTAN
Abstract: "the original method of extracting the square root", which can be attributed to the group of methods "digit by digit". A feature of this method, based on the property of the sum of the members of the arithmetic progression of odd numbers, is that at each cyclically repeated step one correct digit of the result is obtained.
Keywords: method, process, math, logarithm, roots, proportional, calculating, quadratic.
UDC 004.65
In the process of solving practical computational problems quite often there is a need to calculate the roots of varying degrees. Typically, when programming on a computer, standard library functions for calculating the logarithm and exponent or iterative methods are used for this purpose. The analytical methods of successive approximations, often used in calculating arithmetic roots, are universal in nature, but they have some drawbacks, one of which is the dependence of the calculation time on the value of the argument and on the choice of the first approximation. Significantly better characteristics when calculating, for example, the square root, are shown by the method described in the article "The original method of extracting the square root", which can be attributed to the group of methods "digit by digit". A feature of this method, based on the property of the sum of the members of the arithmetic progression of odd numbers, is that at each cyclically repeated step one correct digit of the result is obtained.
During the analysis of this method, the idea arose of extending its concept to the process of calculating the roots of the nth degree, as well as conducting a numerical study of the resulting algorithms. The basis for this approach is the fact that the sequence of odd numbers used to calculate the square root is not only an arithmetic progression with step 2, but, most importantly in this idea, also a series of first finite differences (hereinafter referred to as finite differences) for a quadratic functions with a single step of changing the argument.
We briefly mention the "digit by digit" method used for the "manual" calculation of the square root, which consists in the fact that the expression (20A + B) \ times B (here A is an integer a number made up of already found root digits, B is the desired next root digit, determined by selection), provided that the value of this expression does not exceed the current decrement. This method is described in detail in educational and reference books; now it has only historical significance.
Now we will develop a method for calculating the roots using finite differences.
Definition: For the function y = f (x) we denote \ Delta x - constant finite increment (step). Then \ Delta y = \ Delta f (x) = f (x + \ Delta x) - f (x) is called the finite first-order difference of the function y = f (x).
In order to avoid confusion in terms, the difference obtained by sequentially subtracting the final differences from the radical number will be called the remainder of the subtraction.
We write the expressions for the finite differences of the power function y = x A n, for example, with exponent 1; 2; 3; 5; with a single step.
(1) \begin{equation*} \Delta (xA1) = (x + 1)A1 - xA1 = 1 \end{equation*}
(2) \begin{equation* } \Delta (xA2) = (x + l)A2 - xA2 = 2x + 1 \end{equation*}
(3) \begin{equation*} \Delta (xA3) = (x + 1)a3 - xA3 = 3xA2 + 3x + 1 \end{equation*}
(4) \begin{equation*} \Delta (xA5) = (x + 1)a5 - xA5 = 5xA4 + 10xA3 + 10xA2 + 5x + 1 . \end{equation*}
To simplify further considerations, let us assume for now that the radical number is an integer. Along the way, we note that all computers, regardless of complexity, always process only integers at a low level. Consider, for methodological reasons, the process of computing a "degenerate" root of the 1st degree from, for example, 123. Since the finite difference (1) for the function y = x A 1 for any x is equal to 1, then the calculation of this root reduces to repeatedly subtracting unity from the root number. In our example, we can do 123 subtractions. In fact, the number of subtractions can be significantly reduced by subtracting the finite differences bitwise with a shift. To do this, we divide the radical number into groups, one digit in the group, since the root index is 1, and we subtract the final differences from them, starting from the leftmost group, until the remainder of the subtraction during processing of each group becomes less next deductible. After each cycle of subtractions, the number of subtractions will be the next digit in the root value. At its core, this algorithm is nothing but a well-known way of dividing the "corner" by one. The calculation of the roots of higher degrees is also based on the division algorithm, but only with the difference that now the value of the subtracted finite differences will be a variable, and it must be calculated using the formula corresponding to the degree of the root.
For the square root, the finite differences are calculated using expression (2), where, due to the discreteness of the change in the argument, we can make an integer change x = i
References
1. Shakhno K. U. Spravochnik po elementarnoy matematike. L., 1955.
2. Korn G., Korn T. Spravochnik po matematike dlya nauchnykh rabotnikov i inzhenerov.
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