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goldschmidt division
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Goldschmidt division

 

Goldschmidt (after Robert Elliott Goldschmidt) division uses an iterative process to repeatedly multiply both the dividend and divisor by a common factor Fi to converge the divisor, D, to 1 as the dividend, N, converges to the quotient Q:

 

Q = \frac{N}{D} \frac{F_1}{F_1} \frac{F_2}{F_2}  \frac{F_{\ldots}}{F_{\ldots}}.

The steps for Goldschmidt division are:

         Generate an estimate for the multiplication factor Fi .

         Multiply the dividend and divisor by Fi .

         If the divisor is sufficiently close to 1, return the dividend, otherwise, loop to step 1.

         Assuming N/D has been scaled so that 0 < D < 1, each Fi is based on D:

Fi + 1 = 2 − Di.

         Multiplying the dividend and divisor by the factor yields:

. \frac{N_{i+1}}{D_{i+1}} = \frac{N_i}{D_i}\frac{F_i}{F_i}()

After a sufficient number of iterations k: Q = Nk.

The Goldschmidt method is used in AMD Athlon CPUs and later models

 

 

 ILLUSTRATIONS

 

Following are some examples that describe the rule of

 

5.2.1. Example 1: Division of 2 digit number by

 

a. Division of 12 by 9:

 

12 will be divided into 2 parts by a stroke.

         1|2 9 = Q= 1

                    R= 3

The Quotient will be the first digit of the Dividend

The Remainder will be the sum of the 2 digits of the Dividend.

3 digits

 

b. Division of 113 by 9:

 

We will divide 113 into 2 parts so that the units place is in one division the the other 2 digits in the other division.

    11|3 9 = Q=12

                      R=5

In the Quotient we bring down the first digit of the dividend as the 1st digit of the quotient and then add the first digit of the quotient to the second digit of the dividend to get the second digit of the quotient. In the remainder we add the second digit of the quotient to the third digit of the dividend.

 

5.2.2. Example 2: Division by single digit divisor.

 

a. Division of 111 by 8 :

 

111 will be divided into 2 parts by a diagonal stroke.

http://www.angelfire.com/pq2/vedicmaths/div2.h1.gif

2`is the complement(10-8) of 8(written under 8). 2 written below the 1 before the stroke is 2(complement)x1(first digit). The 6 written below the 1 after the stroke is 2(complement)x3(complement + first digit). The first digit of the quotient 13 is 1 i.e. the first digit of the dividend. The second digit, 3, is 1(second digit)+2(2x1 mentioned above).

The remainder, 7, is 6(2x3 mentioned above)+1(last digit of dividend).

 

5.2.3. Example 3: Division by 2 digit divisors.

 

a. Division of 123 by 88 :

 

123 will be divided into 2 parts by a diagonal stroke. Now leaving 2 digits after the diagonal stroke since there are 2 digits in the divisor.

 

http://www.angelfire.com/pq2/vedicmaths/div3.h1.gif

 

12 written under 88 is the complement of 88(100-88).

1 the quotient is the first digit of the dividend.

12(complement x first digit) written under 23 is added to 23(last 2 digits of dividend) to give the remainder.