# Methods of computing square roots

In numerical analysis, a branch of mathematics, there are several square root algorithms or methods of computing the principal square root of a nonnegative real number. For the square roots of a negative or complex number, see below.

Finding ${\displaystyle {\sqrt {S}}}$ is the same as solving the equation ${\displaystyle f(x)=x^{2}-S=0\,\!}$. Therefore, any general numerical root-finding algorithm can be used. Newton's method, for example, reduces in this case to the so-called Babylonian method:

${\displaystyle x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}=x_{n}-{\frac {x_{n}^{2}-S}{2x_{n}}}={\frac {1}{2}}\left(x_{n}+{\frac {S}{x_{n}}}\right)}$

Generally, these methods yield approximate results. To get a higher precision for the root, a higher precision for the square is required and a larger number of steps must be calculated.

## Rough estimation

Many square root algorithms require an initial seed value. If the initial seed value is far away from the actual square root, the algorithm will be slowed down. It is therefore useful to have a rough estimate, which may be very inaccurate but easy to calculate. With ${\displaystyle S}$ expressed in scientific notation as ${\displaystyle a\times 10^{2n}}$ where ${\displaystyle 1\leq a<100}$ and n is an integer, the square root ${\displaystyle {\sqrt {S}}={\sqrt {a}}\times 10^{n}}$ can be estimated as

${\displaystyle {\sqrt {S}}\approx {\begin{cases}2\cdot 10^{n}&{\text{if }}a<10,\\6\cdot 10^{n}&{\text{if }}a\geq 10.\end{cases}}}$

The factors two and six are used because they approximate the geometric means of the lowest and highest possible values with the given number of digits: ${\displaystyle {\sqrt {{\sqrt {1}}\cdot {\sqrt {10}}}}={\sqrt[{4}]{10}}\approx 2\,}$ and ${\displaystyle {\sqrt {{\sqrt {10}}\cdot {\sqrt {100}}}}={\sqrt[{4}]{1000}}\approx 6\,}$.

When working in the binary numeral system (as computers do internally), by expressing ${\displaystyle S}$ as ${\displaystyle a\times 2^{2n}}$ where ${\displaystyle 0.1_{2}\leq a<10_{2}}$, the square root ${\displaystyle {\sqrt {S}}={\sqrt {a}}\times 2^{n}}$ can be estimated as ${\displaystyle {\sqrt {S}}\approx 2^{n}}$, since the geometric mean of the lowest and highest possible values is ${\displaystyle {\sqrt {{\sqrt {0.1_{2}}}\cdot {\sqrt {10_{2}}}}}={\sqrt[{4}]{1}}=1}$.

These approximations are useful to find better seeds for iterative algorithms, which results in faster convergence.

## Babylonian method

{{#invoke:Hatnote|hatnote}}Template:Main other

Graph charting the use of the Babylonian method for approximating the square root of 100 (10) using starting values x0 = 50, x0 = 1, and x0 = −5. Note that using a negative starting value yields the negative root.

Perhaps the first algorithm used for approximating ${\displaystyle {\sqrt {S}}}$ is known as the "Babylonian method", named after the Babylonians,[1] or "Hero's method", named after the first-century Greek mathematician Hero of Alexandria who gave the first explicit description of the method.[2] It can be derived from (but predates by 16 centuries) Newton's method. The basic idea is that if x is an overestimate to the square root of a non-negative real number S then ${\displaystyle \scriptstyle S/x\,}$ will be an underestimate and so the average of these two numbers may reasonably be expected to provide a better approximation (though the formal proof of that assertion depends on the inequality of arithmetic and geometric means that shows this average is always an overestimate of the square root, as noted in the article on square roots, thus assuring convergence).

More precisely, if ${\displaystyle x}$ is our initial guess of ${\displaystyle {\sqrt {S}}}$ and ${\displaystyle e}$ is the error in our estimate such that ${\displaystyle S=(x+e)^{2}}$, then we can expand the binomial and solve for

${\displaystyle e={\frac {S-x^{2}}{2x+e}}\approx {\frac {S-x^{2}}{2x}}}$ since ${\displaystyle e\ll x}$

Therefore, we can compensate for the error and update our old estimate as

${\displaystyle x:=x+e={\frac {S+x^{2}}{2x}}={\frac {x+{\frac {S}{x}}}{2}}}$

Since the computed error was not exact, this becomes our next best guess. The process of updating is iterated until desired accuracy is obtained. This is a quadratically convergent algorithm, which means that the number of correct digits of the approximation roughly doubles with each iteration. It proceeds as follows:

1. Begin with an arbitrary positive starting value x0 (the closer to the actual square root of S, the better).
2. Let xn+1 be the average of xn and S / xn (using the arithmetic mean to approximate the geometric mean).
3. Repeat step 2 until the desired accuracy is achieved.

It can also be represented as:

${\displaystyle x_{0}\approx {\sqrt {S}}.}$
${\displaystyle x_{n+1}={\frac {1}{2}}\left(x_{n}+{\frac {S}{x_{n}}}\right),}$
${\displaystyle {\sqrt {S}}=\lim _{n\to \infty }x_{n}.}$

This algorithm works equally well in the p-adic numbers, but cannot be used to identify real square roots with p-adic square roots; one can, for example, construct a sequence of rational numbers by this method that converges to +3 in the reals, but to −3 in the 2-adics.

### Example

To calculate ${\displaystyle {\sqrt {S}}}$, where S = 125348, to 6 significant figures, use the rough estimation method above to get

${\displaystyle x_{0}=6\cdot 10^{2}=600.000.\,}$
${\displaystyle x_{1}={\frac {1}{2}}\left(x_{0}+{\frac {S}{x_{0}}}\right)={\frac {1}{2}}\left(600.000+{\frac {125348}{600.000}}\right)=404.457.}$
${\displaystyle x_{2}={\frac {1}{2}}\left(x_{1}+{\frac {S}{x_{1}}}\right)={\frac {1}{2}}\left(404.457+{\frac {125348}{404.457}}\right)=357.187.}$
${\displaystyle x_{3}={\frac {1}{2}}\left(x_{2}+{\frac {S}{x_{2}}}\right)={\frac {1}{2}}\left(357.187+{\frac {125348}{357.187}}\right)=354.059.}$
${\displaystyle x_{4}={\frac {1}{2}}\left(x_{3}+{\frac {S}{x_{3}}}\right)={\frac {1}{2}}\left(354.059+{\frac {125348}{354.059}}\right)=354.045.}$
${\displaystyle x_{5}={\frac {1}{2}}\left(x_{4}+{\frac {S}{x_{4}}}\right)={\frac {1}{2}}\left(354.045+{\frac {125348}{354.045}}\right)=354.045.}$

### Convergence

Let the relative error in xn be defined by

${\displaystyle \varepsilon _{n}={\frac {x_{n}}{\sqrt {S}}}-1}$

and thus

${\displaystyle x_{n}={\sqrt {S}}\cdot (1+\varepsilon _{n}).}$

Then it can be shown that

${\displaystyle \varepsilon _{n+1}={\frac {\varepsilon _{n}^{2}}{2(1+\varepsilon _{n})}}}$

and thus that

${\displaystyle 0\leq \varepsilon _{n+2}\leq \min \left\{{\frac {\varepsilon _{n+1}^{2}}{2}},{\frac {\varepsilon _{n+1}}{2}}\right\}}$

and consequently that convergence is assured provided that x0 and S are both positive.

#### Worst case for convergence

If using the rough estimate above with the Babylonian method, then the worst cases areTemplate:Clarify:

{\displaystyle {\begin{aligned}S&=1;&x_{0}&=2;&x_{1}&=1.250;&\varepsilon _{1}&=0.250.\\S&=10;&x_{0}&=2;&x_{1}&=3.500;&\varepsilon _{1}&<0.107.\\S&=10;&x_{0}&=6;&x_{1}&=3.833;&\varepsilon _{1}&<0.213.\\S&=100;&x_{0}&=6;&x_{1}&=11.333;&\varepsilon _{1}&<0.134.\end{aligned}}}

Thus in any case,

${\displaystyle \varepsilon _{1}\leq 2^{-2}.\,}$
${\displaystyle \varepsilon _{2}<2^{-5}<10^{-1}.\,}$
${\displaystyle \varepsilon _{3}<2^{-11}<10^{-3}.\,}$
${\displaystyle \varepsilon _{4}<2^{-23}<10^{-6}.\,}$
${\displaystyle \varepsilon _{5}<2^{-47}<10^{-14}.\,}$
${\displaystyle \varepsilon _{6}<2^{-95}<10^{-28}.\,}$
${\displaystyle \varepsilon _{7}<2^{-191}<10^{-57}.\,}$
${\displaystyle \varepsilon _{8}<2^{-383}<10^{-115}.\,}$

Remember that rounding errors will slow the convergence. It is recommended to keep at least one extra digit beyond the desired accuracy of the xn being calculated to minimize round off error.

## Digit-by-digit calculation

This is a method to find each digit of the square root in a sequence. It is slower than the Babylonian method (if you have a calculator that can divide in one operation), but it has several advantages:

• It can be easier for manual calculations.
• Every digit of the root found is known to be correct, i.e., it does not have to be changed later.
• If the square root has an expansion that terminates, the algorithm terminates after the last digit is found. Thus, it can be used to check whether a given integer is a square number.
• The algorithm works for any base, and naturally, the way it proceeds depends on the base chosen.

Napier's bones include an aid for the execution of this algorithm. The shifting nth root algorithm is a generalization of this method.

### Basic principle

Suppose we are able to find the square root of N by expressing it as a sum of n positive numbers such that

${\displaystyle N=(a_{1}+a_{2}+a_{3}+\dotsb +a_{n})^{2}.}$

By repeatedly applying the basic identity

${\displaystyle (x+y)^{2}=x^{2}+2xy+y^{2},}$

the right-hand-side term can be expanded as

{\displaystyle {\begin{aligned}&(a_{1}+a_{2}+a_{3}+\dotsb +a_{n})^{2}\\=&\,a_{1}^{2}+2a_{1}a_{2}+a_{2}^{2}+2(a_{1}+a_{2})a_{3}+a_{3}^{2}+\dotsb +a_{n-1}^{2}+2\left(\sum _{i=1}^{n-1}a_{i}\right)a_{n}+a_{n}^{2}\\=&\,a_{1}^{2}+[2a_{1}+a_{2}]a_{2}+[2(a_{1}+a_{2})+a_{3}]a_{3}+\dotsb +\left[2\left(\sum _{i=1}^{n-1}a_{i}\right)+a_{n}\right]a_{n}.\end{aligned}}}

This expression allows us to find the square root by sequentially guessing the values of ${\displaystyle a_{i}}$s. Suppose that the numbers ${\displaystyle a_{1},\ldots ,a_{m-1}}$ have already been guessed, then the m-th term of the right-hand-side of above summation is given by ${\displaystyle Y_{m}=[2P_{m-1}+a_{m}]a_{m},}$ where ${\displaystyle P_{m-1}=\sum _{i=1}^{m-1}a_{i}}$ is the approximate square root found so far. Now each new guess ${\displaystyle a_{m}}$ should satisfy the recursion

${\displaystyle X_{m}=X_{m-1}-Y_{m},}$

such that ${\displaystyle X_{m}\geq 0}$ for all ${\displaystyle 1\leq m\leq n,}$ with initialization ${\displaystyle X_{0}=N.}$ When ${\displaystyle X_{n}=0,}$ the exact square root has been found; if not, then the sum of ${\displaystyle a_{i}}$s gives a suitable approximation of the square root, with ${\displaystyle X_{n}}$ being the approximation error.

For example, in the decimal number system we have

${\displaystyle N=(a_{1}\cdot 10^{n-1}+a_{2}\cdot 10^{n-2}+\cdots +a_{n-1}\cdot 10+a_{n})^{2},}$

where ${\displaystyle 10^{n-i}}$ are place holders and the coefficients ${\displaystyle a_{i}\in \{0,1,2,\ldots ,9\}}$. At any m-th stage of the square root calculation, the approximate root found so far, ${\displaystyle P_{m-1}}$ and the summation term ${\displaystyle Y_{m}}$ are given by

${\displaystyle P_{m-1}=\sum _{i=1}^{m-1}a_{i}\cdot 10^{n-i}=10^{n-m+1}\sum _{i=1}^{m-1}a_{i}\cdot 10^{m-i-1},}$
${\displaystyle Y_{m}=[2P_{m-1}+a_{m}\cdot 10^{n-m}]a_{m}\cdot 10^{n-m}=[20\sum _{i=1}^{m-1}a_{i}\cdot 10^{m-i-1}+a_{m}]a_{m}\cdot 10^{2(n-m)}.}$

Here since the place value of ${\displaystyle Y_{m}}$ is an even power of 10, we only need to work with the pair of most significant digits of the remaining term ${\displaystyle X_{m-1}}$ at any m-th stage. The section below codifies this procedure. It is obvious that a similar method can be used to compute the square root in number systems other than the decimal number system. For instance, finding the digit-by-digit square root in the binary number system is quite efficient since the value of ${\displaystyle a_{i}}$ is searched from a smaller set of binary digits {0,1}. This makes the computation faster since at each stage the value of ${\displaystyle Y_{m}}$ is either ${\displaystyle Y_{m}=0}$ for ${\displaystyle a_{m}=0}$ or ${\displaystyle Y_{m}=2P_{m-1}+1}$ for ${\displaystyle a_{m}=1}$. The fact that we have only two possible options for ${\displaystyle a_{m}}$ also makes the process of deciding the value of ${\displaystyle a_{m}}$ at m-th stage of calculation easier. This is because we only need to check if ${\displaystyle Y_{m}\leq X_{m-1}}$ for ${\displaystyle a_{m}=1.}$ If this condition is satisfied, then we take ${\displaystyle a_{m}=1}$; if not then ${\displaystyle a_{m}=0.}$ Also, the fact that multiplication by 2 is done by left bit-shifts helps in the computation.

### Decimal (base 10)

Write the original number in decimal form. The numbers are written similar to the long division algorithm, and, as in long division, the root will be written on the line above. Now separate the digits into pairs, starting from the decimal point and going both left and right. The decimal point of the root will be above the decimal point of the square. One digit of the root will appear above each pair of digits of the square.

Beginning with the left-most pair of digits, do the following procedure for each pair:

1. Starting on the left, bring down the most significant (leftmost) pair of digits not yet used (if all the digits have been used, write "00") and write them to the right of the remainder from the previous step (on the first step, there will be no remainder). In other words, multiply the remainder by 100 and add the two digits. This will be the current value c.
2. Find p, y and x, as follows:
• Let p be the part of the root found so far, ignoring any decimal point. (For the first step, p = 0).
• Determine the greatest digit x such that ${\displaystyle x(20p+x)\leq c}$. We will use a new variable y = x(20p + x).
• Note: 20p + x is simply twice p, with the digit x appended to the right).
• Note: You can find x by guessing what c/(20·p) is and doing a trial calculation of y, then adjusting x upward or downward as necessary.
• Place the digit ${\displaystyle x}$ as the next digit of the root, i.e., above the two digits of the square you just brought down. Thus the next p will be the old p times 10 plus x.
3. Subtract y from c to form a new remainder.
4. If the remainder is zero and there are no more digits to bring down, then the algorithm has terminated. Otherwise go back to step 1 for another iteration.

#### Examples

Find the square root of 152.2756.

          1  2. 3  4
/
\/  01 52.27 56

01                   1*1 <= 1 < 2*2                 x = 1
01                     y = x*x = 1*1 = 1
00 52                22*2 <= 52 < 23*3              x = 2
00 44                  y = (20+x)*x = 22*2 = 44
08 27             243*3 <= 827 < 244*4           x = 3
07 29               y = (240+x)*x = 243*3 = 729
98 56          2464*4 <= 9856 < 2465*5        x = 4
98 56            y = (2460+x)*x = 2464*4 = 9856
00 00          Algorithm terminates: Answer is 12.34


Find the square root of 2.

          1. 4  1  4  2
/
\/  02.00 00 00 00

02                  1*1 <= 2 < 2*2                 x = 1
01                    y = x*x = 1*1 = 1
01 00               24*4 <= 100 < 25*5             x = 4
00 96                 y = (20+x)*x = 24*4 = 96
04 00            281*1 <= 400 < 282*2           x = 1
02 81              y = (280+x)*x = 281*1 = 281
01 19 00         2824*4 <= 11900 < 2825*5       x = 4
01 12 96           y = (2820+x)*x = 2824*4 = 11296
06 04 00      28282*2 <= 60400 < 28283*3     x = 2
The desired precision is achieved:
The square root of 2 is about 1.4142


### Binary numeral system (base 2)

Inherent to digit-by-digit algorithms is a search and test step: find a digit, ${\displaystyle \,e}$, when added to the right of a current solution ${\displaystyle \,r}$, such that ${\displaystyle \,(r+e)\cdot (r+e)\leq x}$, where ${\displaystyle \,x}$ is the value for which a root is desired. Expanding: ${\displaystyle \,r\cdot r+2re+e\cdot e\leq x}$. The current value of ${\displaystyle \,r\cdot r}$—or, usually, the remainder—can be incrementally updated efficiently when working in binary, as the value of ${\displaystyle \,e}$ will have a single bit set (a power of 2), and the operations needed to compute ${\displaystyle \,2\cdot r\cdot e}$ and ${\displaystyle \,e\cdot e}$ can be replaced with faster bit shift operations.

#### Example

Here we obtain the square root of 81, which when converted into binary gives 1010001. The numbers in the left column gives the option between that number or zero to be used for subtraction at that stage of computation. The final answer is 1001, which in decimal is 9.

             1 0 0 1
---------
√ 1010001

      1      1
1
---------
101     01
0
--------
1001     100
0
--------
10001    10001
10001
-------
0


This gives rise to simple computer implementations:[3]

short isqrt(short num) {
short res = 0;
short bit = 1 << 14; // The second-to-top bit is set: 1 << 30 for 32 bits

// "bit" starts at the highest power of four <= the argument.
while (bit > num)
bit >>= 2;

while (bit != 0) {
if (num >= res + bit) {
num -= res + bit;
res = (res >> 1) + bit;
}
else
res >>= 1;
bit >>= 2;
}
return res;
}


Using the notation above, the variable "bit" corresponds to ${\displaystyle e_{m}^{2}}$ which is ${\displaystyle (2^{m})^{2}=4^{m}}$, the variable "res" is equal to ${\displaystyle 2re_{m}}$, and the variable "num" is equal to the current ${\displaystyle X_{m}}$ which is the difference of the number we want the square root of and the square of our current approximation with all bits set up to ${\displaystyle 2^{m+1}}$. Thus in the first loop, we want to find the highest power of 4 in "bit" to find the highest power of 2 in ${\displaystyle e}$. In the second loop, if num is greater than res + bit, then ${\displaystyle X_{m}}$ is greater than ${\displaystyle 2re_{m}+e_{m}^{2}}$ and we can subtract it. The next line, we want to add ${\displaystyle e_{m}}$ to ${\displaystyle r}$ which means we want to add ${\displaystyle 2e_{m}^{2}}$ to ${\displaystyle 2re_{m}}$ so we want res = res + bit<<1. Then update ${\displaystyle e_{m}}$ to ${\displaystyle e_{m-1}}$ inside res which involves dividing by 2 or another shift to the right. Combining these 2 into one line leads to res = res>>1 + bit. If ${\displaystyle X_{m}}$ isn't greater than ${\displaystyle 2re_{m}+e_{m}^{2}}$ then we just update ${\displaystyle e_{m}}$ to ${\displaystyle e_{m-1}}$ inside res and divide it by 2. Then we update ${\displaystyle e_{m}}$ to ${\displaystyle e_{m-1}}$ in bit by dividing it by 4. The final iteration of the 2nd loop has bit equal to 1 and will cause update of ${\displaystyle e}$ to run one extra time removing the factor of 2 from res making it our integer approximation of the root.

Faster algorithms, in binary and decimal or any other base, can be realized by using lookup tables—in effect trading more storage space for reduced run time.[4]

## Exponential identity

Pocket calculators typically implement good routines to compute the exponential function and the natural logarithm, and then compute the square root of S using the identity found using the properties of logarithms (${\displaystyle \ln x^{n}=n\ln x}$) and exponentials (${\displaystyle e^{\ln x}=x}$):

${\displaystyle {\sqrt {S}}=e^{{\frac {1}{2}}\ln S}.}$

The denominator in the fraction corresponds to the nth root. In the case above the denominator is 2, hence the equation specifies that the square root is to be found. The same identity is used when computing square roots with logarithm tables or slide rules.

## Bakhshali approximation

This method for finding an approximation to a square root was described in an ancient Indian mathematical manuscript called the Bakhshali manuscript. It is equivalent to two iterations of the Babylonian method beginning with N. The original presentation goes as follows: To calculate ${\displaystyle {\sqrt {S}}}$, let N2 be the nearest perfect square to S. Then, calculate:

${\displaystyle d=S-N^{2}\,\!}$
${\displaystyle P={\frac {d}{2N}}}$
${\displaystyle A=N+P\,\!}$
${\displaystyle {\sqrt {S}}\approx A-{\frac {P^{2}}{2A}}}$

This can be also written as:

${\displaystyle {\sqrt {S}}\approx N+{\frac {d}{2N}}-{\frac {d^{2}}{8N^{3}+4Nd}}={\frac {8N^{4}+8N^{2}d+d^{2}}{8N^{3}+4Nd}}={\frac {N^{4}+6N^{2}S+S^{2}}{4N^{3}+4NS}}={\frac {N^{2}(N^{2}+6S)+S^{2}}{4N(N^{2}+S)}}}$

### Example

${\displaystyle N=3\,\!}$
${\displaystyle d=9.2345-3^{2}=0.2345\,\!}$
${\displaystyle P={\frac {0.2345}{2\times 3}}=0.0391}$
${\displaystyle A=3+0.0391=3.0391\,\!}$
${\displaystyle {\sqrt {9.2345}}\approx 3.0391-{\frac {0.0391^{2}}{2\times 3.0391}}\approx 3.0388}$

## Vedic duplex method for extracting a square root

The Vedic duplex method from the book 'Vedic Mathematics' is a variant of the digit by digit method for calculating the square root.[5] The duplex is the square of the central digit plus double the cross-product of digits equidistant from the center. The duplex is computed from the quotient digits (square root digits) computed thus far, but after the initial digits. The duplex is subtracted from the dividend digit prior to the second subtraction for the product of the quotient digit times the divisor digit. For perfect squares the duplex and the dividend will get smaller and reach zero after a few steps. For non-perfect squares the decimal value of the square root can be calculated to any precision desired. However, as the decimal places proliferate, the duplex adjustment gets larger and longer to calculate. The duplex method follows the Vedic ideal for an algorithm, one-line, mental calculation. It is flexible in choosing the first digit group and the divisor. Small divisors are to be avoided by starting with a larger initial group.

### Basic Principle

We proceed as with the digit-by-digit calculation by assuming that we want to express a number N as a square of the sum of n positive numbers as

${\displaystyle N=(a_{0}+a_{1}+\ldots +a_{n-1})^{2}}$
${\displaystyle =a_{0}^{2}+2a_{0}\sum _{i=1}^{n-1}a_{i}+a_{1}^{2}+2a_{1}\sum _{i=2}^{n-1}a_{i}+\cdots +a_{n-1}^{2}.}$

Define divisor as ${\displaystyle q=2a_{0}}$ and the duplex for a sequence of m numbers as

${\displaystyle d_{m}={\begin{cases}a_{\lceil m/2\rceil }^{2}+\sum _{i=1}^{\lfloor m/2\rfloor }2a_{i}a_{m-i+1}&{\mbox{for}}\;m\;{\mbox{odd}}\\\sum _{i=1}^{m/2}2a_{i}a_{m-i+1}&{\mbox{for}}\;m\;{\mbox{even}}.\\\end{cases}}}$

Thus, we can re-express the above identity in terms of the divisor and the duplexes as

${\displaystyle N-a_{0}^{2}=\sum _{i=1}^{n-1}(qa_{i}+d_{i}).}$

Now the computation can proceed by recursively guessing the values of ${\displaystyle a_{m}}$ so that

${\displaystyle X_{m}=X_{m-1}-qa_{m}-d_{m},}$

such that ${\displaystyle X_{m}\geq 0}$ for all ${\displaystyle 1\leq m\leq n-1}$, with initialization ${\displaystyle X_{0}=N-a_{0}^{2}.}$ When ${\displaystyle X_{m}=0}$ the algorithm terminates and the sum of ${\displaystyle a_{i}}$s give the square root. The method is more similar to long division where ${\displaystyle X_{m-1}}$ is the dividend and ${\displaystyle X_{m}}$ is the remainder.

For the case of decimal numbers, if

${\displaystyle N=(a_{0}\cdot 10^{n-1}+a_{1}\cdot 10^{n-2}+\cdots +a_{n-2}\cdot 10+a_{n-1})^{2}}$

where ${\displaystyle a_{i}\in \{0,1,2,\ldots ,9\}}$, then the initiation ${\displaystyle X_{0}=N-a_{0}^{2}\cdot 10^{2(n-1)}}$ and the divisor will be ${\displaystyle q=2a_{0}\cdot 10^{n-1}}$. Also the product at any m-th stage will be ${\displaystyle qa_{m}\cdot 10^{n-m-1}=2a_{0}a_{m}\cdot 10^{2n-m-2}}$ and the duplexes will be ${\displaystyle d_{m}={\tilde {d}}_{m}\cdot 10^{2n-m-3}}$, where ${\displaystyle {\tilde {d}}_{m}}$ are the duplexes of the sequence ${\displaystyle a_{1},a_{2},\ldots ,a_{m}}$. At any m-th stage, we see that the place value of the duplex ${\displaystyle {\tilde {d}}_{m}}$ is one less than the product ${\displaystyle 2a_{0}a_{m}}$. Thus, in actual calculations it is customary to subtract the duplex value of the m-th stage at (m+1)-th stage. Also, unlike the previous digit-by-digit square root calculation, where at any given m-th stage, the calculation is done by taking the most significant pair of digits of the remaining term ${\displaystyle X_{m-1}}$, the duplex method uses only a single most significant digit of ${\displaystyle X_{m-1}}$.

In other words, to calculate the duplex of a number, double the product of each pair of equidistant digits plus the square of the center digit (of the digits to the right of the colon).

Number => Calculation = Duplex
3 ==> 32 = 9
14 ==>2(1·4) = 8
574 ==> 2(5·4) + 72 = 89
1,421 ==> 2(1·1) + 2(4·2) = 2 + 16 = 18
10,523 ==> 2(1·3) + 2(0·2) + 52 = 6+0+25 = 31
406,739 ==> 2(4·9)+ 2(0·3)+ 2(6·7) = 72+0+84  = 156



In a square root calculation the quotient digit set increases incrementally for each step.

### Example

Consider the perfect square 2809 = 532. Use the duplex method to find the square root of 2,809.

• Set down the number in groups of two digits.
• Define a divisor, a dividend and a quotient to find the root.
• Given 2809. Consider the first group, 28.
• Find the nearest perfect square below that group.
• The root of that perfect square is the first digit of our root.
• Since 28 > 25 and 25 = 52, take 5 as the first digit in the square root.
• For the divisor take double this first digit (2 · 5), which is 10.
• Next, set up a division framework with a colon.
• 28: 0 9 is the dividend and 5: is the quotient. (Note: the quotient should always be a single digit number, and it should be such that the dividend in the next stage is non-negative.)
• Put a colon to the right of 28 and 5 and keep the colons lined up vertically. The duplex is calculated only on quotient digits to the right of the colon.
• Calculate the remainder. 28: minus 25: is 3:.
• Append the remainder on the left of the next digit to get the new dividend.
• Here, append 3 to the next dividend digit 0, which makes the new dividend 30. The divisor 10 goes into 30 just 3 times. (No reserve needed here for subsequent deductions.)
• Repeat the operation.
• The zero remainder appended to 9. Nine is the next dividend.
• This provides a digit to the right of the colon so deduct the duplex, 32 = 9.
• Subtracting this duplex from the dividend 9, a zero remainder results.
• Ten into zero is zero. The next root digit is zero. The next duplex is 2(3·0) = 0.
• The dividend is zero. This is an exact square root, 53.
Find the square root of 2809.
Set down the number in groups of two digits.
The number of groups gives the number of whole digits in the root.
Put a colon after the first group, 28, to separate it.
From the first group, 28, obtain the divisor, 10, since
28>25=52 and by doubling this first root, 2x5=10.
Gross dividend:     28:  0  9. Using mental math:
Divisor: 10)     3  0   Square: 10)  28:  30  9
Duplex, Deduction:     25: xx 09  Square root:  5:   3. 0
Dividend:         30 00
Remainder:      3: 00 00
Square Root, Quotient:      5:  3. 0


## A two-variable iterative method

This method is applicable for finding the square root of ${\displaystyle 0 and converges best for ${\displaystyle S\approx 1}$. This, however, is no real limitation for a computer based calculation, as in base 2 floating point and fixed point representations, it is trivial to multiply ${\displaystyle S\,\!}$ by an integer power of 4, and therefore ${\displaystyle {\sqrt {S}}}$ by the corresponding power of 2, by changing the exponent or by shifting, respectively. Therefore, ${\displaystyle S\,\!}$ can be moved to the range ${\displaystyle {\frac {1}{2}}\leq S<2}$. Moreover, the following method does not employ general divisions, but only additions, subtractions, multiplications, and divisions by powers of two, which are again trivial to implement. A disadvantage of the method is that numerical errors accumulate, in contrast to single variable iterative methods such as the Babylonian one.

The initialization step of this method is

${\displaystyle a_{0}=S\,\!}$
${\displaystyle c_{0}=S-1\,\!}$

${\displaystyle a_{n+1}=a_{n}-a_{n}c_{n}/2\,\!}$
${\displaystyle c_{n+1}=c_{n}^{2}(c_{n}-3)/4\,\!}$

Note that the convergence of ${\displaystyle c_{n}\,\!}$, and therefore also of ${\displaystyle a_{n}\,\!}$, is quadratic.

The proof of the method is rather easy. First, rewrite the iterative definition of ${\displaystyle c_{n}\,\!}$ as

${\displaystyle 1+c_{n+1}=(1+c_{n})(1-c_{n}/2)^{2}\,\!}$.

Then it is straightforward to prove by induction that

${\displaystyle S(1+c_{n})=a_{n}^{2}}$

and therefore the convergence of ${\displaystyle a_{n}\,\!}$ to the desired result ${\displaystyle {\sqrt {S}}}$ is ensured by the convergence of ${\displaystyle c_{n}\,\!}$ to 0, which in turn follows from ${\displaystyle -1.

This method was developed around 1950 by M. V. Wilkes, D. J. Wheeler and S. Gill[6] for use on EDSAC, one of the first electronic computers.[7] The method was later generalized, allowing the computation of non-square roots.[8]

## Iterative methods for reciprocal square roots

The following are iterative methods for finding the reciprocal square root of S which is ${\displaystyle 1/{\sqrt {S}}}$. Once it has been found, find ${\displaystyle {\sqrt {S}}}$ by simple multiplication: ${\displaystyle {\sqrt {S}}=S\cdot (1/{\sqrt {S}})}$. These iterations involve only multiplication, and not division. They are therefore faster than the Babylonian method. However, they are not stable. If the initial value is not close to the reciprocal square root, the iterations will diverge away from it rather than converge to it. It can therefore be advantageous to perform an iteration of the Babylonian method on a rough estimate before starting to apply these methods.

${\displaystyle x_{n+1}={\frac {x_{n}}{2}}\cdot (3-S\cdot x_{n}^{2}).}$
${\displaystyle y_{n}=S\cdot x_{n}^{2}\,,\!}$
${\displaystyle x_{n+1}={\frac {x_{n}}{8}}\cdot (15-y_{n}\cdot (10-3\cdot y_{n})).}$

### Goldschmidt’s algorithm

Some computers use Goldschmidt's algorithm to simultaneously calculate ${\displaystyle {\sqrt {S}}}$ and ${\displaystyle 1/{\sqrt {S}}}$. Goldschmidt's algorithm finds ${\displaystyle {\sqrt {S}}}$ faster than Newton-Raphson iteration on a computer with a fused multiply–add instruction and either a pipelined floating point unit or two independent floating-point units. Two ways of writing Goldschmidt's algorithm are:[9]

${\displaystyle b_{0}=S}$
${\displaystyle Y_{0}\approx 1/{\sqrt {S}}}$ (typically using a table lookup)
${\displaystyle y_{0}=Y_{0}}$
${\displaystyle x_{0}=Sy_{0}}$

Each iteration:

${\displaystyle b_{n+1}=b_{n}Y_{n}^{2}}$
${\displaystyle Y_{n+1}=(3-b_{n})/2}$
${\displaystyle x_{n+1}=x_{n}Y_{n+1}}$
${\displaystyle y_{n+1}=y_{n}Y_{n+1}}$

until ${\displaystyle b_{i}}$ is sufficiently close to 1, or a fixed number of iterations.

which causes

${\displaystyle {\sqrt {S}}=\lim _{n\to \infty }x_{n}.}$
${\displaystyle 1/{\sqrt {S}}=\lim _{n\to \infty }y_{n}.}$

Goldschmidt's equation can be rewritten as:

${\displaystyle y_{0}\approx 1/{\sqrt {S}}}$ (typically using a table lookup)
${\displaystyle x_{0}=Sy_{0}}$
${\displaystyle h_{0}=y_{0}/2}$

Each iteration: (All 3 operations in this loop are in the form of a fused multiply–add.)

${\displaystyle r_{n}=(1/2)-x_{n}h_{n}}$
${\displaystyle x_{n+1}=x_{n}+x_{n}r_{n}}$
${\displaystyle h_{n+1}=h_{n}+h_{n}r_{n}}$

until ${\displaystyle r_{i}}$ is sufficiently close to 0, or a fixed number of iterations.

which causes

${\displaystyle {\sqrt {S}}=\lim _{n\to \infty }x_{n}.}$
${\displaystyle 1/{\sqrt {S}}=\lim _{n\to \infty }2h_{n}.}$

## Taylor series

If N is an approximation to ${\displaystyle {\sqrt {S}}}$, a better approximation can be found by using the Taylor series of the square root function:

${\displaystyle {\sqrt {N^{2}+d}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n)!d^{n}}{(1-2n)n!^{2}4^{n}N^{2n-1}}}=N+{\frac {d}{2N}}-{\frac {d^{2}}{8N^{3}}}+{\frac {d^{3}}{16N^{5}}}-{\frac {5d^{4}}{128N^{7}}}+\cdots }$

As an iterative method, the order of convergence is equal to the number of terms used. With 2 terms, it is identical to the Babylonian method; With 3 terms, each iteration takes almost as many operations as the Bakhshali approximation, but converges more slowly. Therefore, this is not a particularly efficient way of calculation. To maximize the rate of convergence, choose N so that ${\displaystyle {\frac {|d|}{N^{2}}}\,}$ is as small as possible.

## Other methods

Other methods are less efficient than the ones presented above.

A completely different method for computing the square root is based on the CORDIC algorithm, which uses only very simple operations (addition, subtraction, bitshift and table lookup, but no multiplication). The square root of S may be obtained as the output ${\displaystyle x_{n}}$ using the hyperbolic coordinate system in vectoring mode, with the following initialization:[10]

${\displaystyle x_{0}=S+1}$
${\displaystyle y_{0}=S-1}$
${\displaystyle \omega _{0}=0}$

## Continued fraction expansion

Quadratic irrationals (numbers of the form ${\displaystyle {\frac {a+{\sqrt {b}}}{c}}}$, where a, b and c are integers), and in particular, square roots of integers, have periodic continued fractions. Sometimes what is desired is finding not the numerical value of a square root, but rather its continued fraction expansion. The following iterative algorithm can be used for this purpose (S is any natural number that is not a perfect square):

${\displaystyle m_{0}=0\,\!}$
${\displaystyle d_{0}=1\,\!}$
${\displaystyle a_{0}=\left\lfloor {\sqrt {S}}\right\rfloor \,\!}$
${\displaystyle m_{n+1}=d_{n}a_{n}-m_{n}\,\!}$
${\displaystyle d_{n+1}={\frac {S-m_{n+1}^{2}}{d_{n}}}\,\!}$
${\displaystyle a_{n+1}=\left\lfloor {\frac {{\sqrt {S}}+m_{n+1}}{d_{n+1}}}\right\rfloor =\left\lfloor {\frac {a_{0}+m_{n+1}}{d_{n+1}}}\right\rfloor \!.}$

Notice that mn, dn, and an are always integers. The algorithm terminates when this triplet is the same as one encountered before. The algorithm can also terminate on ai when ai = 2 a0,[11] which is easier to implement.

The expansion will repeat from then on. The sequence [a0; a1, a2, a3, …] is the continued fraction expansion:

${\displaystyle {\sqrt {S}}=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+\,\ddots }}}}}}}$

### Example, square root of 114 as a continued fraction

Begin with m0 = 0; d0 = 1; and a0 = 10 (102 = 100 and 112 = 121 > 114 so 10 chosen).

{\displaystyle {\begin{aligned}{\sqrt {114}}&={\frac {{\sqrt {114}}+0}{1}}=10+{\frac {{\sqrt {114}}-10}{1}}=10+{\frac {({\sqrt {114}}-10)({\sqrt {114}}+10)}{{\sqrt {114}}+10}}\\&=10+{\frac {114-100}{{\sqrt {114}}+10}}=10+{\frac {1}{\frac {{\sqrt {114}}+10}{14}}}.\end{aligned}}}
${\displaystyle m_{1}=d_{0}\cdot a_{0}-m_{0}=1\cdot 10-0=10\,.}$
${\displaystyle d_{1}={\frac {S-m_{1}^{2}}{d_{0}}}={\frac {114-10^{2}}{1}}=14\,.}$
${\displaystyle a_{1}=\left\lfloor {\frac {a_{0}+m_{1}}{d_{1}}}\right\rfloor =\left\lfloor {\frac {10+10}{14}}\right\rfloor =\left\lfloor {\frac {20}{14}}\right\rfloor =1\,.}$

So, m1 = 10; d1 = 14; and a1 = 1.

${\displaystyle {\frac {{\sqrt {114}}+10}{14}}=1+{\frac {{\sqrt {114}}-4}{14}}=1+{\frac {114-16}{14({\sqrt {114}}+4)}}=1+{\frac {1}{\frac {{\sqrt {114}}+4}{7}}}.}$

Next, m2 = 4; d2 = 7; and a2 = 2.

${\displaystyle {\frac {{\sqrt {114}}+4}{7}}=2+{\frac {{\sqrt {114}}-10}{7}}=2+{\frac {14}{7({\sqrt {114}}+10)}}=2+{\frac {1}{\frac {{\sqrt {114}}+10}{2}}}.}$
${\displaystyle {\frac {{\sqrt {114}}+10}{2}}=10+{\frac {{\sqrt {114}}-10}{2}}=10+{\frac {14}{2({\sqrt {114}}+10)}}=10+{\frac {1}{\frac {{\sqrt {114}}+10}{7}}}.}$