Hi. I only just recently found out about an algorithm for calculating the square roots of a number.(adsbygoogle = window.adsbygoogle || []).push({});

Lets say i want to evaluate [tex]\sqrt {n}[/tex]. I can make an approximation by inspection, and say [tex]\sqrt n \approx \frac{a}{b}[/tex]. Now, using this approximation, i can write:

[tex]

\left[ {\begin{array}{*{20}c}

1 & n \\

1 & 1 \\

\end{array}} \right]\left[ {\begin{array}{*{20}c}

a \\

b \\

\end{array}} \right] = \left[ {\begin{array}{*{20}c}

{a + bn} \\

{a + b} \\

\end{array}} \right]

[/tex]

Treating the resultant matrix as a fraction ([tex]

\frac{{a + bn}}{{a + b}}

[/tex]

), i have a better approximation of [tex]\sqrt {n}[/tex]. If i keep repeating this method with the new approximation, over and over again, i get a more accurate answer. So the next step would be:

[tex]

\left[ {\begin{array}{*{20}c}

1 & n \\

1 & 1 \\

\end{array}} \right]\left[ {\begin{array}{*{20}c}

{a + bn} \\

{a + b} \\

\end{array}} \right] = \left[ {\begin{array}{*{20}c}

{a + an + 2bn} \\

{2a + b + bn} \\

\end{array}} \right]

[/tex]

And [tex]

\frac{{a + an + 2bn}}{{2a + b + bn}}[/tex] would be an even better approximation to [tex]\sqrt {n}[/tex].

Even if the starting approximation is way off, if alot of iterations are complete, the answer will still be accurate.

Im just wondering, why does this method work? Is there a name for this method, or anywhere i can look to find more information?

Thanks,

Dan.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Calculating Square Roots - Matrix Algorithm

**Physics Forums | Science Articles, Homework Help, Discussion**