Simple Ways to Compute Square Root

Published January 3, 2008 math 3 Comments

Here are some methods to compute the square root of a positive number $c$ . Nothing innovative; but classic.

If $c>1$ , then

$\sqrt{c}=\sqrt{\frac{1}{1-q}}=1+\frac{1}{2}q+\frac{3}{8}q^2+\cdots+(-1)^n{-1/2 \choose n}q^n+\cdots$

If $c<1$ , then

$\sqrt{c}=\sqrt{1-q}=1-\frac{1}{2}q+\frac{3}{8}q^2+\cdots+(-1)^n{1/2 \choose n}q^n+\cdots$

Convergence: linear.

$x_{n+1}\leftarrow\frac{1}{2}\left(x_n+\frac{c}{x_n}\right)$

Convergence: quadratic.

$x_{n+1}\leftarrow\dfrac{x_n(x_n^2+3c)}{3x_n^2+c}$

Convergence: cubic.

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1 nenny March 2, 2010 at 3:14 pm

thank you for this post.. this helps me a lot.. really.. thanks..!

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2 Frank April 17, 2010 at 11:40 pm

Hi,

I want to plot a implicit function on matlab, would you please let me know the commands?

Here is the function:

k=[x-k]x/[3x-2k]+[2x-3k][1-x]/[1-x-k]

x in (0,1)

Thanks a lot.

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3 Jie April 19, 2010 at 8:27 pm

Frank, you can do this:

xm = 0:0.01:1;
km = 0:0.01:4;
[x,k] = meshgrid(xm,km);
f = (x-k).*x./(3*x-2*k)+(2*x-3*k).*(1-x)./(1-x-k) – k;
contour(x,k,f,[0,0]);

Btw, your question should be posted here instead..

http://chenfuture.wordpress.com/2007/07/12/implicit-function-plot/

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