Calculating the square root of a

In this learning project, we learn how to approximate the square root of a number numerically.

Newton's method

File:Newton iteration.svg

Newton's method is a basic method in numerical approximations. The idea of Newton's method is to "follow the tangent line" to try to get a better approximation. For more details, see w:Newton's method

Try your hand on Newton's method

Step-by-step

Cut and paste the following wikitext into the sandbox, and enter your favourite number and a first (non-zero!) approximation of its square root. See what happen after multiple edit-and-saves:

{{subst:square root|number|first approximation}}

Sandbox

{{subst:square root|17|4.1231056256177}}
4.1231056256177
4.1231056256177
4.1231056256177
4.1231056256177
4.1231056256177
4.1231056256177
4.12310562562
4.12310562562
4.12310562562
4.12310562562
4.12310562562
4.12310562562
4.12310562562
4.12310562562
4.12310562562
4.12310562562
4.12310562562
4.12310562562
4.12310562562
4.12310562562
4.12310562562
4.12310562562
4.12310562562
4.12310562562
4.12310877528
4.1282051282
4.33333333333
3

What is going on

Our calculator-template Template:Tl implements the formula $x_{n+1}=(x_n+a/x_n)/2$, which is an instant of the general formula for Newton's method $x_{n+1}=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}$ in the case when the function f(x) is given by $f(x)=x^2-a$. It solves the equation $x^2-a=0$ numerically, by successive approximations.

Template:Mathematics-stub