A method to compute roots other than sqrt.

1. Mar 11, 2006

MathematicalPhysicist

is there a method to compute roots other than sqrt?, like 10th root or 13th root of a number?

and, what are they?

2. Mar 11, 2006

VietDao29

This one is a good candidate for Newton's method. We choose an arbitrary value x0, then use:
$$x_{n + 1} = x_n - \frac{f(x_{n})}{f'(x_{n})}$$.
And then let n increases without bound to obtain the answer.
$$x = \lim_{n \rightarrow \infty} x_n$$.
For example:
Find $$\sqrt[3]{4}$$
Let $$x = \sqrt[3]{4} \Rightarrow x ^ 3 = 4 \Rightarrow x ^ 3 - 4 = 0$$
We then define f(x) := x3 - 4.
Say, we choose x0 = 1, plug everything into a calculator, use the formula:
$$x_{n + 1} = x_n - \frac{x_{n} ^ 3 - 4}{3 x_{n} ^ 2}$$.
and we'll have:
x1 = 2
x2 = 1.6666667
x3 = 1.5911111
x4 = 1.5874097
x5 = 1.5874010
x6 = 1.5874010
...
So the value of xn will converge quite fast to $$\sqrt[3]{4}$$, as n tends to infinity.

3. Mar 11, 2006

matt grime

Of course, the most obvious question is 'what is this method you have for computing arbitrary square roots'

4. Mar 13, 2006

jim mcnamara

FWIW - Logarithms work well for this. Especially if you're a programmer, and are happy with the inherent imprecision of floating point numbers.

If you take
result = (log(x) / n)

and then convert the result back ie.,

nth_root = exp(result)

you can generate all roots of x.

5. Mar 13, 2006

AlphaNumeric

If you're doing fractions then you can use Newton's binomial expansion

For instance, work out the fifth root of 31 by expanding $$(1+x)^{\frac{1}{5}}$$ with x = -1/32

$$(1+x)^{\frac{1}{5}} = 1 + \frac{1}{5}x + \frac{1}{5}\left(-\frac{4}{5}\right)\frac{1}{2!}x^{2} + ....$$

Put in x = -1/32 (which gives excellent convergence) to get

$$\left( \frac{31}{32} \right)^{\frac{1}{5}} = 1 + \frac{1}{5}\left(-\frac{1}{32}\right) + \frac{1}{5}\left(-\frac{4}{5}\right)\frac{1}{2!}\left(-\frac{1}{32}\right)^{2} + .... = 1 - \frac{1}{160} - \frac{1}{12800} = \frac{12719}{12800}$$

$$\frac{(31)^{\frac{1}{5}}}{2} = \frac{12719}{12800}$$

$$(31)^{\frac{1}{5}} = \frac{12719}{6400}$$

In decimal form this is 1.9873475. Raise it to the 5th power and get 31.00023361. A nice approximation for 2 minutes work.