nth root algorithm

Out of curiosity, this calculator implements nth root algorithm. This is iterative algorithm which is said to be very fast-converging, so the calculator lists the results of each iteration: current computed root value and current delta from previously computed root value. Calculator stops the iterations when desired precision is obtained, in other words, when the delta between previous and current computed roots is less than a precision. The description of the algorithm can be found below the calculator.

nth root algorithm

Number

Degree of the root

Calculate

Calculation precision

Digits after the decimal point: 4

nth root

 

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nth root algorithm

The algorithm is quite simple.

Step 1. Make an initial guess

In this calculator to get initial guess I simply divide number by root's degree if number is more than 1, and multiply by root's degree otherwise.

Step 2. Set

.

For precision check, it is more practical to compute delta separately

.

Step 3. Repeat step 2 until the desired precision is reached:

.

The algorithm can be derived from Newton's method for

,

where x is the root, and A is positive real number.