Math2.org Math Tables: Recursive Formulas for sqrt(A)

Recursive Formulas for " src="../../art/sqrt.gif">(A)

Explicit form:

Find " src="../../art/sqrt.gif">(A), where A is a real number > 0.

Recursive form:

Convert A to scientific notation base 2 (C++ has function "frexp" for this). Note: mantissa is ³ .5 and < 1.

Let

a = mantissa of A
exp = exponent of a
a = a * 2^{(exp mod 2)}
exp = exp \ 2       (Note:  integer divide)
x_n+1 = (x_n/2)(3 - ax_n²)

Reiterate about 5 or 6 times then do y with that result.

y_i+1 = y_i + (x_n/2)(a - y_i²)

Reiterate until required precision attained.

" src='../../art/sqrt.gif' align=middle>A = y_i * 2^exp

Source: Jeff Yates.