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Square Root

Square roots are taken using the "Karatsuba Square Root" algorithm by Paul Zimmermann (see References). This is expressed in a divide and conquer form, but as noted in the paper it can also be viewed as a discrete variant of Newton's method.

In the Karatsuba multiplication range this is an O(1.5*M(N/2)) algorithm, where M(n) is the time to multiply two numbers of n limbs. In the FFT multiplication range this grows to a bound of O(6*M(N/2)). In practice a factor of about 1.5 to 1.8 is found in the Karatsuba and Toom-3 ranges, growing to 2 or 3 in the FFT range.

The algorithm does all its calculations in integers and the resulting mpn_sqrtrem is used for both mpz_sqrt and mpf_sqrt. The extended precision given by mpf_sqrt_ui is obtained by padding with zero limbs.