[Paul Zimmermann <Paul.Zimmermann@loria.fr>] mpz_cbrtrem
Mon, 4 Nov 2002 13:54:44 +0100
> [zimmerma@ecrouves ~/gmp]$ ./cbrtrem 50000 1
> mpz_root took 29670ms
> mpz_sqrtrem took 2270ms
> mpz_cbrtrem took 1950ms
strange , On my athlon mpz_cbrtrem is 50% slower than mpz_sqrtrem ?
For the same number of limbs?
> while being reasonably fast for one-limb operands:
> [zimmerma@ecrouves ~/gmp]$ ./cbrtrem 1 100000
> mpz_root took 1830ms
> mpz_sqrtrem took 110ms
> mpz_cbrtrem took 2400ms
> The key function is mpz_add_bits: set the low bits of the destination
> to bits n0 to n1-1 from the operand.
I assume this is the karatsuba cube root , does it generilize to k-th root ?
Yes this is a recursive algorithm, with a direct computation of the remainder
together with the cube root. It generalizes to k-th root, but when k increases
the efficiency decreases with respect to first computing the k-th root, then
the remainder by n - s^k.
If this is the karatsuba cbrt-root then I was under the impression that it was
not asymptotically optimal , although faster for pratical ranges .
In fact my "karatsuba square root" is more generally a divide & conquer
algorithm. Unlike divide & conquer division, it is asymptotically optimal
(except perhaps for the multiplicative constant), since it is just a Newton
iteration with incremental remainder computation.
So Paul , if it does generalize to k-th root , do you plan to implement it for
GMP ? , or does anyone else ?
I first want to see if the cube-root implementation is fast enough.
How does it compare to your generic implementation for k=3?