[Paul Zimmermann <Paul.Zimmermann@loria.fr>] mpz_cbrtrem

[Paul Zimmermann]

   > [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.
   >
   > Paul
   >

   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?

Paul