package for fast polynomial evaluation

[gmoroz]

The attached package provides conversion of univariate and multivariate polynomials into object that are optimized for fast evaluation on python object or low-levels c++ classes (see examples at the end).

It could enhanced the fast_callable function for several types, and also enhances in general the evaluation of polynomials on polynomials.

To test it, you can install it with: ./setup.py install This will install the package in $SAGE_ROOT/local/lib/python2.7/site-packages/

Main features:

• handles univariate and multivariate polynomials
• specialized for several low-level types (mpfi, mpz, mpq, boost::interval)
• different evaluation layouts (horner, estrin, expanded, ...)
• easily extensible:
  • add new types (see fast_polynomial/interfaces/README)
  • add new layouts (see docstring of fast_polynomial.method)
• handles generic python/sage objects
• can be multi-threaded

Main limitations:

• only handles polynomial (no evaluation of trigonometric functions,...)
• polynomial needs to be converted to a fast callable object before evaluation (there is room for speed up on conversion time)

Examples and benchmarks:

from fast_polynomial import *
R.<x> = ZZ[x]
p = R.random_element(500,-100,100)

# evaluation of polynomials
q = python_polynomial(p, mode='horner')
r = python_polynomial(p, mode='estrin')
%timeit p(x+1) #5 loops, best of 3: 40.3 ms per loop
%timeit q(x+1) #5 loops, best of 3: 40.3 ms per loop
%timeit r(x+1) #125 loops, best of 3: 2.26 ms per loop
%timeit python_polynomial(p)(x+1) #125 loops, best of 3: 3.2 ms per loop

# evaluation of long integers
q = mpz_polynomial(p, num_threads=1)
r = mpz_polynomial(p, num_threads=2)
%timeit p(100) #625 loops, best of 3: 50.4 µs per loop
%timeit q(100) #625 loops, best of 3: 48.1 µs per loop
%timeit r(100) #625 loops, best of 3: 34.9 µs per loop


# evaluation of mpfi interval with precision 1000
q = mpfi_polynomial(p, 1000)
e = RealIntervalField(1000)(2^500, 2^500+1)
cmp(p(e),q(e)) #0
%timeit p(e)   #125 loops, best of 3: 2.71 ms per loop
%timeit q(e)   #625 loops, best of 3: 513 µs per loop
%timeit mpfi_polynomial(p)(e) #125 loops, best of 3: 1.15 ms per loop

# evaluation of boost interval (précision 53)
q = boost_polynomial(p, mode='horner')
r = boost_polynomial(p, mode='balanced', num_threads=2)
f = fast_callable(p, domain=float)
e = RIF(0.01)
%timeit p(e)    #125 loops, best of 3: 2.14 ms per loop
%timeit f(0.01) #625 loops, best of 3: 9.54 µs per loop
%timeit q(e)    #625 loops, best of 3: 13.4 µs per loop
%timeit r(e)    #625 loops, best of 3: 11.7 µs per loop
# Note that boost_polynomial evaluation offers more guarantees than raw float evaluation

# multivariate polynomials
R20 = PolynomialRing(QQ, 20,'x')
p = R20.random_element(5,100)
q = mpq_polynomial(p)
%timeit p((2/3,)*20) #125 loops, best of 3: 2.06 ms per loop
%timeit q((2/3,)*20) #625 loops, best of 3: 178 µs per loop
%timeit mpq_polynomial(p) #125 loops, best of 3: 1.91 ms per loop

[gmoroz]

fast_polynomial package compatible with sage >= 4.8

[gmoroz]

A minimal spkg (without boost dependency) to make the installation easier.

[gmoroz]

bug fix and add changelog.txt file