Partitions of an Integer

The subject of integer partitions comes up regularly, with multiple threads over the years at PerlMonks, a big chunk of Chapter 5 of Higher-Order Perl, and some pages from Knuth TAOCP volume 4A. Last year I added a partitions function to my ntheory Perl module, for counting integer partitions (OEIS A000041). A few months ago I added a partition iterator (forpart) since I liked the one Pari/GP added last year.

Integer partitions
Solution Impl
Max in 10s Count in 10s
ntheory 0.45 XS yes no yes yes 87 223,000
ntheory 0.45 Perl yes no yes yes 72 7,300
Integer::Partition 0.05 Perl yes yes no no 67 -
(unreleased, from Limbic 2004)
Perl no yes no no 62 6,000
MJD 2013 Perl no no no no 71 -
blokhead 2007 Perl yes no no no 63 -
kvale 2004 Perl yes no no no 62 -
sfink 2004 Perl yes no no no 58 -
tye 2001 Perl no no no no 58 -
(golfed, 73 chrs)
no (73)
no no no 21 -
Pari/GP 2.8.0
(not a Perl module!)
C/Pari no no yes yes 100 34,000,000

A comparison of memory use for primality modules

Performance and memory use are two areas of concern for many libraries. Last December I made a number of changes in my Math::Prime::Util module to reduce memory use, and bulk88 helped really tweak some of the XS. I thought I'd pick a simple task and look at the speed and memory use of a number of solutions.

isprime from 1 to 10 million
Memory Time Solution
2096k 72.6s Perl trial division mod 6
2100k 124.8s Perl trial division
3652k 36.2s Math::GMP
3940k 14.8s Math::GMPz
4040k 1.9s Math::Prime::Util (no GMP backend)
4388k 1.9s Math::Prime::Util (with GMP backend)
4568k 1.4s Math::Prime::Util (GMP + precalc)
4888k 4.4s Math::Prime::XS
5316k 245.1s Math::Primality
5492k 29.8s Math::Pari
6260k 1.5s Math::Prime::FastSieve
~16MB >1 year regex

Math::Prime::Util, May update

Math::Prime::Util 0.41 was released, with new functions (valuation, invmod, vecsum, binomial, forpart), and the usual crop of performance improvements. In particular, primality testing on x86_64 got another big speedup.

Fun with binomials

I decided to add a binomial(n,k) function to Math::Prime::Util, and found some interesting things while doing it. Overflow detection and mitigation in C and Perl were the first thing. Next was looking at negative arguments, which led to finding some differences in various solutions as well as filing a bug report for Math::BigInt.

Math::Prime::Util, April update

Math::Prime::Util, and the optional Math::Prime::Util::GMP back end, are a set of modules for number theory in Perl, with a large overlap of functionality with PARI/GP. This is an update on some of the things that were new in the April release (0.40 of MPU, 0.19 of MPU::GMP).

The usual speed improvements in various areas, some approximation improvements, and new functions. Primality testing benchmarks also show Perl has state of the art solutions.

About Dana Jacobsen

user-pic I've been using Perl since 1991 (4.010 or so). At some point I realized that while I have used many languages, I enjoy using Perl more than any others.