RFR: JDK-8277175 : Add a parallel multiply method to BigInteger

[kabutz]

BigInteger currently uses three different algorithms for multiply. The simple quadratic algorithm, then the slightly better Karatsuba if we exceed a bit count and then Toom Cook 3 once we go into the several thousands of bits. Since Toom Cook 3 is a recursive algorithm, it is trivial to parallelize it. I have demonstrated this several times in conference talks. In order to be consistent with other classes such as Arrays and Collection, I have added a parallelMultiply() method. Internally we have added a parameter to the private multiply method to indicate whether the calculation should be done in parallel.

The performance improvements are as should be expected. Fibonacci of 100 million (using a single-threaded Dijkstra's sum of squares version) completes in 9.2 seconds with the parallelMultiply() vs 25.3 seconds with the sequential multiply() method. This is on my 1-8-2 laptop. The final multiplications are with very large numbers, which then benefit from the parallelization of Toom-Cook 3.  Fibonacci 100 million is a 347084 bit number.

We have also parallelized the private square() method. Internally, the square() method defaults to be sequential.


Benchmark                                          (n)  Mode  Cnt      Score      Error  Units
BigIntegerParallelMultiply.multiply            1000000    ss    4     68,043 ±   25,317  ms/op
BigIntegerParallelMultiply.multiply           10000000    ss    4   1073,095 ±  125,296  ms/op
BigIntegerParallelMultiply.multiply          100000000    ss    4  25317,535 ± 5806,205  ms/op
BigIntegerParallelMultiply.parallelMultiply    1000000    ss    4     56,552 ±   22,368  ms/op
BigIntegerParallelMultiply.parallelMultiply   10000000    ss    4    536,193 ±   37,393  ms/op
BigIntegerParallelMultiply.parallelMultiply  100000000    ss    4   9274,657 ±  826,197  ms/op

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Commit messages:
 - Update comments
 - Added parallelMultiply() method to BigInteger to allow large multiplications to run in parallel
 - Merge remote-tracking branch 'upstream/master'
 - Merge remote-tracking branch 'upstream/master'
 - Merge remote-tracking branch 'origin/master'
 - 8176501: Method Shape.getBounds2D() incorrectly includes Bezier control points in bounding box
 - 8176501: Method Shape.getBounds2D() incorrectly includes Bezier control points in bounding box

Changes: https://git.openjdk.java.net/jdk/pull/6391/files
 Webrev: https://webrevs.openjdk.java.net/?repo=jdk&pr=6391&range=00
  Issue: https://bugs.openjdk.java.net/browse/JDK-8277175
  Stats: 731 lines in 8 files changed: 565 ins; 136 del; 30 mod
  Patch: https://git.openjdk.java.net/jdk/pull/6391.diff
  Fetch: git fetch https://git.openjdk.java.net/jdk pull/6391/head:pull/6391

PR: https://git.openjdk.java.net/jdk/pull/6391