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[numbers] Fraction() and Knuth 4.5.1 -- overflow, BigInteger, long, and rounding


I read Kunth's "Art of Computer Programming 4.5.1" that is referenced many
times in the doc as the guidance for the commons-math/commons-numbers
Fraction class. It is an interesting read. Also, for all the times it is
cited in the doc, it is interesting that Fraction doesn't really use it as
implemented. Here is one example.

Knuth is concerned about overflow in multiplication and division, because
numerator of f1 is multiplied by denominator of f2 and so forth, so he
suggests a technique called "mediant rounding" that allows for intermediate
quantities in fraction multiplication to be rounded.

It is a clever technique and probably works well, however the current
Fraction class cites this chapter, then implements multiplication with
BigInteger instead, ignoring this suggestion.

First of all, the doc should be clear that the code is NOT following 4.5.1,
while it gives the opposite impression. And that's ok but the use of
BigInteger creates additional inconsistency: Multiply and divide are
accomplished using ArithmeticUtils.addAndCheck and
ArithmeticUtils.mulAndCheck . These convert the relevant ints to longs,
then perform the operation, then if the resulting long is greater than the
range of an int, throw an OverflowException. So some parts of Fraction
check for overflow using longs and others use BigInteger.

It seems to me that BigInteger is overkill here for the vast majority of
practical uses of Fraction in a way that could be damaging for performance.
And furthermore, we already have a BigFraction class to handle cases that
require BigInteger.

So, I propose rewriting the doc to say the opposite of what it currently
says when appropriate, and get usages of BigInteger out of Fraction, use
them only in BigFraction, and use the long-based ArithmeticUtils methods to
check for overflow and underflow in fraction addition and subtraction.

Eric