Scheme defines a “numerical tower” of numerical types:
number, complex, real, rational, and integer.
Java has primitive “unboxed” number types (such as
`int`), just like C, and also has some
“wrapper” classes that are basically boxed versions
of the unboxed number types. Specifically, the standard Java
number classes are not organized in any particularly useful hierarchy,
except that they all inherit from `Number`.
Kawa implements the full “tower” of Scheme number types,
using its own set of sub-classes of the abstract
class `Quantity`, a sub-class of
`Number` we will discuss later.

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`Complex` is the class of abstract complex numbers.
It has three subclasses: the abstract class `RealNum`
of real numbers; the general class `CComplex` where the
components are arbitrary `RealNum` fields; and the
optimized `DComplex` where the components
are represented by `double` fields.

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Concrete class for double-precision (64-bit) floating-point real numbers.

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`RatNum`, the abstract class for exact rational numbers,
has two sub-classes: `IntFraction`
and `IntNum`.

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The `IntFraction` class implements fractions in the
obvious way. Exact real infinities are identified with the
fractions `1/0` and `-1/0`.

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The `IntNum` concrete class implements infinite-precision integers.
The value is stored in the first `ival` elements of `words`,
in 2's complement form (with the low-order bits in `word[0]`).

There are already many bignum packages, including one that Sun added for JDK 1.1. What are the advantages of this one?

A complete set of operations, including gcd and lcm; logical, bit, and shift operations; power by repeated squaring; all of the division modes from Common Lisp (floor, ceiling, truncate, and round); and exact conversion to

`double`.Consistency and integration with a complete “numerical tower.” Specifically, consistency and integration with “fixnum” (see below).

Most bignum packages use a signed-magnitude representation, while Kawa uses 2's complement. This makes for easier integration with fixnums, and also makes it cheap to implement logical and bit-fiddling operations.

Use of all 32 bits of each “big-digit” word, which is the “expected” space-efficient representation. More importantly, it is compatible with the

`mpn`routines from the Gnu Multi-Precision library (`gmp`) [gmp]. The`mpn`routines are low-level algorithms that work on unsigned pre-allocated bignums; they have been transcribed into Java in the`MPN`class. If better efficiency is desired, it is straight-forward to replace the`MPN`methods with native ones that call the highly-optimized`mpn`functions.

If the integer value fits within a signed 32-bit `int`,
then it is stored in `ival` and `words` is null.
This avoids the need for extra memory allocation for the `words`
array, and also allows us to special-case the common case.

As a further optimization, the integers in the range -100 to 1024 are pre-allocated.

Many operations are overloaded to have different
definitions depending on the argument types.
The classic examples are the functions of arithmetic
such as “`+`”, which needs to use different
algorithms depending on the argument types.
If there is a fixed and reasonably small set of number types
(as is the case with standard Scheme), then we can just
enumerate each possibility. However, the Kawa system is
meant to be more extensible and support adding new
number types.

The solution is straight-forward in the case of a one-operand
function such as “negate”, since we can use method
overriding and virtual method calls to dynamically
select the correct method. However, it is more difficult
in the case of a binary method like “`+`,” since
classic object-oriented languages (including Java) only
support dynamic method selection using the type of the
first argument (“`this`”).
Common Lisp and some Scheme dialects support dynamic method
selection using all the arguments, and in fact the
problem of binary arithmetic operations is probably the
most obvious example where “multi-dispatch” is useful.

Since Java does not have multi-dispatch, we have to solve the problem in other ways. Smalltalk has the same problems, and solved it using “coercive generality”: Each number class has a generality number, and operands of lower generality are converted to the class with the higher generality. This is inefficient because of all the conversions and temporary objects (see [Budd91Arith]), and it is limited to what extent you can add new kinds of number types.

In “double dispatch” [Ingalls86]
the expression `x-y`
is implemented as `x.sub(y)`. Assuming the (run-time)
class of `x` is `Tx` and that of `y` is `Ty`,
this causes the `sub` method defined in `Tx` to be
invoked, which just does `y.subTx(x)`. That invokes
the `subTx` method defined in `Ty` which can without
further testing do the subtraction for types `Tx` and `Ty`.

The problem with this approach is that it is difficult to add
a new `Tz` class, since you have to also add `subTz`
methods in all the existing number classes, not to mention
`addTz` and all the other operations.

In Kawa, `x-y` is also implemented by `x.sub(y)`.
The `sub` method of `Tx` checks if `Ty` is one
of the types it knows how to handle. If so, it does the
subtraction and returns the result itself. Otherwise,
`Tx.sub` does `y.subReversed(x)`.
This invokes `Ty.subReversed` (or
`subReversed` as defined in
a super-class of `Ty`).
Now `Ty` (or one of its
super-classes) gets a chance to see if it knows how to
subtract itself from a `Tx` object.

The advantage of this scheme is flexibility. The knowledge of
how to handle a binary operation for types `Tx` and
`Ty` can be in either of `Tx` or
`Ty` or either of their
super-classes. This makes is easier to add new classes without
having to modify existing ones.

The DSSSL language [DSSSL] is a dialect of Scheme used
to process SGML documents. DSSSL has “quantities” in addition
to real and integer numbers. Since DSSSL is used to format documents,
it provides length values that are a multiple of a meter
(*e.g.* `0.2m`),
as well as derived units like `cm` and
`pt` (point).
A DSSSL quantity is a product of a dimension-less number with an integral
power of a length unit (the meter). A (pure) number is a quantity where
the length power is zero.

For Kawa, I wanted to merge the Scheme number types with the DSSSL
number types, and also generalize the DSSSL quantities to support
other dimensions (such as mass and time) and units (such as `kg`
and seconds). Quantities are implemented by the abstract class
`Quantity`. A *quantity* is a
product of a `Unit` and a pure number.
The number can be an arbitrary complex number.

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A `CQuantity` is a concrete class that implements general
`Quantities`. But usually we don't need that much generality,
and instead use `DQuanity`.

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A `Unit` is a product of a floating-point `factor`
and one or more primitive units, combined into a `Dimensions` object.
The `Unit` may have a name (such as “`kg`”), which is
used for printing, and when parsing literals.

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A `BaseUnit` is a primitive unit that is not
defined in terms of any other `Unit`, for example the meter.
Each `BaseUnit` has a different `index`, which is used
for identification and comparison purposes. Two `BaseUnit`s
have the same `index` if and only if they are the same `BaseUnit`.

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A `Dimensions` object is a product and/or ratio of `BaseUnit`s.
You can think of it as a data structure that maps every `BaseUnit`
to an integer power. The `bases` array is a list of the `BaseUnit`s
that have a non-zero power, in order of the `index` of the `BaseUnit`.
The `powers` array gives the power (exponent) of the `BaseUnit`
that has the same index in the `bases` array.

Two `Dimensions` objects are equal if they have the same list of
`bases` and `powers`. `Dimensions` objects are “interned”
(using a global hash table) so that they are equal only if they are
the same object. This makes it easy to implement addition and
subtraction:

public static DQuantity add (DQuantity x, DQuantity y) { if (x.unit().dims != y.unit().dims) throw new ArithmeticException ("units mis-match"); double r = y.unit().factor / x.unit().factor; double s = x.factor + r * y.factor; return new DQuantity (s, x.unit()); } |

The `Unit` of the result of an addition or subtraction
is the `Unit` of the first operand.
This makes it easy to convert units:

(+ 0cm 2.5m) ==> 250cm |

Because Kawa represents quantities relative to user-specified units,
instead of representing them relative to primitive base units,
it can display quantities using the user's preferred units,
rather than having to use prmitive units.
However, this does make multiplication and division a problem
The actual calculation (finding the right `Dimensions` and
multiplying the constant factors) is straight-forward.
The difficulty is that we have to generate a new compound
`Unit`, and print it out in a reasonable fashion.
Exactly how this should best be done is not obvious.