GNU Free Documentation License, Version 1.2

- Examples
- User documentation
- Maintainer Documentation
- Bugs, Shortcomings and other ideas
- Main changes

The class `BigRat`

is intended to represent (exact) rational numbers
of practically unlimited range; it is currently based on the
implementation in the GMP *multiple precision* library. This code forms
the interface between CoCoALib and the big integer/rational library
upon which it relies. It seems most unlikely that GMP will be
displaced from its position as the foremost library of this type; as a
consequence the class `BigRat`

may eventually be replaced by GMP's
own C++ interface.

The usual arithmetic operations are available with standard C++ syntax
but generally these incur run-time overhead since results are returned
through temporaries which are created and destroyed silently by the
compiler. Thus if the variables `a`

, `b`

and `c`

are each of
type `BigRat`

then `a = b+c;`

is a valid C++ statement for placing
the sum of `b`

and `c`

in `a`

, **but** the sum is first computed
into a hidden temporary which is then copied to `a`

, and then
finally the temporary is destroyed. As a general principle, the type
`BigRat`

is provided for convenience of representing rational
values rather than for rapid computation.

There is an important exception to the natural syntax: `^`

does **not**
denote exponentiation; you must use the function `power`

instead.
We have chosen not to define `operator^`

to perform exponentiation
because it is too easy to write misleading code: for instance,
`a*b^2`

is interpreted by the compiler as `(a*b)^2`

. There is no
way to make the C++ compiler use the expected interpretation.

Arithmetic may also be performed between a `BigRat`

and a machine
integer or a `BigInt`

. The result is always of type `BigRat`

(even if the value turns out to be an integer). Do remember, though,
that operations between two machine integers are handled directly by
C++, and problems of overflow can occur.

It is important not to confuse values of type `BigRat`

with values of type
`RingElem`

which happen to belong to the ring `RingQQ`

. The distinction
is analogous to that between values of type `BigInt`

and value of type
`RingElem`

which happen to belong to the ring `RingZZ`

. In summary, the
operations available for `RingElem`

are those applicable to elements of
any ordered commutative ring, whereas the range of operations on `BigRat`

values is wider (since we have explicit knowledge of the type).

A value of type `BigRat`

may be created from:

- nothing, in which case the value is zero
- another value of type
`BigRat`

(its value is copied) `BigRat(n,d)`

a pair of integers (machine integers or`BigInt`

s) specifying numerator and denominator in that order`BigRat(str)`

where`str`

is a string of the form`N`

or`N/D`

where`N`

is the decimal representation of the numerator and`D`

that of the denominator`BigRat(mpq_value)`

copy a GMP rational (of type`mpq_t`

) into a`BigRat`

; helps interfacing between CoCoALib and code using GMP directly.

The ctors `BigRat(n,d)`

and `BigRat(str)`

accept an optional arg
`BigRat::AlreadyReduced`

which asserts that the value is already reduced
(*i.e.* positive denominator, and numerator and denominator are coprime).
Use this feature only if you are **absolutely certain** that there is no
common factor between the given numerator and denominator.

See **Bugs** section for why there is no ctor from a single integer, and
also for why `BigRat(0)`

is accepted by the compiler (but crashes at run-time).

**NOTE:** similar to operations on `BigInt`

-- see `IntOperations`

- normal arithmetic (potentially inefficient because of temporaries)
`+`

the sum`-`

the difference`*`

the product`/`

quotient`=`

assignment

- arithmetic and assignment
`+=`

,`-=`

,`*=`

,`/=`

-- definitions as expected; LHS must be of type`BigRat`

- arithmetic ordering
`==`

,`!=`

`<`

,`<=`

,`>`

,`>=`

-- comparison (using the normal arithmetic ordering) -- see also the`cmp`

function below.

- increment/decrement
`++`

,`--`

(prefix, e.g.`++a`

) use these if you can`++`

,`--`

(postfix, e.g.`a++`

) avoid these if you can, as they create temporaries

- query functions (all take 1 argument)
`IsZero(q)`

-- true iff`q`

is zero`IsOne(q)`

-- true iff`q`

is 1`IsMinusOne(q)`

-- true iff`q`

is -1`IsOneNum(q)`

-- true iff`num(q)`

is 1`IsOneDen(q)`

-- true iff`den(q)`

is 1`sign(q)`

-- gives -1 (machine integer) to mean`q`

is negative, 0 (machine integer) to mean`q`

is zero, +1 (machine integer) to mean`q`

is positive.

- Exponentiation
`power(a, b)`

-- returns`a`

to the power`b`

(result is always a`BigRat`

)

- The cmp function (three way comparison)
`cmp(a,b)`

-- returns an`int`

which is`< 0`

if`a < b`

, or`== 0`

if`a == b`

, or`> 0`

if`a > b`

.`CmpAbs(a,b)`

-- equivalent to`cmp(abs(a),abs(b))`

- Other functions
`abs(q)`

-- gives the absolute value of`q`

`floor(q)`

-- returns a`BigInt`

for the greatest integer`<= q`

`ceil(q)`

-- returns a`BigInt`

for the least integer`>= q`

`round(q)`

-- returns a`BigInt`

which is the nearest to`q`

(halves round the same way as in`RoundDiv`

, see`IntOperations`

)`num(q)`

-- returns a`BigInt`

which is the numerator of`q`

`den(q)`

-- returns a positive`BigInt`

which is the denominator of`q`

`log(q)`

-- returns a double whose value is (approx) the natural logarithm of`q`

`ILogBase(q,base)`

-- returns largest integer`k`

such that`power(base,k) <= abs(q)`

`mantissa(q)`

-- returns a`double`

between 0.5 and 1 (excluded)`exponent(q)`

--

- Functions violating encapsulation
`mpqref(n)`

-- this gives a (const) reference to the`mpq_t`

value inside a`BigRat`

object. You should use this accessor very sparingly!

Nothing very clever. Conversion from a string was a bit tedious.

Note that the ctor call `BigRat(0)`

actually calls the ctor from a string.
Unfortunately, this a C++ "feature". It will result in a run-time error.

I have replaced the bodies of the `BigRat`

ctors which take two integers
as arguments by a call to the common body `BigRat::myAssign`

. This does
mean that some wasteful temporaries are created when either of the
arguments is a machine integer. Time will tell whether this waste is
intolerable.

This code is probably not *exception safe*; I do not know what the `mpq_*`

functions do when there is insufficient memory to proceed. Making the
code *exception safe* could well be non-trivial: I suspect a sort of
`auto_ptr`

to an `mpq_t`

value might be needed.

Removed `BigRat`

ctors from a single (machine) integer because too often
I made the mistake of writing something like `BigRat(1/2)`

instead of
`BigRat(1,2)`

.

Should the `BigRat`

ctor from string also accept numbers with decimal points?
*e.g.* `BigRat("3.14159")`

? We'll wait and see whether there is demand for this
before implementing; note that GMP does **not** offer this capability.

**2011**

- August (v0.9950):
class
`QQ`

renamed into`BigRat`

: to avoid confusion with`RingQQ`

and its name in CoCoA system