GNU Free Documentation License, Version 1.2

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

- ex-RingElem1.C
- ex-RingFp1.C
- ex-RingFp2.C
- ex-RingQ1.C
- ex-RingTwinFloat1.C
- ex-RingWeyl1.C
- ex-RingZZ1.C
- ex-PolyRing3.C
- ex-SparsePolyOps1.C
- ex-NF.C

The file `ring.H`

introduces several classes used for representing
rings and their elements. A normal user of the CoCoA library will use
principally the classes `ring`

and `RingElem`

: an object of type
`ring`

represents a mathematical ring with unity, and objects of
type `RingElem`

represent values from some ring. To make the
documentation more manageable it has been split into two: this file
describes operations on a `RingElem`

, whereas a separate file
describes the operations directly applicable to `ring`

s.
Documentation about the creation and use of homomorphisms between
rings can be found in `RingHom`

.

An object of type `RingElem`

comprises two internal parts: the ring
to which the value belongs, and the value itself. For instance, this
means that the zero elements of different rings are quite different
objects.

Normally when creating a new `RingElem`

we specify both the ring to
which it belongs, and its initial value in that ring. Let `R`

be a
`ring`

. Let `n`

be a machine integer or a `BigInt`

. Let `q`

be a rational, *i.e.* a value of type `BigRat`

. Let `r2`

be a
ring element.

`RingElem r;` |
the zero element of `RingZZ()` (special case) |

`RingElem r(R);` |
an element of `R` , initially 0 |

`RingElem r(R, n);` |
an element of `R` , initially the image of `n` |

`RingElem r(R, q);` |
an element of `R` , initially the image of `q` (or error) |

`RingElem r(R, str);` |
an element of `R` , initially the value of string `str` |

`RingElem r(R, s);` |
an element of `R` , initially the value of `symbol` `s` |

`RingElem r(R, r2);` |
an element of `R` , maps `r2` into `R` via `CanonicalHom` |

`RingElem r(R, MPZ);` |
an element of `R` , initially the value of `mpz_t MPZ` |

`RingElem r(R, MPQ);` |
an element of `R` , initially the value of `mpq_t MPQ` (or error) |

**Note 1:** To create a `RingElem`

from a value of type `mpz_class`

you must do `RingElem(R, MPZ.get_mpz_t())`

; analogously for `mpq_class`

.

**Note 2:** Construction from a rational may fail, *e.g.* if the denominator is a zero divisor in the ring; if it does fail then an exception is thrown (with code `ERR::DivByZero`

).

**Note 3:** There is also a *string constructor* for making vectors:

`RingElems(R, str)`

-- a`vector<RingElem>`

of elements of`R`

, initially the values in string`str`

(separated by commas)

You can create a **copy** of a ring element in the usual way:

`RingElem r(r2);` |
a copy of `r2` , element of the same ring |

`RingElem r = r2;` |
(alternative syntax, discouraged) |

Naturally the last constructor works only if the denominator of the rational `q`

is 1.

These are not really constructors: you can get the zero and one of a
`ring`

directly using the following:

`zero(R)` |
the zero element of `R` |

`one(R)` |
the one element of `R` |

`RingElem`

s are designed to be easy and safe to use; the various checks
do incur a certain run-time overhead, so a faster alternative is offered
(see below in the section *Fast and Ugly Code*). Arithmetic operations
between `RingElem`

s will fail if they do not belong to the same ring
(the exception has code `ERR::MixedRings`

).

Assigning an integer or rational to a `RingElem`

wil automatically map
the value into the ring to which the `RingElem`

belongs.

`r = n;` |
map `n` into `owner(r)` and assign the result |

`r = q;` |
map `q` into `owner(r)` and assign the result |

`r = r2;` |
`r` becomes a copy of `r2` -- afterwards `owner(r)==owner(r2)` |

`swap(r,s);` |
exchange the values (and the owning rings) |

Arithmetic operations between `RingElem`

s will fail if they do not belong
to the same ring (the exception has code `ERR::MixedRings`

). You may
perform arithmetic between a `RingElem`

and a machine integer, a `BigInt`

value
or a `BigRat`

value -- the integer/rational is automatically mapped into the
same ring as the `RingElem`

.

Let `r`

be a non-const `RingElem`

,
and `r1`

, `r2`

be potentially const `RingElem`

s.
Assume they are all associated to the same ring.
Then the operations available are: (meanings are obvious)

`cout << r1`

-- output value of`r1`

(decimal only, see notes)`r1 == r2`

-- equality test`r1 != r2`

-- not-equal test`-r1`

-- negation (unary minus)`r1 + r2`

-- sum`r1 - r2`

-- difference`r1 * r2`

-- product`r1 / r2`

-- quotient, division must be exact (see`IsDivisible`

)`r += r1`

-- equivalent to`r = r + r1`

`r -= r1`

-- equivalent to`r = r - r1`

`r *= r1`

-- equivalent to`r = r * r1`

`r /= r1`

-- equivalent to`r = r / r1`

division must be exact (see`IsDivisible`

)`power(r1, n)`

--`n`

-th power of`r1`

;`n`

any integer,**NB**`power(0,0)`

gives 1`r^n`

--**THIS DOES NOT WORK!!!**it does not even compile, you must use`power`

Attempting to compute a `gcd`

or `lcm`

in a ring which not an
effective GCD domain will produce an exception with code `ERR::NotTrueGCDDomain`

.
If `r1`

or `r2`

is a `BigRat`

then an error is signalled at compile time.

`gcd(r1, r2)`

-- an associate of the gcd`IsCoprime(r1, r2)`

-- true iff`IsInvertible(gcd(r1,r2))`

; for polynomials (over`ZZ`

) consider instead`IsConstant(gcd(r1,r2))`

`lcm(r1, r2)`

-- an associate of the lcm`GcdQuot(&gcd, "1, "2, r1, r2)`

-- procedure computes`gcd`

and`quot1=r1/gcd`

and`quot2=r2/gcd`

, here`r1`

and`r2`

must be`RingElem`

.

There is a class called `GCDFreeBasis_RingElem`

for computing a factor base
of a set of `RingElem`

from an effective GCD domain:

`GCDFreeBasis_RingElem()`

default ctor; base is initially empty`GFB.myAddInfo(x)`

use also`RingElem x`

when computing he factor base`GFB.myAddInfo(v)`

use also the elements of`std::vector<RingElem> v`

when computing the factor base`FactorBase(GFB)`

returns the factor base obtained so far

CoCoALib offers functions for querying various properties of `RingElem`

s,
and about relationships between `RingElems`

.

Let `r1`

and `r2`

be a (possibly const) `RingElem`

s, and
let `N`

be a variable of type `BigInt`

, and `q`

a variable of type `BigRat`

`owner(r1)`

-- the ring to which`r1`

is associated`IsZero(r1)`

-- true iff`r1`

is zero`IsOne(r1)`

-- true iff`r1`

is one`IsMinusOne(r1)`

-- true iff`-r1`

is one`IsInvertible(r1)`

-- true iff`r1`

has a multiplicative inverse`IsZeroDivisor(r1)`

-- true iff`r1`

is zero-divisor`IsDivisible(r1, r2)`

-- true iff`r1`

is divisible by`r2`

(throws`ERR::DivByZero`

if`r2`

is a zero-divisor)`IsDivisible(r, r1, r2)`

--`r = r1/r2`

and returns true iff`r1`

is divisible by`r2`

`IsDivisible_AllowFields(r1, r2)`

-- true iff`r1`

is divisible by`r2`

(throws`ERR::DivByZero`

if`r2`

is a zero-divisor)`IsDivisible_AllowFields(r, r1, r2)`

--`r = r1/r2`

and returns true iff`r1`

is divisible by`r2`

`IsInteger(N, r1)`

-- true iff`r1`

is the image of an integer (if true, a preimage is placed in`N`

, otherwise it is left unchanged)`IsRational(q, r1)`

-- true iff`r1`

is the image of a rational (if true, a preimage is placed in`q`

, otherwise it is left unchanged)`IsDouble(d, r1)`

-- true iff`r1`

is the image of a rational whose approx is put into d (false if overflow and`d`

unchanged)

Note that `IsDivisible`

tests divisibility in the ring containing the
values: so 1 is not divisible by 2 in `RingZZ`

, but their images in
`RingQQ`

would be divisible. Also note that `IsDivisible`

throws an
exception if the test is in a field; use `IsDivisible_AllowFields`

if
you want to allow divisibility testing also in a field.

If the ring is an ordered domain then these functions may also be
used. You can discover whether CoCoALib thinks that the ring `R`

is
arithmetically ordered by calling `IsOrderedDomain(R)`

: the value is
`true`

iff `R`

is arithmetically ordered.

Note that comparison operations between `RingElem`

s will fail if they do not belong
to the same ring (the exception has code `ERR::MixedRings`

). You may
perform comparisons between a `RingElem`

and an integer or a rational -- the integer/rational is automatically mapped into the
same ring as the `RingElem`

.

Let `r1`

and `r2`

belong to an ordered ring. Trying to use any
of these functions on elements belonging to a ring which is not ordered
will produce an exception with code `ERR::NotOrdDomain`

.

`sign(r1)`

-- value is -1, 0 or +1 according as`r1`

is negative, zero or positive`abs(r1)`

-- absolute value of`r1`

`floor(r1)`

-- greatest integer`<= r1`

`ceil(r1)`

-- least integer`>= r1`

`NearestInteger(r1)`

-- returns nearest integer (`BigInt`

) to`r1`

(halves round as in`round`

, see`BigRat`

).`cmp(r1, r2)`

-- returns a value <0, =0, >0 according as`r1-r2`

is <0, =0, >0`CmpAbs(r1, r2)`

-- equiv to`cmp(abs(r1),abs(r2))`

`CmpDouble(r1, z)`

-- compare a ring elem with a`double`

, result is <0, =0, >0 according as`r1-z`

is <0, =0, >0`r1 < r2`

-- standard inequalities`r1 > r2`

-- ...`r1 <= r2`

-- ...`r1 >= r2`

-- ...

If `owner(r)`

is a finite field, or a polynomial ring whose coeffs are
ini a finite field then the following functions may be used. The `P`

in the names of these function refers to the characteristic of the ring.

`IsPthPower(r1)`

-- true iff`r1`

has a p-th root where p is the ring characteristic (see also`PthRoot`

)`PthRoot(r1)`

-- returns the p-th root of`r1`

(error if no p-th root exists)

If `owner(r)`

is a fraction field then the following functions may
be used. You can find out whether CoCoALib thinks that the ring `R`

is a fraction field by calling `IsFractionField(R)`

: the result is
`true`

iff `R`

is a fraction field.

Let `K`

denote a `FractionField`

Let `r`

denote an element of `K`

.

`num(r)`

-- gives a copy of the numerator of`r`

as an element of`BaseRing(K)`

`den(r)`

-- gives a copy of the denominator of`r`

as an element of`BaseRing(K)`

**Note:** the numerator and denominator are defined only upto
multiples by a unit: so it is (theoretically) possible to have two
elements of `FrF`

which are equal but which have different
numerators and denominators, for instance, (x-1)/(x-2) = (1-x)/(2-x)

If `owner(r)`

is a quotient ring then the following function may be
called. You can find out whether CoCoALib thinks that the ring `R`

is a quotient ring by calling `IsQuotientRing(R)`

: the result is
`true`

iff `R`

is a quotient ring.

In addition to the standard `RingElem`

operations, elements of a
`QuotientRing`

may be used in other functions.

Let `RmodI`

denote a quotient ring.
Let `r`

denote a non-const element of `RmodI`

.

`CanonicalRepr(r)`

-- produces a`RingElem`

belonging to`BaseRing(RmodI)`

whose image under`QuotientingHom(RmodI)`

is`r`

. For instance, if`r`

= -3 in ZZ/(10) then this function could give 7 as an element of`RingZZ`

.

You can determine if an element belongs to a *twin-float* ring by
calling `IsRingTwinFloat(owner(r))`

: this yields `true`

iff `r`

belongs to a twin-float ring.

Let `x`

, `y`

be `RingElem`

belonging to a `RingTwinFloat`

`MantissaAndExponent2(x)`

-- produce a`MantExp2`

structure representing the approximate value of`x`

(for`MantExp2`

see`FloatApprox`

)`DebugPrint(out, x)`

-- print out both components of`x`

`IsPracticallyEqual(x, y)`

-- returns true if`IsZero(x-y)`

otherwise false.

In contrast the test `x==y`

may throw a
`RingTwinFloat::InsufficientPrecision`

while
`IsPracticallyEqual`

will never throw this exception.
`IsPracticallyEqual`

is intended for use in a termination
criterion for an iterative approximation algorithm (*e.g.* see
`test-RingTwinFloat4.C`

).

You can determine whether an element belongs to a *PolyRing* by
calling `IsPolyRing(owner(r))`

: the result is `true`

iff `r`

belongs to a poly ring.

Let `P`

denote a polynomial ring.
Let `f`

denote a non-const element of `P`

.
Let `f1`

, `f2`

denote const elements of `P`

.
Let `v`

denote a const vector of elements of `P`

.

`IsMonomial(f)`

-- true iff`f`

is non zero and of the form coeff*pp`AreMonomials(v)`

-- true iff`v`

is non zero and of the form coeff*pp (if`v`

is empty it returns true)`IsConstant(f)`

-- true iff`f`

is "constant", i.e. the image of an element of the coeff ring.`IsIndet(f)`

-- equivalent to`f`

== x[i] for some index i`IsIndet(index, f)`

-- equivalent to`f`

== x[i]; and sets index = i`IsIndetPosPower(f)`

-- true iff there is index`i`

and exponent`e`

such that`f == x[i]^e`

`IsEvenPoly(f)`

-- true iff`f`

is even as a function`IsOddPoly(f)`

-- true iff`f`

is odd as a function`IsIrred(f)`

-- true iff`f`

is irreducible in P;**important**for polys applies only to "primtiive part" (ignores content)`IsPrimitivePoly(f)`

-- true iff`f`

is irred (over finite field) and its roots are group gens of the field extn`owner(f1)`

-- the owner of`f`

as a`ring`

.`NumTerms(f1)`

-- the number of terms in`f1`

.`StdDeg(f1)`

-- the standard degree of`f1`

(deg(x[i])=1);**error**if`f1`

is 0.`deg(f1)`

-- same as`StdDeg(f1)`

.`deg(f1, var)`

-- maximum degree of`var`

-th indet in`f1`

where`var`

is the index of the indet in`P`

(result is of type long).`LC(f1)`

-- the leading coeff of`f1`

; it is an element of CoeffRing(P).`ConstantCoeff(f)`

-- the constant coeff of`f`

; may be zero, belongs to CoeffRing(P).`monic(f1)`

-- same as`f1/LC(f1)`

`content(f1)`

-- gcd of the coeffs of`f1`

; it is an element of CoeffRing(P). Content of zero poly is 0. If coeffs are in a fraction field then

content is`c`

such that`f1/c`

is a primitive polynomial.

`FixedDivisor(f1)`

-- computes gcd of`f1(n)`

as`n`

ranges over all integers (aka. intrinsic content).`CommonDenom(f1)`

-- the simplest common denominator for the coeffs of`f1`

; it is an element of BaseRing(CoeffRing(P)); throws if CoeffRing is not a FractionField of a GCD domain.`ClearDenom(f1)`

--`f1`

*`CommonDenom(f1)`

(same restrictions as above)`ClearDenom(Rx, f1)`

-- like`ClearDenom(f1)`

but puts result in (SparsePoly)ring`Rx`

`prim(f1)`

-- same as`g = ClearDenom(f1); return g/content(g);`

`deriv(f1, var)`

-- formal derivative of`f1`

wrt. indet having index`var`

.`deriv(f1, x)`

-- derivative of`f1`

w.r.t.`x`

,`x`

must be an indeterminate (also works for f1 in`FractionField`

of a`PolyRing`

)

NOTE: to compute the *weighted degree* of a polynomial use the function
`wdeg`

defined for `RingElem`

of a `SparsePolyRing`

(see below).

You can determine whether an element belongs to a *sparse poly ring* by
calling `IsSparsePolyRing(owner(r))`

: the result is `true`

iff `r`

belongs to a (sparse) poly ring.

Let `P`

denote a `SparsePolyRing`

.
Let `f`

denote a non-const element of `P`

.
Let `f1`

, `f2`

denote const elements of `P`

.
Let `expv`

be a `vector<long>`

of size equal to the number of indeterminates.

`owner(f1)`

-- the owner of`f1`

as a`ring`

`NumTerms(f1)`

-- the number of terms in`f1`

with non-zero coefficient.`UnivariateIndetIndex(f)`

-- if`f`

is univariate in j-th indet returns j, o/w returns -1; throws error if`f`

is constant`LPP(f1)`

-- the leading PP of`f1`

; it is an element of PPM(P). Also known as*LT(f)*or*in(f)*`LF(f1)`

-- the leading form of`f1`

; sum of all summands of highest weighted`degree`

; gives error if`f1`

is zero.`CutLF(f)`

-- like`LF(f)`

, but also modifies`f`

to become`f-LF(f)`

.`HomogCompt(f,d)`

-- return homogeneous part of`f`

of deg`d`

(does not modify`f`

)`wdeg(f1)`

-- the weighted`degree`

of the leading PP of`f1`

(see [KR] Sec.4.3);**error**if`f1`

is 0.**NB**result is of type`CoCoA::degree`

(see`degree`

). (contrast with`StdDeg(f1)`

and`deg(f1)`

defined for general`PolyRing`

)`CmpWDeg(f1, f2)`

-- compare the weighted degrees of the LPPs of`f1`

and`f2`

; result is <0 =0 >0 according as deg(`f1`

) < = > deg(`f2`

)`IsHomog(f)`

-- says whether`f`

is homogeneous wrt weighted`degree`

.`homog(f, h)`

-- returns`f`

homogenized with indet`h`

(requires`GrDim`

=1 and`wdeg(h)`

=1)`NR(f, v)`

-- returns the (normal) remainder of the Groebner Division Algorithm by`v`

. If`v`

is a GBasis this is the Normal Form (see`NF`

in the doc for`ideal`

).`indet(P,v)`

-- returns the`v`

-th indet of`P`

as a`RingElem`

`IndetPower(P,v,e)`

-- returns the`e`

-th power of the`v`

-th indet of`P`

as a`RingElem`

`monomial(P,pp)`

-- returns`pp`

as an element of`P`

`monomial(P,c,pp)`

-- returns`c`

*`pp`

as an element of`P`

where`c`

is an integer, rational or is in`CoeffRing(P)`

and`pp`

is in`PPM(P)`

.`monomial(P,c,expv)`

-- returns`c*x[0]^expv[0]*x[1]^expv[1]*...`

where`c`

is an integer, rational or is in`CoeffRing(P)`

, and x[i] are the indets of`P`

.

Let `X`

be an indet (*i.e.* a `RingElem`

in `P`

)
or a vector of indices (`vector<long>`

)

`IndetsProd(f)`

-- monomial which is product of all indets actually in`f`

`ContentWRT(f, X)`

-- the content of`f`

wrt the indet(s)`X`

; result is a`RingElem`

in`P`

`CoefficientsWRT(f, X)`

-- returns a`vector<CoeffPP>`

: each`CoeffPP`

has fields`myCoeff`

and`myPP`

where`myCoeff`

is an element of`P`

and`myPP`

is in`PPM(P)`

being a power product of the indets in`X`

; the entries are in**decreasing order**of`myPP`

.`CoeffVecWRT(f, x)`

--`x`

must be an indet; returns a`vector<RingElem>`

whose k-th entry contains the coeff of`x^k`

as an element of`P`

; NB the coeff may be zero!`CoeffVecWRTSupport(f, B)`

-- returns a`vector<RingElem>`

being the representation of`f`

in the basis given by**REVERSE**of`support(B)`

; error if`f`

is not in the span of the basis`CoeffHeight(f)`

-- returns the max of the absolute values of the coeffs of`f`

`IsPalindromic(f)`

-- returns`true`

iff`f`

is palindromic; error if`f`

is not univariate`reverse(f)`

-- returns`x^deg(f)*f(1/x)`

; error if`f`

is not univariate`reverse(f,t)`

-- returns`f`

with each power product`PP`

replaced`t/PP`

`graeffe(f)`

-- returns graeffe transformation of univariate`f`

; its roots are the squares of the roots of`f`

.`graeffe3(f)`

-- returns cubic graeffe transformation of univariate`f`

; its roots are the cubes of the roots of`f`

.`MinPoly`

-- [SparsePolyOps-MinPoly].

**NB** For running through the summands (or terms) of a
polynomial use `SparsePolyIter`

s (see `SparsePolyRing`

).

We have still doubts on the usefulness of these two functions:

`CmpWDegPartial(f1, f2, i)`

-- compare the first`i`

weighted degrees of the LPPs of`f1`

and`f2`

; result is <0 =0 >0 according as`deg(f1)`

< = >`deg(f2)`

`IsHomogPartial(f,i)`

-- says whether`f`

is homogeneous wrt the first`i`

components of the weighted degree

Use the following two functions with great care: they throw an **error**
if the `PPOrdering`

is not respected: (the coefficient `c`

may be 0)

`PushFront(f, c, t)`

-- add to`f`

the term`c`

*t where t is a PP belonging to`PPM(owner(f))`

and**assuming**that t >`LPP(f)`

or`f`

==0`PushBack(f, c, t)`

-- add to`f`

the term`c`

*t where t is a PP belonging to`PPM(owner(f))`

and**assuming**that t < t' for all t' appearing in`f`

.`PushFront(f, c, expv)`

-- add to`f`

the term`c`

*t where t is the PP with exponent vector`expv`

, and**assuming**that t >`LPP(f)`

or`f`

==0`PushBack(f, c, expv)`

-- add to`f`

the term`c`

*t where t is the PP with exponent vector`expv`

, and**assuming**that t < t' for all t' appearing in`f`

.

The corresponding member functions `myPushFront/myPushBack`

will not
check the validity of these assumptions, so they are sometimes considerably
faster. Since they do not check, use them with extra special care: they do
have a `CoCoA_ASSERT`

to perform checks when in DEBUG mode.

You can determine whether an element belongs to a *DenseUPolyRing* by
calling `IsDenseUPolyRing(owner(r))`

: the result is `true`

iff `r`

belongs to a poly ring.

Let `P`

denote a `DenseUPolyRing`

.
Let `f`

denote an element of `P`

.

`monomial(P,c,exp)`

--`c`

*x^`exp`

as an element of`P`

with`c`

an integer or in`CoeffRing(P)`

`exp`

a`MachineInt`

`coeff(f,d)`

-- the`d`

-th coefficient of`f`

(as a`ConstRingElem`

, read-only)

**WARNING** Use this functions with great care: no checks on size and degree

Let `f`

denote a *non-const* element of `P`

.

`myAssignCoeff(f,c,d)`

-- assigns the`d`

-th coefficient in`f`

to`c`

`myAssignZeroCoeff(f,d)`

`myAssignNonZeroCoeff(f,c,d)`

Operations combining elements of different rings will cause a run-time
**error**.

In all functions involving two `RingElem`

s either `r1`

or `r2`

may be replaced by a machine integer, or by a big integer (an element
of the class `BigInt`

). The integer value is automatically mapped into
the ring owning the `RingElem`

in the same expression.

The exponent `n`

in the power function may be zero or negative, but a
run-time error will be signalled if one attempts to compute a negative
power of a non-invertible element.
**NB** You cannot use `^`

to compute powers -- see *Bugs* section.

An attempt to perform an inexact division or to compute a GCD not in a GCD domain will produce a run-time error.

The printing of ring elements is always in decimal regardless of the
`ostream`

settings (this is supposed to be a feature rather than a bug).

At this point, if you are new to CoCoALib, you should probably look
at some of the example programs in the `examples/`

directory.

One would normally expect to use the type `const RingElem&`

for
read-only arguments which are `RingElem`

s, and `RingElem&`

for
read-write arguments. Unfortunately, doing so would lead to problems
with the CoCoA library. **INSTEAD** you should use the types:

`ConstRefRingElem x` |
for read-only arguments: morally `const RingElem& x` |

`RingElem& x` |
for read-write arguments |

`RingElem x` |
for read-only arguments which make a local copy |

If you are curious to know why this non-standard quirk has to be used, read on.

When accessing matrix elements or coefficients in a polynomial
CoCoALib uses *proxies*: these are objects which should behave much
like `const RingElem`

values. To allow easy use of such proxies in
functions which want a read-only `RingElem`

we use the type
`ConstRefRingElem`

(which is actually `const RingElemAlias&`

) for
the formal parameter.

Internally, ring element values are really smart pointers to the true
value. Now the `const`

keyword in C++ when applied to a pointer
makes the pointer const while the pointed-to value remains alterable
-- this is not the behaviour we want for `const RingElem&`

. To get
the desired behaviour we have to use another type: the type we have
called `ConstRefRingElem`

.

The rest of this section is for more advanced use of `ring`

s and
`RingElem`

s (*e.g.* by CoCoA library contributors). If you are new to
CoCoA, you need not read beyond here.

**WE DO NOT RECOMMEND** that you use what is described in this section.
If you are curious to know a bit more how rings are implemented, you
might find this section informative.

`RingElem`

s are designed to be easy and pleasant to use, but this
convenience has a price: a run-time performance penalty (and a memory
space penalty too).
Both penalities may be avoided by using *raw values* but at a
considerable loss of programming convenience and safety. You should
consider using raw values only if you are desperate for speed; even
so, performance gains may be only marginal except perhaps for
operations on elements of a simple ring (e.g. a small finite field).

A `RingElem`

object contains within itself an indication of the owning
ring, and a *raw value* which is a pointer to where the real
representation of the ring element value lies. These raw values may be
accessed via the *raw* function. They may be combined arithmetically by
calling member functions of the owning ring. For instance, if x,y,z are
all RingElem objects all BELONGING TO EXACTLY THE SAME RING then we can
achieve

x = y+z;

slightly faster by calling

owner(x)->my.Add(raw(x), raw(y), raw(z));

It should now be clear that the syntax involved is cumbersome and
somewhat obscure. For the future maintainability of the code the
simpler `x = y+z;`

has many advantages. Furthermore, should x,y,z
somehow happen not all to lie in the same ring then `x = y+z;`

will act
in a reasonable way, whereas the supposedly faster call will likely lead
to many hours of debugging grief. The member functions for arithmetic
(e.g. `myAdd`

) **DO NOT PERFORM** sanity checks on their arguments:
e.g. attempting to divide by zero could well crash the program.

If you use a *debugging version* of the CoCoA Library then some
member functions do use assertions to check their arguments. This is
useful during development, but must not be relied upon since the
checks are absent from the *non-debugging version* of the CoCoA
Library. See the file config.txt for more information.

This fast, ugly, unsafe way of programming is made available for those who desperately need the speed. If you're not desperate, don't use it!

Read the section *Fast and Ugly Code* before using any of these!

Let `r`

be a non-const raw value (*e.g.* `raw(x)`

, with `x`

a
`RingElem`

), and `r1`

, `r2`

potentially const raw values.
Assume they are all owned by the ring `R`

.
Then the functions available are:

`R->myNew()`

--*construct*a new element of R, value=0`R->myNew(n)`

--*construct*a new element of R, value=n`R->myNew(N)`

--*construct*a new element of R, value=N`R->myNew(r1)`

--*construct*a new element of R, value=r1`R->myDelete(r)`

--*destroy*r, element of R (frees resources)`R->mySwap(r, s)`

-- swaps the two values (s is non-const raw value)`R->myAssignZero(r)`

-- r = 0`R->myAssign(r, r1)`

-- r = r1`R->myAssign(r, n)`

-- r = n (n is a long)`R->myAssign(r, N)`

-- r = n (N is a`BigInt`

)`R->myNegate(r, r1)`

-- r = -r1`R->myAdd(r, r1, r2)`

-- r = r1+r2`R->mySub(r, r1, r2)`

-- r = r1-r2`R->myMul(r, r1, r2)`

-- r = r1*r2`R->myDiv(r, r1, r2)`

-- r = r1/r2 (division must be exact)`R->myIsDivisible(r, r1, r2)`

-- r = r1/r2, and returns true iff division was exact`R->myIsZeroDivisor(r)`

-- returns true iff r is a zero-divisor`R->myIsUnit(r1)`

-- IsUnit(r1)`R->myGcd(r, r1, r2)`

-- r = gcd(r1, r2)`R->myLcm(r, r1, r2)`

-- r = lcm(r1, r2)`R->myPower(r, r1, n)`

-- r = power(r1, n) BUT n MUST be non-negative!!`R->myIsZero(r1)`

-- r1 == 0`R->myIsZeroAddMul(r, r1, r2)`

-- ((r += r1*r2) == 0)`R->myIsEqual(r1, r2)`

-- r1 == r2`R->myIsPrintAtom(r1)`

-- true iff`r1`

does not need brackets when a num or denom of a fraction`R->myIsPrintedWithMinus(r1)`

-- true iff the printed form of`r1`

begins with a minus sign`R->myOutput(out, r1)`

-- out << r1`R->mySequentialPower(r, r1, n)`

-- normally it is better to use R->myPower(r, r1, n)`R->myBinaryPower(r, r1, n)`

-- normally it is better to use R->myPower(r, r1, n)

(**NB** consider consulting also `QuotientRing`

, `FractionField`

and `PolyRing`

)

The design underlying rings and their elements is more complex than I would have liked, but it is not as complex as the source code may make it appear. The guiding principles are that the implementation should be flexible and easy/pleasant to use while offering a good degree of safety; extreme speed of execution was not a goal (as it is usually contrary to good flexibility) though an interface offering slightly better run-time efficiency remains.

Regarding flexibility: in CoCoALib we want to handle polynomials whose
coefficients reside in (almost) any commutative ring. Furthermore, the
actual rings to be used will be decided at run-time, and cannot
restricted to a given finite set. We have chosen to use C++ inheritance
to achieve the implementation: the abstract class `RingBase`

defines the
interface that every concrete ring class must offer.

Regarding ease of use: since C++ allows the common arithmetic operators
to be overloaded, it is essential that these work as expected for
elements of arbitrary rings -- with the caveat that `/`

means exact
division, being the only reasonable interpretation. Due to problems of
ambiguity arithmetic between elements of different rings is forbidden:
e.g. let f in Q[x,y] and g in Z[y,x], where should f+g reside?

The classes in the file ring.H are closely interrelated, and there is no obvious starting point for describing them -- you may find that you need to read the following more than once to comprehend it. Here is a list of the classes:

`ring` |
value represents a ring; it is a smart pointer |

`RingBase` |
abstract class defining what a ring is |

`RingElem` |
value represents an element of a ring |

`RingElemAlias` |
reference to a `RingElem` belonging to someone else |

`ConstRefRingElem` |
C++ const-reference to a `RingElemAlias` |

`RingElemConstRawPtr` |
raw pointer to a const ring value |

`RingElemRawPtr` |
raw pointer to a ring value |

For the first two see `ring`

.

The classes `RingElem`

and `RingElemAlias`

are
related by inheritance: they are very similar but differ in one important
way. The base class `RingElemAlias`

defines the data members
which are inherited by `RingElem`

. The essential difference is that
a `RingElem`

owns the value whereas a `RingElemAlias`

does not.
The two data members are `myR`

and `myRawValue`

: the first is the
identity of ring to which the element belongs, and the second is the
value in that ring (the value is stored in a format that only the
owning ring can comprehend). All operations on ring elements are
effected by member functions of the ring to which the value belongs.

The differing ownership inherent in `RingElemAlias`

and `RingElem`

lead to several consequences. The destructor of a `RingElem`

will
destroy in the internal representation of the value; in contrast, the
destructor of a `RingElemAlias`

does nothing. A `RingElemAlias`

object becomes meaningless (& dangerous) if the owner of the value it
aliases is destroyed.

Why did I create `RingElemAlias`

? The main reason was to allow
matrices and iterators of polynomials to be implemented cleanly and
efficiently. Clearly a `matrix`

should be the owner of the values
appearing as its entries, but we also want a way of reading the matrix
entries without having to copy them. Furthermore, the matrix can use
a compact representation: the ring to which its elements belong is
stored just once, and not once for each element. Analogous comments
apply to the coefficients of a polynomial.

As already stated above, the internal data layouts for objects of
types `RingElem`

and `RingElemAlias`

are identical -- this is
guaranteed by the C++ inheritance mechanism. The subfield indicating
the ring to which the value belongs is simply a `ring`

, which is
just a reference counting smart pointer. The subfield indicating the
value is a raw pointer of type `void*`

; however, when the raw
pointer value is to be handled outside a ring element object then it
is wrapped up as a `RingElemRawPtr`

or `RingElemConstRawPtr`

--
these are simply wrapped copies of the `void*`

.

The classes `RingElemRawPtr`

and `RingElemConstRawPtr`

are used
for two reasons. One is that if a naked `void*`

were used outside
the ring element objects then C++ would find the call
`RingElem(R,0)`

ambiguous because the constant `0`

can be
interpreted either as an integer constant or as a null pointer: there
are two constructors which match the call equally well. The other
reason is that it discourages accidentally creating a ring element
object from any old pointer; it makes the programmer think -- plus I
feel uneasy when there are naked `void*`

pointers around. Note that
the type of the data member `RingElemConstRawPtr::myPtr`

is simply
`void*`

as opposed to `void const*`

which one might reasonably
expect. I implemented it this way as it is simpler to add in the
missing constness in the member function
`RingElemConstRawPtr::myRawPtr`

than it would be to cast it away in
the `myRawPtr`

function of `RingElemRawPtr`

.

The class `RingBase`

declares a number of pure virtual functions for
computing with ring elements. Since these functions are pure they
must all be fully defined in any instantiable ring class
(e.g. `RingZZImpl`

or `RingFpImpl`

). These member functions follow
certain conventions:

**RETURN VALUES**:- most arithmetic functions return no value, instead the result is placed in one of the arguments (normally the first argument is the one in which the result is placed), but functions which return particularly simple values (e.g. booleans or machine integers) do indeed return the values by the usual function return mechanism.
**ARG TYPES**:-
ring element values are passed as
*raw pointers*(i.e. a wrapped`void*`

pointing to the actual value). A read-only arg is of type`RingElemConstRawPtr`

, while a writable arg is of type`RingElemRawPtr`

. When there are writable args they normally appear first. For brevity there are typedefs`ConstRawPtr`

and`RawPtr`

in the scope of`RingBase`

or any derived class. **ARG CHECKS**:- sanity checks on the arguments are NOT CONDUCTED (e.g. the division function assumes the divisor is non-zero). These member functions are supposed to be fast rather than safe.

In a few cases there are non-pure virtual member functions in
`RingBase`

. They exist either because there is a simple universal
definition or merely to avoid having to define inappropriate member
functions (*e.g.* gcd functions when the ring cannot be a gcd domain).
Here is a list of them:

`myIsUnit(x)`

-- default checks that 1 is divisible by`x`

`myIsZeroDivisor(x)`

-- special implementation in [QuotientRing] for setting primality flag to defining ideal`myGcd(lhs, x, y)`

-- gives an error: either`NotGcdDom`

or`NYI`

`myLcm(lhs, x, y)`

-- gives an error: either`NotGcdDom`

or`NYI`

`myGcdQuot(lhs, xquot, yquot, x, y)`

-- gives an error: either`NotGcdDom`

or`NYI`

`myExgcd(lhs, xcofac, ycofac, x, y)`

-- gives an error: either`NotGcdDom`

or`NYI`

`myIsPrintAtom(x)`

-- defaults to false`myIsPrintedWithMinus(x)`

-- gives`ShouldNeverGetHere`

error`myIsMinusOne(x)`

-- defaults to myIsOne(-x); calculates -x`myIsZeroAddMul(lhs, y, z)`

-- computes lhs += y*z in the obvious way, and calls`myIsZero`

`myCmp(x, y)`

-- gives`NotOrdDom`

error`myCmpAbs(x, y)`

-- tries to compute`cmp(abs(x),abs(y))`

so may give`NotOrdDom`

error`mySign(x)`

-- simply calls`myCmp(x, 0)`

, then returns -1,0,1 accordingly There are three non-virtual member functions for calculating powers: one uses the sequential method, the other two implement the repeated squaring method (one is an entry point, the other an implementation detail). These are non-virtual since they do not need to be redefined; they are universal for all rings. For the moment I shall assume that the intended meaning of the pure virtual functions is obvious (given the comments in the source code).

Recall that arithmetic operations on objects of type
`ConstRefRingElem`

(which matches `RingElem`

too) are converted
into member function calls of the corresponding owning ring. Here is
the source code for addition of ring elements -- it typifies the
implementation of operations on ring elements.

RingElem operator+(ConstRefRingElem x, ConstRefRingElem y) { const ring& Rx = owner(x); const ring& Ry = owner(y); if (Rx != Ry) error(CoCoAError(ERR::MixedRings, "RingElem + RingElem")); RingElem ans(Rx); Rx->myAdd(raw(ans), raw(x), raw(y)); return ans; }

The arguments are of type `ConstRefRingElem`

since they are
read-only, and the return type is `RingElem`

since it is new
self-owning value (it does not refer to a value belonging to some
other structure). Inside the function we check that the rings of the
arguments are compatible, and report an error if they are not.
Otherwise a temporary local variable is created for the answer, and
the actual computation is effected via a member function call to the
ring in which the values lie. Note the use of the `raw`

function
for accessing the raw pointer of a ring element. In summary, an
operation on ring elements intended for public use should fully check
its arguments for compatibility and correctness (*e.g.* to avoid
division by zero); if all checks pass, the result is computed by
passing raw pointers to the appropriate member functions of the ring
involved -- this member function assumes that the values handed to it
are compatible and valid; if not, *undefined behaviour* will result
(*i.e.* a crash if you are lucky).

Most of the member functions of a ring are for manipulating raw values
from that same ring, a few permit one to query properties of the ring.
The type of a raw value is `RingBase::RawValue`

, which helpfully
abbreviates to RawValue inside the namespace of `RingBase`

. Wherever
possible the concrete implementations should be **exception safe**, *i.e.* they
should offer either the strong exception guarantee or the no-throw
guarantee (according to the definitions in *Exceptional C++* by Sutter).

I have chosen not to use `operator^`

for computing powers because of a
significant risk of misunderstanding between programmer and compiler.
The syntax/grammar of C++ cannot be changed, and `operator^`

binds less
tightly than (binary) `operator*`

, so any expression of the form `a*b^c`

will be parsed as `(a*b)^c`

; this is almost certainly not what the
programmer intended. To avoid such problems of misunderstanding I
have preferred not to define `operator^`

; it seems too dangerous.

Note about comparison operators (<,<=,>,>=, and !=). The C++ STL
does have templates which will define all the relational operators
efficiently assuming the existence of `operator<`

and `operator==`

.
These are defined in the namespace `std::rel_ops`

in the standard
header file `<utility>`

. I have chosen NOT to use these because they can
define only *homogeneous* comparisons; so the comparisons between
`ConstRefRingElem`

and `int`

or `BigInt`

would still have to be written out
manually, and I prefer the symmetry of writing them all out.
See p.69ff of Josuttis for details.

The function `myAssignZero`

was NECESSARY because `myAssign(x, 0)`

was
ambiguous (ambiguated by the assignment from an `mpz_t`

). It is no longer
necessary, but I prefer to keep it (for the time being).

The requirement to use the type `ConstRefRingElem`

for function arguments
(which should normally be `const RingElem&`

is not ideal, but it seems hard
to find a better way. It is not nice to expect users to use a funny type
for their function arguments. How else could I implement (noncopying) access to
coefficients in a polynomial via an iterator, or access to matrix elements?

Would we want ++ and -- operators for `RingElem`

s???

Should (some of) the query functions return `bool3`

values?
What about properties which are hard to determine?

How to generate random elements from a ring?

Anna thinks that `NearestInteger`

could handle specially elements of
`RingZZ`

rather than doing the full wasteful computation. Not sure
if the extra code and complication would really make a difference in
practice.

`gcd`

and `lcm`

: there is no guarantee on sign/monic because it may
be costly to compute and generally useless.

**2013**

- May (v0.9953):
- added IsZeroDivisor -