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AlgebraicStructureTraits::DivMod

Definition

AdaptableFunctor computes both integral quotient and remainder of division with remainder. The quotient q and remainder r are computed such that x = q*y + r and |r| < |y| with respect to the proper integer norm of the represented ring. 1 In particular, r is chosen to be 0 if possible. Moreover, we require q to be minimized with respect to the proper integer norm.

Note that the last condition is needed to ensure a unique computation of the pair (q,r). However, an other option is to require minimality for |r|, with the advantage that a mod(x,y) operation would return the unique representative of the residue class of x with respect to y, e.g. mod(2,3) should return -1. But this conflicts with nearly all current implementation of integer types. From there, we decided to stay conform with common implementations and require q to be computed as x/y rounded towards zero.

The following table illustrates the behavior for integers:

x y q r
3 3 1 0
2 3 0 2
1 3 0 1
0 3 0 0
-1 3 0 -1
-2 3 0 -2
-3 3 -1 0
-
x y q r
3 -3 -1 0
2 -3 0 2
1 -3 0 1
0 -3 0 0
-1 -3 0 -1
-2 -3 0 -2
-3 -3 1 0

Refines

AdaptableFunctor

Types

AlgebraicStructureTraits::DivMod::result_type
Is void.

AlgebraicStructureTraits::DivMod::first_argument_type
Is AlgebraicStructureTraits::Type.

AlgebraicStructureTraits::DivMod::second_argument_type
Is AlgebraicStructureTraits::Type.

AlgebraicStructureTraits::DivMod::third_argument_type
Is AlgebraicStructureTraits::Type&.

AlgebraicStructureTraits::DivMod::fourth_argument_type
Is AlgebraicStructureTraits::Type&.

Operations

result_type
div_mod ( first_argument_type x ,
second_argument_type y ,
third_argument_type q ,
fourth_argument_type r )
computes the quotient q and remainder r, such that x = q*y + r and r minimal with respect to the Euclidean Norm on Type.

template <class NT1, class NT2>
result_type div_mod ( NT1 x , NT2 y , third_argument_type q , fourth_argument_type r )
This operator is well defined if NT1 and NT2 are ExplicitInteroperable with coercion type AlgebraicStructureTraits::Type.

See Also

AlgebraicStructureTraits
AlgebraicStructureTraits::Mod
AlgebraicStructureTraits::Div

Footnotes

 1  For integers this norm is the absolute value.
For univariate polynomials this norm is the degree.
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Navigation: Up   Table of Contents   Package Overview   Bibliography   Index   Title   Acknowledging CGAL
CGAL Open Source Project. Release 3.5. 1 October 2009.