mirror of https://github.com/CGAL/cgal
rename IntegralSqrt to IsSquare
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Michael Hemmer
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@ -4,8 +4,8 @@ Algebraic_foundations/doc_tex/Algebraic_foundations_ref/AlgebraicStructureTraits
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Algebraic_foundations/doc_tex/Algebraic_foundations_ref/AlgebraicStructureTraits_DivMod.tex -text
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Algebraic_foundations/doc_tex/Algebraic_foundations_ref/AlgebraicStructureTraits_Gcd.tex -text
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Algebraic_foundations/doc_tex/Algebraic_foundations_ref/AlgebraicStructureTraits_IntegralDivision.tex -text
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Algebraic_foundations/doc_tex/Algebraic_foundations_ref/AlgebraicStructureTraits_IntegralSqrt.tex -text
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Algebraic_foundations/doc_tex/Algebraic_foundations_ref/AlgebraicStructureTraits_IsOne.tex -text
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Algebraic_foundations/doc_tex/Algebraic_foundations_ref/AlgebraicStructureTraits_IsSquare.tex -text
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Algebraic_foundations/doc_tex/Algebraic_foundations_ref/AlgebraicStructureTraits_IsZero.tex -text
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Algebraic_foundations/doc_tex/Algebraic_foundations_ref/AlgebraicStructureTraits_KthRoot.tex -text
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Algebraic_foundations/doc_tex/Algebraic_foundations_ref/AlgebraicStructureTraits_Mod.tex -text
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@ -84,22 +84,19 @@ In case \ccc{Algebraic_structure} is a model of \ccc{IntegralDomainWithoutDivisi
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this is not \ccc{CGAL::Null_functor}.
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}
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\ccNestedType{Integral_division}{
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This is either \ccc{CGAL::Null_functor} or a model of
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\ccc{AlgebraicStructureTraits::IntegralDivision}.\\
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In case \ccc{Algebraic_structure} is a model of \ccc{IntegralDomain}
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this is not \ccc{CGAL::Null_functor}.
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}
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\begin{ccAdvanced}
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\ccNestedType{Integral_sqrt}{
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\ccNestedType{Is_square}{
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This is either \ccc{CGAL::Null_functor} or a model of
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\ccc{AlgebraicStructureTraits::IntegralSqrt}.\\
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\ccc{AlgebraicStructureTraits::IsSquare}.\\
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In case \ccc{Algebraic_structure} is a model of \ccc{IntegralDomain}
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this is not \ccc{CGAL::Null_functor}.
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}
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\end{ccAdvanced}
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\ccNestedType{Gcd}{
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This is either \ccc{CGAL::Null_functor} or a model of
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@ -2,14 +2,15 @@
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\begin{ccRefConcept}{AlgebraicStructureTraits::IsSquare}
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\begin{ccAdvanced}
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\ccDefinition
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\ccc{AdaptableBinaryFunction} providing an integral square root.
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An ring element $x$ is said to be a square if there exists a ring element $y$ such
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that $x= y*y$. Since the ring represented is an integral domain,
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$y$ is uniquely defined up to multiplication by units.
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An ring element $x$ is said to be a square iff there exists a ring element $y$ such
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that $x= y*y$. In case the ring is a \ccc{UniqueFactorizationDomain},
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$y$ is uniquely defined up to multiplication by units. \\
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%Note that this is an optional functor and not required by any algebraic structure concept.
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\ccRefines
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@ -27,8 +28,8 @@ $y$ is uniquely defined up to multiplication by units.
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\ccSetThreeColumns{xxxxxxxxxx}{xxxxx}{}
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\ccMethod{result_type operator()(first_argument_type x,
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second_argument_type y);}
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{ returns {\tt true} in case $x$ is a perfect square, i.e. $x = y*y$.
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\ccPostcond $unit\_part(y) == 1$. ( $y$ is unit normal.)
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{ returns {\tt true} in case $x$ is a square, i.e. $x = y*y$.
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\ccPostcond $unit\_part(y) == 1$.
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}
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%\ccHasModels
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@ -36,6 +37,5 @@ $y$ is uniquely defined up to multiplication by units.
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\ccSeeAlso
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\ccRefIdfierPage{AlgebraicStructureTraits}
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\end{ccAdvanced}
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\end{ccRefConcept}
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@ -2,23 +2,12 @@
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\ccRefChapter{Algebraic Foundations}
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%\label{ChapterRefAlgebraicFoundations}
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\ccChapterAuthor{Michael Hemmer and Who Soever}
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\ccChapterAuthor{Michael Hemmer}
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\section{todo:}
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\begin{itemize}
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\item add models to 'HasModels'
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\item add models to 'has models '
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\item Check types in functors
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\item Introduce a IsSquare rm IntegralSqrt\\
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\ccc{AlgebraicStructureTraits::Sqrt} has a special definition for Integer. \\
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A type may provide this functor even if the set of
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numbers it models does not contain real square roots in general.
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The most important example are number types modeling the integers.
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For them, Sqrt()(x) has to return the largest integer not exceeding
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the square root of x.\\
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In EXACUS this is used to test an integer to be an perfect square. \\
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Question: Should we introduce an extra functor for that?
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see: \ccc{AlgebraicStructureTraits::Integral_sqrt}
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\item TODO: solve this by using unit normal within text?
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The concept \ccc{FieldWithSqrt} (also FieldWithKthRoot,FieldWithRootOf)
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implies that the number type is RealEmbeddable, in particular RootOf talks
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@ -56,6 +45,17 @@ That is: an exact \ccc{Field} may have a Sqrt functor that is not exact. \\
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Proposal: choose the unit normal (positive) remainder. In other words div is rounded to -oo.\\
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Example: -5 div 3 is -2 rest 1 \\
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This also coincides with the definition of the gcd.
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\item \ccc{AlgebraicStructureTraits::IsSquare}: (is optional)\\
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\ccc{AlgebraicStructureTraits::Sqrt} had a special definition for Integer. \\
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A type may provide this functor even if the set of
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numbers it models does not contain real square roots in general.
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The most important example are number types modeling the integers.
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For them, Sqrt()(x) has to return the largest integer not exceeding
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the square root of x.\\
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In EXACUS this is used to test an integer to be an perfect square. \\
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\ccc{AlgebraicStructureTraits::IsSquare} has been introduced to avoid this obscure
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usage of Sqrt.
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\end{itemize}
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@ -117,7 +117,7 @@ That is: an exact \ccc{Field} may have a Sqrt functor that is not exact. \\
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\ccRefConceptPage{AlgebraicStructureTraits::Div}\\
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\ccRefConceptPage{AlgebraicStructureTraits::Mod}\\
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\ccRefConceptPage{AlgebraicStructureTraits::Sqrt}\\
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\ccRefConceptPage{AlgebraicStructureTraits::IntegralSqrt}\\
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\ccRefConceptPage{AlgebraicStructureTraits::IsSquare}\\
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\ccRefConceptPage{AlgebraicStructureTraits::KthRoot}\\
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\ccRefConceptPage{AlgebraicStructureTraits::RootOf}\\
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@ -141,7 +141,7 @@ That is: an exact \ccc{Field} may have a Sqrt functor that is not exact. \\
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\ccRefIdfierPage{simplify}\\
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\ccRefIdfierPage{unit_part}\\
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\ccRefIdfierPage{integral_division}\\
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%\ccRefIdfierPage{integral_sqrt}\\
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%\ccRefIdfierPage{is_square}\\
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\ccRefIdfierPage{gcd}\\
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\ccRefIdfierPage{div_mod}\\
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\ccRefIdfierPage{div}\\
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@ -25,7 +25,7 @@
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\input{Algebraic_foundations_ref/AlgebraicStructureTraits_Simplify.tex}
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\input{Algebraic_foundations_ref/AlgebraicStructureTraits_UnitPart.tex}
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\input{Algebraic_foundations_ref/AlgebraicStructureTraits_IntegralDivision.tex}
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\input{Algebraic_foundations_ref/AlgebraicStructureTraits_IntegralSqrt.tex}
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\input{Algebraic_foundations_ref/AlgebraicStructureTraits_IsSquare.tex}
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\input{Algebraic_foundations_ref/AlgebraicStructureTraits_Gcd.tex}
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\input{Algebraic_foundations_ref/AlgebraicStructureTraits_DivMod.tex}
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\input{Algebraic_foundations_ref/AlgebraicStructureTraits_Div.tex}
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