Defactoring: Difference between revisions

Line 167:

It is not possible for an RREF to be IREF without also being IRREF.

'''[https://en.wikipedia.org/wiki/Hermite_normal_form Hermite Normal Form]''', or '''HNF''', is the last form to discuss. This one's constraints begin with echelon form and integer, therefore every HNF is also IREF. But HNF is not exactly reduced; instead, it is ''normalized'', which — similarly to reduced — is a two-part constraint. Where reduced requires that all pivots be exactly equal to 1, normalized requires only that all pivots be positive (positive integers, of course, due to the other integer constraint). And where reduced requires that all entries in pivot columns besides the pivots are exactly equal to 0, normalized requires only that all entries in pivot columns below the pivots are exactly equal to 0, while entries in pivot columns above the pivots only have to be strictly less than the pivot in the respective column (while still being non-negative). The normalization HNF uses is cool because this constraint, while strictly less strict than the reduced constraint used by RREF, is still strict enough to ensure uniqueness, but loose enough to ensure the integer constraint can be simultaneously satisfied, where RREF cannot ensure that.

'''[https://en.wikipedia.org/wiki/Hermite_normal_form Hermite Normal Form]''', or '''HNF''', is the last form to discuss. This one's constraints begin with echelon form and integer, therefore every HNF is also IREF. But HNF is not exactly reduced; instead, it is ''normalized'', which — similarly to reduced — is a two-part constraint. Where reduced requires that all pivots be exactly equal to 1, normalized requires only that all pivots be positive (positive integers, of course, due to the other integer constraint). And where reduced requires that all entries in pivot columns besides the pivots are exactly equal to 0, normalized requires only that all entries in pivot columns below the pivots are exactly equal to 0, while entries in pivot columns above the pivots only have to be strictly less than the pivot in the respective column (while still being non-negative).<ref>The exact criteria for HNF are not always consistently agreed upon, however.</ref> The normalization HNF uses is cool because this constraint, while strictly less strict than the reduced constraint used by RREF, is still strict enough to ensure uniqueness, but loose enough to ensure the integer constraint can be simultaneously satisfied, where RREF cannot ensure that.

So HNF has a lot in common with IRREF, which is the IREF you find by converting the RREF, but it is not always the same as IRREF.

@@ Line 167: / Line 167: @@
 It is not possible for an RREF to be IREF without also being IRREF.
-'''[https://en.wikipedia.org/wiki/Hermite_normal_form Hermite Normal Form]''', or '''HNF''', is the last form to discuss. This one's constraints begin with echelon form and integer, therefore every HNF is also IREF. But HNF is not exactly reduced; instead, it is ''normalized'', which — similarly to reduced — is a two-part constraint. Where reduced requires that all pivots be exactly equal to 1, normalized requires only that all pivots be positive (positive integers, of course, due to the other integer constraint). And where reduced requires that all entries in pivot columns besides the pivots are exactly equal to 0, normalized requires only that all entries in pivot columns below the pivots are exactly equal to 0, while entries in pivot columns above the pivots only have to be strictly less than the pivot in the respective column (while still being non-negative). The normalization HNF uses is cool because this constraint, while strictly less strict than the reduced constraint used by RREF, is still strict enough to ensure uniqueness, but loose enough to ensure the integer constraint can be simultaneously satisfied, where RREF cannot ensure that.
+'''[https://en.wikipedia.org/wiki/Hermite_normal_form Hermite Normal Form]''', or '''HNF''', is the last form to discuss. This one's constraints begin with echelon form and integer, therefore every HNF is also IREF. But HNF is not exactly reduced; instead, it is ''normalized'', which — similarly to reduced — is a two-part constraint. Where reduced requires that all pivots be exactly equal to 1, normalized requires only that all pivots be positive (positive integers, of course, due to the other integer constraint). And where reduced requires that all entries in pivot columns besides the pivots are exactly equal to 0, normalized requires only that all entries in pivot columns below the pivots are exactly equal to 0, while entries in pivot columns above the pivots only have to be strictly less than the pivot in the respective column (while still being non-negative).<ref>The exact criteria for HNF are not always consistently agreed upon, however.</ref> The normalization HNF uses is cool because this constraint, while strictly less strict than the reduced constraint used by RREF, is still strict enough to ensure uniqueness, but loose enough to ensure the integer constraint can be simultaneously satisfied, where RREF cannot ensure that.
 So HNF has a lot in common with IRREF, which is the IREF you find by converting the RREF, but it is not always the same as IRREF.