Difference between revisions of "IsUPB"

	IsUPB
	Determines whether or not a set of product vectors form a UPB
Other toolboxes required	none
Related functions	MinUPBSize; UPB; UPBSepDistinguishable
Function category	Distinguishing objects

Latest revision as of 18:26, 21 October 2014

IsUPB is a function that determines whether or not a given set of product vectors forms an unextendible product basis (UPB).

Syntax

IU = IsUPB(U,V,W,...)

Argument descriptions

Input arguments

U,V,W,...: Matrices, each with the same number of columns as each other, whose columns are the local vectors of the supposed UPB.

Output arguments

IU: A flag (either 1 or 0) specifying that the set of vectors described by U,V,W,... is or is not a UPB.
WIT (optional): If IU = 0 then this is a cell containing local vectors for a product vector orthogonal to every member of the non-UPB described by U,V,W,... (thus, it acts as a witness to the fact that it really is not a UPB). If IU = 1 then WIT is meaningless.

Examples

The "Tiles" UPB

The following code verifies that the well-known "Tiles" UPB is indeed a UPB:

>> [u,v] = UPB('Tiles') % generate the local vectors of the "Tiles" UPB

u =

    1.0000    0.7071         0         0    0.5774
         0   -0.7071         0    0.7071    0.5774
         0         0    1.0000   -0.7071    0.5774


v =

    0.7071         0         0    1.0000    0.5774
   -0.7071         0    0.7071         0    0.5774
         0    1.0000   -0.7071         0    0.5774

>> IsUPB(u,v)

ans =

     1

However, if we remove the fifth vector from this set, then it is no longer a UPB:

>> [iu,wit] = IsUPB(u(:,1:4),v(:,1:4));
>> iu

iu =

     0

>> celldisp(wit) % display the witness
 
wit{1} =
 
     0
     0
     1

 
 
wit{2} =
 
    0.0000
    0.7071
    0.7071

Indeed, it is not difficult to verify that wit{1} is orthogonal to u(:,1) and u(:,2), and wit{2} is orthogonal to v(:,3) and v(:,4), so the first four columns of u,v do not specify a UPB.

Source code

Click here to view this function's source code on github.

@@ Line 1: / Line 1: @@
 {{Function
 |name=IsUPB
-|desc=Determines whether or a set of product vectors form a UPB
+|desc=Determines whether or not a set of product vectors form a UPB
 |rel=[[MinUPBSize]]<br />[[UPB]]<br />[[UPBSepDistinguishable]]
 |cat=[[List of functions#Distinguishing_objects|Distinguishing objects]]
-|upd=October 18, 2014
+|upd=October 21, 2014
 |v=0.50}}
 <tt>'''IsUPB'''</tt> is a [[List of functions|function]] that determines whether or not a given set of product vectors forms an unextendible product basis (UPB).
@@ Line 12: / Line 12: @@
 ==Argument descriptions==
+===Input arguments===
 * <tt>U,V,W,...</tt>: Matrices, each with the same number of columns as each other, whose columns are the local vectors of the supposed UPB.
+===Output arguments===
+* <tt>IU</tt>: A flag (either 1 or 0) specifying that the set of vectors described by <tt>U,V,W,...</tt> is or is not a UPB.
+* <tt>WIT</tt> (optional): If <tt>IU = 0</tt> then this is a cell containing local vectors for a product vector orthogonal to every member of the non-UPB described by <tt>U,V,W,...</tt> (thus, it acts as a witness to the fact that it really is not a UPB). If <tt>IU = 1</tt> then <tt>WIT</tt> is meaningless.
 ==Examples==
@@ Line 42: / Line 47: @@
 However, if we remove the fifth vector from this set, then it is no longer a UPB:
 <syntaxhighlight>
->> IsUPB(u(:,1:4),v(:,1:4))
+>> [iu,wit] = IsUPB(u(:,1:4),v(:,1:4));
+>> iu
+iu =
-ans =
+>> celldisp(wit) % display the witness
+wit{1} =
+wit{2} =
+.0000
+.7071
+.7071
 </syntaxhighlight>
+Indeed, it is not difficult to verify that <tt>wit{1}</tt> is orthogonal to <tt>u(:,1)</tt> and <tt>u(:,2)</tt>, and <tt>wit{2}</tt> is orthogonal to <tt>v(:,3)</tt> and <tt>v(:,4)</tt>, so the first four columns of <tt>u,v</tt> do not specify a UPB.
 {{SourceCode|name=IsUPB}}