Re: [Om3] OpenMath Symbols for Symbolic Computation

Thu, 11 Sep 2008 06:22:50 -0700 

Hi!


(1) There is no reasonable representation for matrices.
   (We developed a 'matrix1' CD, if you are interested, we'd love to
moot it.)
David Carlisle commented on this, but I'd like to see your matrix1 to see 
what sort of functionality you thought was wanted.



Ooooops! I just discovered that the matrix1.html-file I sent yesterday  
was a pretty outdated preliminary version. Please apologize and find  
the correct version attached.


Best regards, Peter
--
Peter   Horn,   University  of  Kassel          *
Computational    Mathematics     Group      *       *
Heinrich Plett Str. 40,  34132  Kassel    *  [S] [C] [I] [E] [n] [c] [e]
Phone: + 4 9 - 5 6 1 - 8 0 4 - 4 1 9 2   *   http://www.symcomp.org
Fax:   + 4 9 - 5 6 1 - 8 0 4 - 4 6 4 6    *
E-Mail: [EMAIL PROTECTED]      *       *
            http://kassel.symcomp.org/          *




Title: matrix1





OpenMath Content Dictionary: matrix1




Canonical URL:


http://www.symcomp.org/cd/matrix1.ocd


CD File:


matrix1.ocd
  


CD as XML Encoded OpenMath:


matrix1.omcd
  


Defines:



banded, block, column_dimension, dense, diagonal, entry_domain, lower_band, matrix, matrix_domain, row_dimension, sparse, sparse_entry, upper_band



Date:


 2008-04-10 


Version:


 0 


Review Date:


  2008-04-10  


Status:


 experimental 


Uses CD:



alg1, ringname1, linalg2, linalg3






  
  
  
  
  
  
  
  

  
    This CD holds a collection of matrix constructors over arbitrary rings.
  

  

    
 entry_domain 

    
      This symbol is a unary function, whose argument should be a ring r.
      When applied to r, it represents the matrix-algebra ground domain (MAD).
    

    


Example:



      
      
<OMOBJ>
	<OMA>
	  <OMS name="entry_domain" cd="matrix1"/>
	  <OMS name="Z" cd="ringname1"/>
	</OMA>
      </OMOBJ>



	entry_domain
(Z)

      

    



  







      [Next: matrix_domain]
    
      [Last: lower_band]
    
[Top]


  

    
 matrix_domain 

    
      This symbol is a ternary function, whose first argument should be a
      matrix1.entry_domain application.
      The second and third arguments must be matrix1.row_dimension and matrix1.column_dimension.

      When applied to these arguments this `creates' the domain of linear mappings
      between modules of specified dimensions over a common ground domain,
      conveniently represented by matrices.
    

    


Example:



      
      
<OMOBJ>
	<OMA>
	  <OMS name="matrix_domain" cd="matrix1"/>
	  <OMA>
	    <OMS name="entry_domain" cd="matrix1"/>
	    <OMS name="Z" cd="ringname1"/>
	  </OMA>
	  <OMA>
	    <OMS name="row_dimension" cd="matrix1"/>
	    <OMI>12</OMI>
	  </OMA>
	  <OMA>
	    <OMS name="column_dimension" cd="matrix1"/>
	    <OMI>10</OMI>
	  </OMA>
	</OMA>
      </OMOBJ>



	matrix_domain
(entry_domain
(Z)
, row_dimension
(12)
, column_dimension
(10)
)

      

    



  







      [Next: row_dimension]
    
      [Previous: entry_domain]
    
[Top]


  

    
 row_dimension 

    
      This symbol is a unary function whose first argument must be either a
      non-negative OpenMath integer or nums1.infinity.

      When applied this creates an object that denotes the dimension of the
      codomain of the linear mapping represented by the matrix.
    
  







      [Next: column_dimension]
    
      [Previous: matrix_domain]
    
[Top]


  

    
 column_dimension 

    
      This symbol is a unary function whose first argument must be either a
      non-negative OpenMath integer or nums1.infinity.

      When applied this creates an object that denotes the dimension of the
      domain of the linear mapping represented by the matrix.
    
  







      [Next: matrix]
    
      [Previous: row_dimension]
    
[Top]


  

    
 matrix 

    
      This symbol is a binary function whose first argument must be a
      matrix algebra constructor and the second argument can be any of the
      below matrix entry constructors.

      Additionally it is possible to use the matrix constructors of the
      linalg2 or linalg3 CDs.
    

    


Example:



      
<OMOBJ>
	  <OMSTRING>Todo</OMSTRING>
	  </OMOBJ>



	  Todo
	  

    



  







      [Next: dense]
    
      [Previous: column_dimension]
    
[Top]


  

    
 dense 

    
      This symbol is an $(m \cdot n)$-ary function whose arguments specify
      the entries of the matrix, where $m$ is the dimension of the codomain
      and $n$ is the dimension of the domain.

      The matrix (or block) must be filled row-wise, that is the first argument
      denotes the entry in row 1, column 1 of the matrix (or block), the second
      argument denotes the entry at row 1, column 2, and so forth.

      The number of arguments MUST match the dimensions of either the matrix
      algebra or the surrounding block (see below).
    

    


Example:



    
<OMOBJ>
	<OMA>
	  <OMS name="matrix" cd="matrix1"/>
	  <OMA>
	    <OMS name="matrix_domain" cd="matrix1"/>
	    <OMA><OMS name="entry_domain" cd="matrix1"/>
	      <OMS name="Z" cd="ringname1"/>
	    </OMA>
	    <OMA><OMS name="row_dimension" cd="matrix1"/><OMI>3</OMI></OMA>
	    <OMA><OMS name="column_dimension" cd="matrix1"/><OMI>3</OMI></OMA>
	  </OMA>
	  <OMA>
	    <OMS cd="matrix1" name="dense"/>
	    <OMI>1</OMI><OMI>2</OMI><OMI>3</OMI>
	    <OMI>4</OMI><OMI>5</OMI><OMI>6</OMI>
	    <OMI>7</OMI><OMI>8</OMI><OMI>9</OMI>
	  </OMA>
	</OMA>
      </OMOBJ>



	matrix
(matrix_domain
(entry_domain
(Z)
, row_dimension
(3)
, column_dimension
(3)
)
, dense
(1, 2, 3, 4, 5, 6, 7, 8, 9)
)

      

    



  







      [Next: sparse]
    
      [Previous: matrix]
    
[Top]


  

    
 sparse 

    
      The constructor for sparse matrices without any indication of dimension
      or domain for the coefficients.
      Its arguments are just matrix1.sparse_entrys.

      Attention: No two matrix1.sparse_entrys must specify the same location,
      it is however as of yet unspecified what happens in such a case.
	  A separate Content Dictionary for errors of this kind (e.g. matrix_error)
	  might be a suitable solution.
    
  







      [Next: sparse_entry]
    
      [Previous: dense]
    
[Top]


  

    
 sparse_entry 

    
      This symbol denotes a ternary function whose first two arguments specify
      the location of an entry inside the matrix, and whose final argument is
      the entry itself.

      The entry MUST be either from the specified ground domain directly,
      or be a diagonal constructor as described below or be a block constructor
      as described below.

      In the block case, the dimensions of the block MUST NOT exceed the total
      dimensions of the matrix algebra.

      In the diagonal case, the dimension of the diagonal MUST NOT exceed the
      total dimensions of the matrix algebra.
    

    


Example:



      
<OMOBJ>
	<OMA>
	  <OMS name="matrix" cd="matrix1"/>
	  <OMA>
	    <OMS name="matrix_domain" cd="matrix1"/>
	    <OMA><OMS name="entry_domain" cd="matrix1"/>
	      <OMS name="Q" cd="fieldname1"/>
	    </OMA>
	    <OMA><OMS name="row_dimension" cd="matrix1"/><OMI>3</OMI></OMA>
	    <OMA><OMS name="column_dimension" cd="matrix1"/><OMI>3</OMI></OMA>
	  </OMA>
	  <OMA>
	    <OMS cd="matrix1" name="sparse"/>
	    <OMA>
	      <OMS cd="matrix1" name="sparse_entry"/>
	      <OMI>1</OMI><OMI>2</OMI>
	      <OMI>12</OMI>
	    </OMA>
	    <OMA>
	      <OMS cd="matrix1" name="sparse_entry"/>
	      <OMI>2</OMI><OMI>1</OMI>
	      <OMI>21</OMI>
	    </OMA>
	    <OMA>
	      <OMS cd="matrix1" name="sparse_entry"/>
	      <OMI>3</OMI><OMI>3</OMI>
	      <OMI>33</OMI>
	    </OMA>
	  </OMA>
	</OMA>
      </OMOBJ>



	matrix
(matrix_domain
(entry_domain
(Q)
, row_dimension
(3)
, column_dimension
(3)
)
, sparse
(sparse_entry
(1, 2, 12)
, sparse_entry
(2, 1, 21)
, sparse_entry
(3, 3, 33)
)
)

      

    



  







      [Next: diagonal]
    
      [Previous: sparse]
    
[Top]


  

    
 diagonal 

    
      This symbol is an $m$-ary function whose arguments specify the entries of
      a (generalised) matrix diagonal.

      The diagonal must be filled from top-left to bottom-right.  That is: the
      first argument represents the entry in row 1, column 1 of the matrix,
      the second argument denotes the entry at row 2, column 2, and so forth.
      If used inside a sparse_entry object at location $(i, j)$, the first entry
      is offset accordingly.

      If not used inside a sparse_entry object, the number of arguments MUST
      match the dimensions of either the matrix algebra (the smaller of $m$ and
      $n$).

      If used inside a sparse_entry object, the number of entries MUST NOT
      exceed the total matrix dimensions.
    

    


Example:



      
<OMOBJ>
	<OMA>
	  <OMS name="matrix" cd="matrix1"/>
	  <OMA>
	    <OMS name="matrix_domain" cd="matrix1"/>
	    <OMA><OMS name="entry_domain" cd="matrix1"/>
	      <OMS name="C" cd="fieldname1"/>
	    </OMA>
	    <OMA><OMS name="row_dimension" cd="matrix1"/><OMI>3</OMI></OMA>
	    <OMA><OMS name="column_dimension" cd="matrix1"/><OMI>3</OMI></OMA>
	  </OMA>
	  <OMA>
	    <OMS cd="matrix1" name="diagonal"/>
	    <OMA>
	      <OMS cd="complex1" name="complex_cartesian"/>
	      <OMI>1</OMI><OMI>1</OMI>
	    </OMA>
	    <OMA>
	      <OMS cd="complex1" name="complex_cartesian"/>
	      <OMI>2</OMI><OMI>2</OMI>
	    </OMA>
	    <OMA>
	      <OMS cd="complex1" name="complex_cartesian"/>
	      <OMI>3</OMI><OMI>3</OMI>
	    </OMA>
	  </OMA>
	</OMA>
      </OMOBJ>



	matrix
(matrix_domain
(entry_domain
(C)
, row_dimension
(3)
, column_dimension
(3)
)
, diagonal
(complex_cartesian
(1, 1)
, complex_cartesian
(2, 2)
, complex_cartesian
(3, 3)
)
)

      

    



  







      [Next: block]
    
      [Previous: sparse_entry]
    
[Top]


  

    
 block 

    
      This symbol is like the matrix constructor as described above, but
      intended for use inside matrix to form ``submatrices''.  The symbol
      takes at least two arguments: a column_dimension and a row_dimension
      object which denote the total extent of the block.
    

    


Example:



      
<OMOBJ>
	<OMA>
	  <OMS name="matrix" cd="matrix1"/>
	  <OMA>
	    <OMS name="matrix_domain" cd="matrix1"/>
	    <OMA><OMS name="entry_domain" cd="matrix1"/>
	      <OMS name="Q" cd="fieldname1"/>
	    </OMA>
	    <OMA><OMS name="row_dimension" cd="matrix1"/><OMI>30</OMI></OMA>
	    <OMA><OMS name="column_dimension" cd="matrix1"/><OMI>30</OMI></OMA>
	  </OMA>
	  <OMA>
	    <OMS cd="matrix1" name="sparse"/>
	    <OMA>
	      <OMS cd="matrix1" name="sparse_entry"/>
	      <!-- something starts in row 10, column 20 -->
	      <OMI>10</OMI><OMI>20</OMI>
	      <OMA>
		<!-- It's a block of rectangle shape -->
		<OMS cd="matrix1" name="block"/>
		<!-- of dimensions 2 by 2 -->
		<OMA><OMS name="row_dimension" cd="matrix1"/><OMI>2</OMI></OMA>
		<OMA><OMS name="column_dimension" cd="matrix1"/><OMI>2</OMI></OMA>
		<!-- and it's got dense entries -->
		<OMA>
		  <OMS cd="matrix1" name="dense"/>
		  <OMI>11</OMI><OMI>12</OMI>
		  <OMI>21</OMI><OMI>22</OMI>
		</OMA>
	      </OMA>
	    </OMA>
	  </OMA>
	</OMA>
      </OMOBJ>



	matrix
(matrix_domain
(entry_domain
(Q)
, row_dimension
(30)
, column_dimension
(30)
)
, sparse
(sparse_entry
(10, 20, block
(row_dimension
(2)
, column_dimension
(2)
, dense
(11, 12, 21, 22)
)
)
)
)

      

    




    


Example:



      
      
<OMOBJ>
	<OMA>
	  <OMS name="matrix" cd="matrix1"/>
	  <OMA>
	    <OMS name="matrix_domain" cd="matrix1"/>
	    <OMA><OMS name="entry_domain" cd="matrix1"/>
	      <OMS name="Z" cd="ringname1"/>
	    </OMA>
	    <OMA>
	      <OMS name="row_dimension" cd="matrix1"/>
	      <OMA>
		<OMS cd="stupid1" name="busy_beaver"/>
		<OMI>12000</OMI>
	      </OMA>
	      <OMA>
		<OMS name="column_dimension" cd="matrix1"/>
		<OMA>
		  <OMS cd="stupid1" name="ackermann"/>
		  <OMI>499</OMI>
		  <OMI>12000</OMI>
		</OMA>
	      </OMA>
	    </OMA>
	  </OMA>
	  <OMA>
	    <OMS cd="matrix1" name="sparse"/>
	    <OMA>
	      <OMS cd="matrix1" name="sparse_entry"/>
	      <!-- something starts in row 24800, column 26133 -->
	      <OMI>24800</OMI><OMI>26133</OMI>
	      <OMA>
		<!-- again, it's a block -->
		<OMS cd="matrix1" name="block"/>
		<!-- dimensions 99999 by 99999 -->
		<OMA><OMS name="row_dimension" cd="matrix1"/><OMI>99999</OMI></OMA>
		<OMA><OMS name="column_dimension" cd="matrix1"/><OMI>99999</OMI></OMA>
		<!-- and it's got sparse entries. -->
		<!-- Yes, it IS redundant, but this is an example after all -->
		<OMA>
		  <OMS cd="matrix1" name="sparse"/>
		  <OMA>
		    <OMS cd="matrix1" name="sparse_entry"/>
		    <!-- position is relative to the top-left of block -->
		    <OMI>4</OMI><OMI>15</OMI>
		    <!-- more redundancy: we put a zero here -->
		    <OMI>0</OMI>
		  </OMA>
		</OMA>
	      </OMA>
	    </OMA>
	  </OMA>
	</OMA>
      </OMOBJ>



	matrix
(matrix_domain
(entry_domain
(Z)
, row_dimension
(busy_beaver
(12000)
, column_dimension
(ackermann
(499, 12000)
)
)
)
, sparse
(sparse_entry
(24800, 26133, block
(row_dimension
(99999)
, column_dimension
(99999)
, sparse
(sparse_entry
(4, 15, 0)
)
)
)
)
)

      

    



  







      [Next: banded]
    
      [Previous: diagonal]
    
[Top]


  

    
 banded 

    
      This symbol is a constructor for banded matrices.  It takes at least
      2 arguments, the first of which being the number of upper bands and
      the second being the number of lower bands.

      Amongst the further arguments you can specify AT MOST one
      diagonal object.  You MUST exactly as many
      upper_band objects as you specified upper bands, and
      you MUST specify as many lower_band objects as you
      specified lower bands.

      FYI: This symbol facilitates the use of blas based systems which expect
      to know the bands structure upfront.
    

    


Example:



      
<OMOBJ>
	<OMA>
	  <OMS name="matrix" cd="matrix1"/>
	  <OMA>
	    <OMS name="matrix_domain" cd="matrix1"/>
	    <OMA><OMS name="entry_domain" cd="matrix1"/>
	      <OMA><OMS name="Zm" cd="ringname1"/><OMI>7</OMI></OMA>
	    </OMA>
	    <OMA><OMS name="row_dimension" cd="matrix1"/><OMI>3</OMI></OMA>
	    <OMA><OMS name="column_dimension" cd="matrix1"/><OMI>3</OMI></OMA>
	  </OMA>
	  <OMA><OMS cd="matrix1" name="banded"/>
	    <OMI>1</OMI><!-- number of upper bands -->
	    <OMI>1</OMI><!-- number of lower bands -->
	    <OMA><OMS cd="matrix1" name="diagonal"/>
	      <!-- takes 3 args -->
	      <OMI>111</OMI><OMI>222</OMI><OMI>333</OMI>
	    </OMA>
	    <OMA><OMS cd="matrix1" name="upper_band"/>
	      <!-- first upper band -->
	      <OMI>1</OMI>
	      <OMA><OMS cd="matrix1" name="diagonal"/>
		    <!-- takes 2 args -->
		    <OMI>4</OMI><OMI>5</OMI>
	      </OMA>
	    </OMA>
	    <OMA><OMS cd="matrix1" name="lower_band"/>
	      <!-- first lower band -->
	      <OMI>1</OMI>
	      <OMA><OMS cd="matrix1" name="diagonal"/>
		    <!-- takes 2 args -->
		    <OMI>1</OMI><OMI>2</OMI>
	      </OMA>
	    </OMA>
	  </OMA>
	</OMA>
      </OMOBJ>



	matrix
(matrix_domain
(entry_domain
(Zm
(7)
)
, row_dimension
(3)
, column_dimension
(3)
)
, banded
(1, 1, diagonal
(111, 222, 333)
, upper_band
(1, diagonal
(4, 5)
)
, lower_band
(1, diagonal
(1, 2)
)
)
)

      

    



  







      [Next: upper_band]
    
      [Previous: block]
    
[Top]


  

    
 upper_band 

    
      This symbol is a binary function whose first argument is a non-negative
      OpenMath integer which denotes the index of the upper band which is
      specified in the second argument.

      Hereby the first upper band is the one immediately above the main
      (generalised) diagonal, its starting coordinates relative to the top-left
      of the matrix thus are (1, 2).
    
  







      [Next: lower_band]
    
      [Previous: banded]
    
[Top]


  

    
 lower_band 

    
      This symbol is a binary function whose first argument is a non-negative
      OpenMath integer which denotes the index of the lower band which is
      specified in the second argument.

      Hereby the first lower band is the one immediately below the main
      (generalised) diagonal, its starting coordinates relative to the top-left
      of the matrix thus are (2, 1).
    
  







      [First: entry_domain]
    
      [Previous: upper_band]
    
[Top]





_______________________________________________
Om3 mailing list
[email protected]
http://openmath.org/mailman/listinfo/om3






















					Reply via email to