OpenMath Content Dictionary: linalgspec1

Canonical URL:

http://www.openmath.org/cd/linalgspec1.ocd

CD Base:

http://www.openmath.org/cd

CD File:

linalgspec1.ocd

CD as XML Encoded OpenMath:

linalgspec1.omcd

Defines:

lower_triangular, tridiagonal, upper_triangular

Date:

2004-11-30

Version:

4 (Revision 2)

Review Date:

2006-03-30

Status:

experimental

---

     This document is distributed in the hope that it will be useful, 
     but WITHOUT ANY WARRANTY; without even the implied warranty of 
     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

     The copyright holder grants you permission to redistribute this 
     document freely as a verbatim copy. Furthermore, the copyright
     holder permits you to develop any derived work from this document
     provided that the following conditions are met.
       a) The derived work acknowledges the fact that it is derived from
          this document, and maintains a prominent reference in the 
          work to the original source.
       b) The fact that the derived work is not the original OpenMath 
          document is stated prominently in the derived work.  Moreover if
          both this document and the derived work are Content Dictionaries
          then the derived work must include a different CDName element,
          chosen so that it cannot be confused with any works adopted by
          the OpenMath Society.  In particular, if there is a Content 
          Dictionary Group whose name is, for example, `math' containing
          Content Dictionaries named `math1', `math2' etc., then you should 
          not name a derived Content Dictionary `mathN' where N is an integer.
          However you are free to name it `private_mathN' or some such.  This
          is because the names `mathN' may be used by the OpenMath Society
          for future extensions.
       c) The derived work is distributed under terms that allow the
          compilation of derived works, but keep paragraphs a) and b)
          intact.  The simplest way to do this is to distribute the derived
          work under the OpenMath license, but this is not a requirement.
     If you have questions about this license please contact the OpenMath
     society at http://www.openmath.org.

This CD contains symbols which represent a number of special types of matrix.

---

upper_triangular

Role:

application

Description:

This symbol represents an upper-triangular matrix, it takes one argument. The argument should be a vector of vectors of elements of the matrix.

Commented Mathematical property (CMP):

the product of two upper-triangular matrices is upper-triangular

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="linalgspec1" name="upper_triangular"/>
      <OMV name="VV1"/>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="times"/>
      <OMA>
        <OMS cd="linalgspec1" name="upper_triangular"/>
	<OMV name="VV2"/>
      </OMA>
      <OMA>
        <OMS cd="linalgspec1" name="upper_triangular"/>
	<OMV name="VV3"/>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="linalgspec1">upper_triangular</csymbol><ci>VV1</ci></apply>
  <apply><csymbol cd="arith1">times</csymbol>
   <apply><csymbol cd="linalgspec1">upper_triangular</csymbol><ci>VV2</ci></apply>
   <apply><csymbol cd="linalgspec1">upper_triangular</csymbol><ci>VV3</ci></apply>
  </apply>
 </apply>
</math>

Prefix

eq (upper_triangular ( VV1) , times (upper_triangular ( VV2) , upper_triangular ( VV3) ) )

Popcorn

linalgspec1.upper_triangular($VV1) = linalgspec1.upper_triangular($VV2) * linalgspec1.upper_triangular($VV3)

Rendered Presentation MathML

$$\mathrm{upper\_triangular}\left({\mathrm{VV}}_1\right)=\mathrm{upper\_triangular}\left({\mathrm{VV}}_2\right)\mathrm{upper\_triangular}\left({\mathrm{VV}}_3\right)$$

Example:

An example to describe the upper triangular matrix: [[1,2,3] [0,4,5] [0,0,6]]

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="linalgspec1" name="upper_triangular"/>
    <OMA>
      <OMS cd="linalg2" name="vector"/>
      <OMA>
        <OMS cd="linalg2" name="vector"/>
	<OMI> 1 </OMI><OMI> 2 </OMI><OMI> 3 </OMI>
      </OMA>
      <OMA>
        <OMS cd="linalg2" name="vector"/>
	<OMI> 4 </OMI><OMI> 5 </OMI>
      </OMA>
      <OMA>
        <OMS cd="linalg2" name="vector"/>
	<OMI> 6 </OMI>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="linalgspec1">upper_triangular</csymbol>
  <apply><csymbol cd="linalg2">vector</csymbol>
   <apply><csymbol cd="linalg2">vector</csymbol>
    <cn type="integer">1</cn>
    <cn type="integer">2</cn>
    <cn type="integer">3</cn>
   </apply>
   <apply><csymbol cd="linalg2">vector</csymbol>
    <cn type="integer">4</cn>
    <cn type="integer">5</cn>
   </apply>
   <apply><csymbol cd="linalg2">vector</csymbol><cn type="integer">6</cn></apply>
  </apply>
 </apply>
</math>

Prefix

upper_triangular (vector (vector ( 1 , 2 , 3 ) , vector ( 4 , 5 ) , vector ( 6 ) ) )

Popcorn

linalgspec1.upper_triangular(linalg2.vector(linalg2.vector(1, 2, 3), linalg2.vector(4, 5), linalg2.vector(6)))

Rendered Presentation MathML

$$\mathrm{upper\_triangular}\left(((1,2,3),(4,5),\left(6\right))\right)$$

Signatures:

sts

---

[Next: lower_triangular] [Last: tridiagonal] [Top]

---

lower_triangular

Role:

application

Description:

This symbol represents a lower-triangular matrix, it takes one argument. The argument should be a vector of vectors of elements of the matrix.

Commented Mathematical property (CMP):

the product of two lower-triangular matrices is lower-triangular

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="linalgspec1" name="lower_triangular"/>
      <OMV name="VV1"/>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="times"/>
      <OMA>
        <OMS cd="linalgspec1" name="lower_triangular"/>
	<OMV name="VV2"/>
      </OMA>
      <OMA>
        <OMS cd="linalgspec1" name="lower_triangular"/>
	<OMV name="VV3"/>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="linalgspec1">lower_triangular</csymbol><ci>VV1</ci></apply>
  <apply><csymbol cd="arith1">times</csymbol>
   <apply><csymbol cd="linalgspec1">lower_triangular</csymbol><ci>VV2</ci></apply>
   <apply><csymbol cd="linalgspec1">lower_triangular</csymbol><ci>VV3</ci></apply>
  </apply>
 </apply>
</math>

Prefix

eq (lower_triangular ( VV1) , times (lower_triangular ( VV2) , lower_triangular ( VV3) ) )

Popcorn

linalgspec1.lower_triangular($VV1) = linalgspec1.lower_triangular($VV2) * linalgspec1.lower_triangular($VV3)

Rendered Presentation MathML

$$\mathrm{lower\_triangular}\left({\mathrm{VV}}_1\right)=\mathrm{lower\_triangular}\left({\mathrm{VV}}_2\right)\mathrm{lower\_triangular}\left({\mathrm{VV}}_3\right)$$

Example:

An example to describe the lower triangular matrix: [[1,0,0] [2,3,0] [4,5,6]]

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="linalgspec1" name="lower_triangular"/>
    <OMA>
      <OMS cd="linalg2" name="vector"/>
      <OMA>
        <OMS cd="linalg2" name="vector"/>
	<OMI> 1 </OMI>
      </OMA>
      <OMA>
        <OMS cd="linalg2" name="vector"/>
	<OMI> 2 </OMI><OMI> 3 </OMI>
      </OMA>
      <OMA>
        <OMS cd="linalg2" name="vector"/>
	<OMI> 4 </OMI><OMI> 5 </OMI><OMI> 6 </OMI>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="linalgspec1">lower_triangular</csymbol>
  <apply><csymbol cd="linalg2">vector</csymbol>
   <apply><csymbol cd="linalg2">vector</csymbol><cn type="integer">1</cn></apply>
   <apply><csymbol cd="linalg2">vector</csymbol>
    <cn type="integer">2</cn>
    <cn type="integer">3</cn>
   </apply>
   <apply><csymbol cd="linalg2">vector</csymbol>
    <cn type="integer">4</cn>
    <cn type="integer">5</cn>
    <cn type="integer">6</cn>
   </apply>
  </apply>
 </apply>
</math>

Prefix

lower_triangular (vector (vector ( 1 ) , vector ( 2 , 3 ) , vector ( 4 , 5 , 6 ) ) )

Popcorn

linalgspec1.lower_triangular(linalg2.vector(linalg2.vector(1), linalg2.vector(2, 3), linalg2.vector(4, 5, 6)))

Rendered Presentation MathML

$$\mathrm{lower\_triangular}\left((\left(1\right),(2,3),(4,5,6))\right)$$

Signatures:

sts

---

[Next: tridiagonal] [Previous: upper_triangular] [Top]

---

tridiagonal

Role:

application

Description:

This symbol represents a tridiagonal matrix, it takes one argument which should be a vector of vectors which should have three elements. These should be vectors representing the sub-diagonal, the diagonal and the super-diagonal in that order.

Commented Mathematical property (CMP):

a tridiagonal matrix is a (1,1) banded matrix

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="logic1" name="and"/>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="linalg4" name="size"/>
	<OMV name="VV"/>
      </OMA>
      <OMI> 3 </OMI>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="linalg4" name="size"/>
	<OMA>
	  <OMS cd="linalg1" name="vector_selector"/>
	  <OMI> 2 </OMI>
	  <OMV name="VV"/>
	</OMA>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="plus"/>
	<OMA>
          <OMS cd="linalg4" name="size"/>
	  <OMA>
	    <OMS cd="linalg1" name="vector_selector"/>
	    <OMI> 1 </OMI>
	    <OMV name="VV"/>
	  </OMA>
	</OMA>
	<OMI> 1 </OMI>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="linalgspec1" name="tridiagonal"/>
	<OMV name="VV"/>
      </OMA>
      <OMA>
        <OMS cd="linalgspec1" name="banded"/>
	<OMV name="VV"/>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">and</csymbol>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="linalg4">size</csymbol><ci>VV</ci></apply>
   <cn type="integer">3</cn>
  </apply>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="linalg4">size</csymbol>
    <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">2</cn><ci>VV</ci></apply>
   </apply>
   <apply><csymbol cd="arith1">plus</csymbol>
    <apply><csymbol cd="linalg4">size</csymbol>
     <apply><csymbol cd="linalg1">vector_selector</csymbol><cn type="integer">1</cn><ci>VV</ci></apply>
    </apply>
    <cn type="integer">1</cn>
   </apply>
  </apply>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="linalgspec1">tridiagonal</csymbol><ci>VV</ci></apply>
   <apply><csymbol cd="linalgspec1">banded</csymbol><ci>VV</ci></apply>
  </apply>
 </apply>
</math>

Prefix

and (eq (size ( VV) , 3 ) , eq (size (vector_selector ( 2 , VV) ) , plus (size (vector_selector ( 1 , VV) ) , 1 ) ) , eq (tridiagonal ( VV) , banded ( VV) ) )

Popcorn

linalg4.size($VV) = 3 and linalg4.size(linalg1.vector_selector(2, $VV)) = linalg4.size(linalg1.vector_selector(1, $VV)) + 1 and linalgspec1.tridiagonal($VV) = linalgspec1.banded($VV)

Rendered Presentation MathML

$$\mathrm{size}\left(\mathrm{VV}\right)=3\wedge \mathrm{size}\left({\mathrm{VV}}_2\right)=\mathrm{size}\left({\mathrm{VV}}_1\right)+1\wedge \mathrm{tridiagonal}\left(\mathrm{VV}\right)=\mathrm{banded}\left(\mathrm{VV}\right)$$

Commented Mathematical property (CMP):

The product of two tridiagonal matrices is tridiagonal

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="arith1" name="times"/>
      <OMA>
        <OMS cd="linalgspec1" name="tridiagonal"/>
	<OMV name="VV1"/>
      </OMA>
      <OMA>
        <OMS cd="linalgspec1" name="tridiagonal"/>
	<OMV name="VV2"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="linalgspec1" name="tridiagonal"/>
      <OMV name="VV3"/>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="arith1">times</csymbol>
   <apply><csymbol cd="linalgspec1">tridiagonal</csymbol><ci>VV1</ci></apply>
   <apply><csymbol cd="linalgspec1">tridiagonal</csymbol><ci>VV2</ci></apply>
  </apply>
  <apply><csymbol cd="linalgspec1">tridiagonal</csymbol><ci>VV3</ci></apply>
 </apply>
</math>

Prefix

eq (times (tridiagonal ( VV1) , tridiagonal ( VV2) ) , tridiagonal ( VV3) )

Popcorn

linalgspec1.tridiagonal($VV1) * linalgspec1.tridiagonal($VV2) = linalgspec1.tridiagonal($VV3)

Rendered Presentation MathML

$$\mathrm{tridiagonal}\left({\mathrm{VV}}_1\right)\mathrm{tridiagonal}\left({\mathrm{VV}}_2\right)=\mathrm{tridiagonal}\left({\mathrm{VV}}_3\right)$$

Signatures:

sts

---

[First: upper_triangular] [Previous: lower_triangular] [Top]