[om] OpenMath bugs
Bill Naylor
Bill.Naylor at mcs.vuw.ac.nz
Thu Aug 7 03:20:58 CEST 2003
Hi David, Mike,
I have just come across some new (old!) bugs in the OpenMath content
dictionaries.
1) linalg5 contains six symbol names which contain the '-' character. This
is not allowed by the grammer in table 4.2 of the OpenMath standard. I
suggest these are replaced by '_'. I have attached linalg5.ocd and
linalg5.sts with these changes.
2) fns2 contains two references to the non-existent make_list from list1
cheers,
Bill
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<CD>
<CDComment>
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.
</CDComment>
<CDName> linalg5 </CDName>
<CDURL> http://www.openmath.org/cd/linalg5.ocd </CDURL>
<CDReviewDate> 2003-04-01 </CDReviewDate>
<CDDate> 1999-07-15 </CDDate>
<CDVersion> 2 </CDVersion>
<CDRevision> 0 </CDRevision>
<CDStatus> experimental </CDStatus>
<CDUses>
<CDName>alg1</CDName>
<CDName>arith1</CDName>
<CDName>linalg1</CDName>
<CDName>nums1</CDName>
<CDName>logic1</CDName>
<CDName>relation1</CDName>
</CDUses>
<Description>
This CD contains symbols which represent a number of special types of
matrix.
</Description>
<CDDefinition>
<Name> identity </Name>
<Description>
This symbol denotes a unary function which is used to construct an
(nxn) identity matrix where n is the single positive integral argument.
</Description>
<CMP>
for all M | identity(rowcount M) * M = M * identity(columncount M) = M
</CMP>
<FMP>
<OMOBJ>
<OMBIND>
<OMS cd="quant1" name="forall"/>
<OMBVAR>
<OMV name="M"/>
</OMBVAR>
<OMA>
<OMS cd="logic1" name="and"/>
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMS cd="arith1" name="times"/>
<OMA>
<OMS cd="linalg5" name="identity"/>
<OMA>
<OMS cd="linalg4" name="rowcount"/>
<OMV name="M"/>
</OMA>
</OMA>
<OMV name="M"/>
</OMA>
<OMV name="M"/>
</OMA>
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMS cd="arith1" name="times"/>
<OMV name="M"/>
<OMA>
<OMS cd="linalg5" name="identity"/>
<OMA>
<OMS cd="linalg4" name="columncount"/>
<OMV name="M"/>
</OMA>
</OMA>
</OMA>
<OMV name="M"/>
</OMA>
</OMA>
</OMBIND>
</OMOBJ>
</FMP>
<Example>
A representation of the 2x2 identity matrix [[1,0],[0,1]]
<OMOBJ>
<OMA>
<OMS cd="linalg5" name="identity"/>
<OMI> 2 </OMI>
</OMA>
</OMOBJ>
</Example>
</CDDefinition>
<CDDefinition>
<Name> zero </Name>
<Description>
This symbol denotes a function with two integral arguments m,n which
is used to construct an (mxn) zero matrix.
</Description>
<CMP>
for all M | M + zero(rowcount M,columncount M) = zero(rowcount M,columncount M)
</CMP>
<FMP>
<OMOBJ>
<OMBIND>
<OMS cd="quant1" name="forall"/>
<OMBVAR>
<OMV name="M"/>
</OMBVAR>
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMS cd="arith1" name="plus"/>
<OMV name="M"/>
<OMA>
<OMS cd="linalg5" name="zero"/>
<OMA>
<OMS cd="linalg4" name="rowcount"/>
<OMV name="M"/>
</OMA>
<OMA>
<OMS cd="linalg4" name="columncount"/>
<OMV name="M"/>
</OMA>
</OMA>
</OMA>
<OMA>
<OMS cd="linalg5" name="zero"/>
<OMA>
<OMS cd="linalg4" name="rowcount"/>
<OMV name="M"/>
</OMA>
<OMA>
<OMS cd="linalg4" name="columncount"/>
<OMV name="M"/>
</OMA>
</OMA>
</OMA>
</OMBIND>
</OMOBJ>
</FMP>
<CMP>
for all M |
zero(rowcount M,rowcount M) * M = M * zero(columncount M,columncount M) =
zero(rowcount M,columncount M)
</CMP>
<FMP>
<OMOBJ>
<OMBIND>
<OMS cd="quant1" name="forall"/>
<OMBVAR>
<OMV name="M"/>
</OMBVAR>
<OMA>
<OMS cd="logic1" name="and"/>
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMS cd="arith1" name="times"/>
<OMA>
<OMS cd="linalg5" name="zero"/>
<OMA>
<OMS cd="linalg4" name="rowcount"/>
<OMV name="M"/>
</OMA>
<OMA>
<OMS cd="linalg4" name="rowcount"/>
<OMV name="M"/>
</OMA>
</OMA>
<OMV name="M"/>
</OMA>
<OMA>
<OMS cd="linalg5" name="zero"/>
<OMA>
<OMS cd="linalg4" name="rowcount"/>
<OMV name="M"/>
</OMA>
<OMA>
<OMS cd="linalg4" name="columncount"/>
<OMV name="M"/>
</OMA>
</OMA>
</OMA>
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMS cd="arith1" name="times"/>
<OMV name="M"/>
<OMA>
<OMS cd="linalg5" name="zero"/>
<OMA>
<OMS cd="linalg4" name="columncount"/>
<OMV name="M"/>
</OMA>
<OMA>
<OMS cd="linalg4" name="columncount"/>
<OMV name="M"/>
</OMA>
</OMA>
</OMA>
<OMA>
<OMS cd="linalg5" name="zero"/>
<OMA>
<OMS cd="linalg4" name="rowcount"/>
<OMV name="M"/>
</OMA>
<OMA>
<OMS cd="linalg4" name="columncount"/>
<OMV name="M"/>
</OMA>
</OMA>
</OMA>
</OMA>
</OMBIND>
</OMOBJ>
</FMP>
<Example>
A representation of the 2x2 zero matrix [[0,0],[0,0]]
<OMOBJ>
<OMA>
<OMS cd="linalg5" name="zero"/>
<OMI> 2 </OMI>
<OMI> 2 </OMI>
</OMA>
</OMOBJ>
</Example>
</CDDefinition>
<CDDefinition>
<Name> diagonal_matrix </Name>
<Description>
This symbol denotes an n_ary function which is used to construct an
(nxn) diagonal matrix, that is a matrix where every non-diagonal
element is zero, the diagonal elements are equal to the n arguments.
</Description>
<CMP> given a diagonal matrix, it is equal to its transpose </CMP>
<Example>
The diagonal matrix with diagonal elements [1,2,3]
<OMOBJ>
<OMA>
<OMS cd="linalg5" name="diagonal_matrix"/>
<OMI> 1 </OMI>
<OMI> 2 </OMI>
<OMI> 3 </OMI>
</OMA>
</OMOBJ>
</Example>
</CDDefinition>
<CDDefinition>
<Name> scalar </Name>
<Description>
This symbol represents a matrix which is a scalar constant times the
identity matrix. It should take three arguments, the first and second
specify the number of rows and columns int he matrix respectively and
the third specifies the scalar multiplier.
</Description>
<CMP>
the scalar matrix of size n, where the scalar multiple is s
= s * identity(n)
</CMP>
<FMP>
<OMOBJ>
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMS cd="linalg5" name="scalar"/>
<OMV name="n"/>
<OMV name="s"/>
</OMA>
<OMA>
<OMS cd="arith1" name="times"/>
<OMV name="s"/>
<OMA>
<OMS cd="linalg5" name="identity"/>
<OMV name="n"/>
</OMA>
</OMA>
</OMA>
</OMOBJ>
</FMP>
</CDDefinition>
<CDDefinition>
<Name> constant </Name>
<Description>
This symbol represents a matrix which has all entries of the same
value. It takes two arguments, the first is the size of the matrix,
the second is the constant which determines every element.
</Description>
<CMP> the rank of a non-zero constant matrix = 1 </CMP>
<FMP>
<OMOBJ>
<OMA>
<OMS cd="logic1" name="implies"/>
<OMA>
<OMS cd="relation1" name="neq"/>
<OMV name="v"/>
<OMS cd="alg1" name="zero"/>
</OMA>
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMS cd="linalg4" name="rank"/>
<OMA>
<OMS cd="linalg5" name="constant"/>
<OMV name="n"/>
<OMV name="v"/>
</OMA>
</OMA>
<OMS cd="alg1" name="one"/>
</OMA>
</OMA>
</OMOBJ>
</FMP>
</CDDefinition>
<CDDefinition>
<Name> banded </Name>
<Description>
This symbol represents a (p,q) banded matrix, it takes one
argument. A (p,q) banded matrix should always be square. The lower non-zero
subdiagonal is the first element of the argument, whilst the highest non-zero
super-diagonal is given by the last element of the argument. The
argument determines the band of possibly non-zero entries which
are positioned around the diagonal. It should be a vector of vectors,
we note that they will not all be the same length, however the length
of the vectors determine p and q. The longest element specifies the
diagonal of the matrix and hence the size of the matrix. Every element
not in the band is zero.
</Description>
<Example>
A specification of the (2,1) banded matrix:
[
[1 2 3 0 0]
[4 5 6 7 0]
[0 8 9 10 11]
[0 0 12 13 14]
[0 0 0 15 16]]
<OMOBJ>
<OMA>
<OMS cd="linalg5" name="banded"/>
<OMA>
<OMS cd="linalg2" name="vector"/>
<OMA>
<OMS cd="linalg2" name="vector"/>
<OMI>4</OMI> <OMI>8</OMI> <OMI>12</OMI> <OMI>15</OMI>
</OMA>
<OMA>
<OMS cd="linalg2" name="vector"/>
<OMI>1</OMI> <OMI>5</OMI> <OMI>9</OMI> <OMI>13</OMI> <OMI>16</OMI>
</OMA>
<OMA>
<OMS cd="linalg2" name="vector"/>
<OMI>2</OMI> <OMI>6</OMI> <OMI>10</OMI> <OMI>14</OMI>
</OMA>
<OMA>
<OMS cd="linalg2" name="vector"/>
<OMI>3</OMI> <OMI>7</OMI> <OMI>11</OMI>
</OMA>
</OMA>
</OMA>
</OMOBJ>
</Example>
</CDDefinition>
<CDDefinition>
<Name> symmetric </Name>
<Description>
This symbol represents a symmetric matrix, it takes one argument. The
argument should be a vector of vectors of elements of the matrix. For
j>=i the ij'th element of the matrix is the (j-i+1)'th element of the i'th
element of the argument. This determines the upper triangle of the
matrix, the lower triangle is specified by the rule M = transpose M.
</Description>
<CMP>
the sum of a symmetric matrix and its transpose is symmetric
</CMP>
<FMP>
<OMOBJ>
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMS cd="arith1" name="plus"/>
<OMA>
<OMS cd="linalg5" name="symmetric"/>
<OMV name="VV1"/>
</OMA>
<OMA>
<OMS cd="linalg1" name="transpose"/>
<OMA>
<OMS cd="linalg5" name="symmetric"/>
<OMV name="VV1"/>
</OMA>
</OMA>
</OMA>
<OMA>
<OMS cd="linalg5" name="symmetric"/>
<OMV name="VV2"/>
</OMA>
</OMA>
</OMOBJ>
</FMP>
<CMP> for a symmetric matrix M, M = transpose M </CMP>
<FMP>
<OMOBJ>
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMS cd="linalg5" name="symmetric"/>
<OMV name="VV"/>
</OMA>
<OMA>
<OMS cd="linalg1" name="transpose"/>
<OMA>
<OMS cd="linalg5" name="symmetric"/>
<OMV name="VV"/>
</OMA>
</OMA>
</OMA>
</OMOBJ>
</FMP>
<CMP>
the dimension of a symmetric matrix = the length of the vector
which defines it
</CMP>
<FMP>
<OMOBJ>
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMS cd="linalg4" name="rowcount"/>
<OMA>
<OMS cd="linalg5" name="symmetric"/>
<OMV name="VV"/>
</OMA>
</OMA>
<OMA>
<OMS cd="linalg4" name="size"/>
<OMV name="VV"/>
</OMA>
</OMA>
</OMOBJ>
</FMP>
<Example>
An example to represent the symmetric matrix:
[[1,2,3,4]
[2,5,6,7]
[3,6,8,9]
[4,7,9,10]]
<OMOBJ>
<OMA>
<OMS cd="linalg5" name="symmetric"/>
<OMA>
<OMS cd="linalg2" name="vector"/>
<OMA>
<OMS cd="linalg2" name="vector"/>
<OMI> 1 </OMI>
<OMI> 2 </OMI>
<OMI> 3 </OMI>
<OMI> 4 </OMI>
</OMA>
<OMA>
<OMS cd="linalg2" name="vector"/>
<OMI> 5 </OMI>
<OMI> 6 </OMI>
<OMI> 7 </OMI>
</OMA>
<OMA>
<OMS cd="linalg2" name="vector"/>
<OMI> 8 </OMI>
<OMI> 9 </OMI>
</OMA>
<OMA>
<OMS cd="linalg2" name="vector"/>
<OMI> 10 </OMI>
</OMA>
</OMA>
</OMA>
</OMOBJ>
</Example>
</CDDefinition>
<CDDefinition>
<Name> skew_symmetric </Name>
<Description>
This symbol represents a skew-symmetric matrix, it takes one
argument. The argument should be a vector of vectors of elements of
the matrix. For j>i the ij'th element of the matrix is the (j-i+1)'th
element of the i'th element of the argument. This determines the
elements above the diagonal of the matrix, the elements below the
diagonal of the matrix must conform to the rule M = - transpose
M. This rule implies that the elements on the diagonal must be equal
to 0, therefore we do not include these in the argument.
</Description>
<CMP> The elements on the diagonal of a skew-symmetric matrix are zero </CMP>
<FMP>
<OMOBJ>
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMS cd="linalg1" name="matrix_selector"/>
<OMV name="i"/>
<OMV name="i"/>
<OMA>
<OMS cd="linalg5" name="skew_symmetric"/>
<OMV name="VV"/>
</OMA>
</OMA>
<OMS cd="alg1" name="zero"/>
</OMA>
</OMOBJ>
</FMP>
<CMP> for a skew-symmetric matrix M, M = - transpose M </CMP>
<FMP>
<OMOBJ>
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMS cd="linalg5" name="skew_symmetric"/>
<OMV name="VV"/>
</OMA>
<OMA>
<OMS cd="arith1" name="unary_minus"/>
<OMA>
<OMS cd="linalg1" name="transpose"/>
<OMA>
<OMS cd="linalg5" name="skew_symmetric"/>
<OMV name="VV"/>
</OMA>
</OMA>
</OMA>
</OMA>
</OMOBJ>
</FMP>
<Example>
An example to represent the skew-symmetric matrix:
[[ 0, 2, 3, 4]
[-2, 0, 6, 7]
[-3,-6, 0, 9]
[-4,-7,-9, 0]]
<OMOBJ>
<OMA>
<OMS cd="linalg5" name="skew_symmetric"/>
<OMA>
<OMS cd="linalg2" name="vector"/>
<OMA>
<OMS cd="linalg2" name="vector"/>
<OMI> 2 </OMI>
<OMI> 3 </OMI>
<OMI> 4 </OMI>
</OMA>
<OMA>
<OMS cd="linalg2" name="vector"/>
<OMI> 6 </OMI>
<OMI> 7 </OMI>
</OMA>
<OMA>
<OMS cd="linalg2" name="vector"/>
<OMI> 9 </OMI>
</OMA>
</OMA>
</OMA>
</OMOBJ>
</Example>
</CDDefinition>
<CDDefinition>
<Name> Hermitian </Name>
<Description>
This symbol represents a Hermitian matrix, it takes one
argument. The argument should be a vector of vectors of values which
determine the upper triangle of the matrix. The lower triangle of the
matrix is specified by the following relation: M^* = transpose(M),
were M^* denotes the matrix consisting of all the complex conjugates
of M.
</Description>
<CMP>
The complex conjugate of a Hermitian matrix equals its transpose
</CMP>
<CMP> The diagonal elements of a Hermitian matrix will be real </CMP>
<FMP>
<OMOBJ>
<OMA>
<OMS cd="set1" name="in"/>
<OMA>
<OMS cd="linalg1" name="matrix_selector"/>
<OMV name="i"/>
<OMV name="i"/>
<OMA>
<OMS cd="linalg5" name="Hermitian"/>
<OMV name="VV"/>
</OMA>
</OMA>
<OMS cd="setname1" name="R"/>
</OMA>
</OMOBJ>
</FMP>
<Example>
An example to describe the Hermitian matrix:
[[1+i , 2+2i]
[2-2i, 3+3i]]
<OMOBJ>
<OMA>
<OMS cd="linalg5" name="Hermitian"/>
<OMA>
<OMS cd="linalg2" name="vector"/>
<OMA>
<OMS cd="linalg2" name="vector"/>
<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>
<OMA>
<OMS cd="linalg2" name="vector"/>
<OMA>
<OMS cd="complex1" name="complex_cartesian"/>
<OMI> 3 </OMI><OMI> 3 </OMI>
</OMA>
</OMA>
</OMA>
</OMA>
</OMOBJ>
</Example>
</CDDefinition>
<CDDefinition>
<Name> anti_Hermitian </Name>
<Description>
This symbol represents an anti-Hermitian matrix, it takes one
argument. The argument should be a vector of vectors of values which
determine the upper triangle of the matrix. The lower triangle of the
matrix is specified by the following relation: - M^* = transpose(M),
were M^* denotes the matrix consisting of all the complex conjugates
of M. This rules implies that the main diagonal is zero, therefore the
argument should not include it.
</Description>
<CMP>
The complex conjugate of an anti-Hermitian matrix equals minus its transpose
</CMP>
<CMP> an anti-hermitian matrix will have zero on the diagonal </CMP>
<FMP>
<OMOBJ>
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMS cd="linalg1" name="matrix_selector"/>
<OMV name="i"/>
<OMV name="i"/>
<OMA>
<OMS cd="linalg5" name="anti_Hermitian"/>
<OMV name="VV"/>
</OMA>
</OMA>
<OMS cd="alg1" name="zero"/>
</OMA>
</OMOBJ>
</FMP>
<Example>
An example to describe the anti-Hermitian matrix:
[[0 , 1+i]
[-1+i , 0 ]]
<OMOBJ>
<OMA>
<OMS cd="linalg5" name="anti_Hermitian"/>
<OMA>
<OMS cd="linalg2" name="vector"/>
<OMA>
<OMS cd="complex1" name="complex_cartesian"/>
<OMI> 1 </OMI><OMI> 1 </OMI>
</OMA>
</OMA>
</OMA>
</OMOBJ>
</Example>
</CDDefinition>
<CDDefinition>
<Name> upper_triangular </Name>
<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.
</Description>
<CMP> the product of two upper-triangular matrices is upper-triangular </CMP>
<FMP>
<OMOBJ>
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMS cd="linalg5" name="upper_triangular"/>
<OMV name="VV1"/>
</OMA>
<OMA>
<OMS cd="arith1" name="times"/>
<OMA>
<OMS cd="linalg5" name="upper_triangular"/>
<OMV name="VV2"/>
</OMA>
<OMA>
<OMS cd="linalg5" name="upper_triangular"/>
<OMV name="VV3"/>
</OMA>
</OMA>
</OMA>
</OMOBJ>
</FMP>
<Example>
An example to describe the upper triangular matrix:
[[1,2,3]
[0,4,5]
[0,0,6]]
<OMOBJ>
<OMA>
<OMS cd="linalg5" 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>
</Example>
</CDDefinition>
<CDDefinition>
<Name> lower_triangular </Name>
<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.
</Description>
<CMP> the product of two lower-triangular matrices is lower-triangular </CMP>
<FMP>
<OMOBJ>
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMS cd="linalg5" name="lower_triangular"/>
<OMV name="VV1"/>
</OMA>
<OMA>
<OMS cd="arith1" name="times"/>
<OMA>
<OMS cd="linalg5" name="lower_triangular"/>
<OMV name="VV2"/>
</OMA>
<OMA>
<OMS cd="linalg5" name="lower_triangular"/>
<OMV name="VV3"/>
</OMA>
</OMA>
</OMA>
</OMOBJ>
</FMP>
<Example>
An example to describe the lower triangular matrix:
[[1,0,0]
[2,3,0]
[4,5,6]]
<OMOBJ>
<OMA>
<OMS cd="linalg5" 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>
</Example>
</CDDefinition>
<CDDefinition>
<Name> upper_Hessenberg </Name>
<Description>
This symbol represents an upper-Hessenberg matrix, it takes one argument,
the argument is a vector of vectors representing the non-zero
elements. The first element of the argument specifies the value of the
first subdiagonal, the subsequent elements specify the value of the
diagonal and subsequent super-diagonals, all other elements are zero.
</Description>
<Example>
A specification of an upper-Hessenberg matrix of dimension 5:
[[1 2 3 0 0]
[4 5 6 7 0]
[0 8 9 10 11]
[0 0 12 13 14]
[0 0 0 15 16]]
<OMOBJ>
<OMA>
<OMS cd="linalg5" name="upper_Hessenberg"/>
<OMA>
<OMS cd="linalg2" name="vector"/>
<OMA>
<OMS cd="linalg2" name="vector"/>
<OMI>4</OMI> <OMI>8</OMI> <OMI>12</OMI> <OMI>15</OMI>
</OMA>
<OMA>
<OMS cd="linalg2" name="vector"/>
<OMI>1</OMI> <OMI>5</OMI> <OMI>9</OMI> <OMI>13</OMI> <OMI>16</OMI>
</OMA>
<OMA>
<OMS cd="linalg2" name="vector"/>
<OMI>2</OMI> <OMI>6</OMI> <OMI>10</OMI> <OMI>14</OMI>
</OMA>
<OMA>
<OMS cd="linalg2" name="vector"/>
<OMI>3</OMI> <OMI>7</OMI> <OMI>11</OMI>
</OMA>
</OMA>
</OMA>
</OMOBJ>
</Example>
<CMP> the transpose of an upper-Hessenberg matrix is lower-Hessenberg </CMP>
<FMP>
<OMOBJ>
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMS cd="linalg1" name="transpose"/>
<OMA>
<OMS cd="linalg5" name="upper_Hessenberg"/>
<OMV name="VV1"/>
</OMA>
</OMA>
<OMA>
<OMS cd="linalg5" name="lower_Hessenberg"/>
<OMV name="VV2"/>
</OMA>
</OMA>
</OMOBJ>
</FMP>
</CDDefinition>
<CDDefinition>
<Name> lower_Hessenberg </Name>
<Description>
This symbol represents a lower-Hessenberg matrix, it takes one argument,
the argument is a vector of vectors representing the non-zero
elements. The first element of the argument specifies the value of the
first super-diagonal, the subsequent elements specify the value of the
diagonal and subsequent subdiagonals, all other elements are zero.
</Description>
<Example>
A specification of a lower-Hessenberg matrix of dimension 5:
[[1 2 0 0 0]
[3 4 5 0 0]
[6 7 8 9 0]
[0 10 11 12 13]
[0 0 14 15 16]]
<OMOBJ>
<OMA>
<OMS cd="linalg5" name="lower_Hessenberg"/>
<OMA>
<OMS cd="linalg2" name="vector"/>
<OMA>
<OMS cd="linalg2" name="vector"/>
<OMI>2</OMI> <OMI>5</OMI> <OMI>9</OMI> <OMI>13</OMI>
</OMA>
<OMA>
<OMS cd="linalg2" name="vector"/>
<OMI>1</OMI> <OMI>4</OMI> <OMI>8</OMI> <OMI>12</OMI> <OMI>16</OMI>
</OMA>
<OMA>
<OMS cd="linalg2" name="vector"/>
<OMI>3</OMI> <OMI>7</OMI> <OMI>11</OMI> <OMI>15</OMI>
</OMA>
<OMA>
<OMS cd="linalg2" name="vector"/>
<OMI>6</OMI> <OMI>10</OMI> <OMI>14</OMI>
</OMA>
</OMA>
</OMA>
</OMOBJ>
</Example>
<CMP> the transpose of a lower-Hessenberg matrix is upper-Hessenberg </CMP>
<FMP>
<OMOBJ>
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMS cd="linalg1" name="transpose"/>
<OMA>
<OMS cd="linalg5" name="lower_Hessenberg"/>
<OMV name="VV1"/>
</OMA>
</OMA>
<OMA>
<OMS cd="linalg5" name="upper_Hessenberg"/>
<OMV name="VV2"/>
</OMA>
</OMA>
</OMOBJ>
</FMP>
</CDDefinition>
<CDDefinition>
<Name> tridiagonal </Name>
<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.
</Description>
<CMP> a tridiagonal matrix is a (1,1) banded matrix </CMP>
<FMP>
<OMOBJ>
<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="linalg5" name="tridiagonal"/>
<OMV name="VV"/>
</OMA>
<OMA>
<OMS cd="linalg5" name="banded"/>
<OMV name="VV"/>
</OMA>
</OMA>
</OMA>
</OMOBJ>
</FMP>
<CMP> The product of two tridiagonal matrices is tridiagonal </CMP>
<FMP>
<OMOBJ>
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMS cd="arith1" name="times"/>
<OMA>
<OMS cd="linalg5" name="tridiagonal"/>
<OMV name="VV1"/>
</OMA>
<OMA>
<OMS cd="linalg5" name="tridiagonal"/>
<OMV name="VV2"/>
</OMA>
</OMA>
<OMA>
<OMS cd="linalg5" name="tridiagonal"/>
<OMV name="VV3"/>
</OMA>
</OMA>
</OMOBJ>
</FMP>
</CDDefinition>
</CD>
-------------- next part --------------
<CDSignatures type="sts" cd="linalg5">
<CDSComment> Written by W. Naylor 2000-03-14 </CDSComment>
<Signature name="diagonal_matrix">
<OMOBJ>
<OMA>
<OMS name="mapsto" cd="sts"/>
<OMA>
<OMS name="nary" cd="sts"/>
<OMS name="Object" cd="sts"/>
</OMA>
<OMV name="Matrix"/>
</OMA>
</OMOBJ>
</Signature>
<Signature name="identity">
<OMOBJ>
<OMA>
<OMS name="mapsto" cd="sts"/>
<OMS name="N" cd="setname1"/>
<OMV name="Matrix"/>
</OMA>
</OMOBJ>
</Signature>
<Signature name="zero">
<OMOBJ>
<OMA>
<OMS name="mapsto" cd="sts"/>
<OMS name="N" cd="setname1"/>
<OMS name="N" cd="setname1"/>
<OMV name="Matrix"/>
</OMA>
</OMOBJ>
</Signature>
<Signature name="scalar">
<OMOBJ>
<OMA>
<OMS name="mapsto" cd="sts"/>
<OMS name="N" cd="setname1"/>
<OMS name="NumericalValue" cd="sts"/>
<OMV name="Matrix"/>
</OMA>
</OMOBJ>
</Signature>
<Signature name="constant">
<OMOBJ>
<OMA>
<OMS name="mapsto" cd="sts"/>
<OMS name="N" cd="setname1"/>
<OMS name="NumericalValue" cd="sts"/>
<OMV name="Matrix"/>
</OMA>
</OMOBJ>
</Signature>
<Signature name="banded">
<OMOBJ>
<OMA>
<OMS name="mapsto" cd="sts"/>
<OMV name="VectorOfVector"/>
<OMV name="Matrix"/>
</OMA>
</OMOBJ>
</Signature>
<Signature name="symmetric">
<OMOBJ>
<OMA>
<OMS name="mapsto" cd="sts"/>
<OMV name="VectorOfVector"/>
<OMV name="Matrix"/>
</OMA>
</OMOBJ>
</Signature>
<Signature name="skew_symmetric">
<OMOBJ>
<OMA>
<OMS name="mapsto" cd="sts"/>
<OMV name="VectorOfVector"/>
<OMV name="Matrix"/>
</OMA>
</OMOBJ>
</Signature>
<Signature name="hermitian">
<OMOBJ>
<OMA>
<OMS name="mapsto" cd="sts"/>
<OMV name="VectorOfVector"/>
<OMV name="Matrix"/>
</OMA>
</OMOBJ>
</Signature>
<Signature name="anti_hermitian">
<OMOBJ>
<OMA>
<OMS name="mapsto" cd="sts"/>
<OMV name="VectorOfVector"/>
<OMV name="Matrix"/>
</OMA>
</OMOBJ>
</Signature>
<Signature name="upper_triangular">
<OMOBJ>
<OMA>
<OMS name="mapsto" cd="sts"/>
<OMV name="VectorOfVector"/>
<OMV name="Matrix"/>
</OMA>
</OMOBJ>
</Signature>
<Signature name="lower_triangular">
<OMOBJ>
<OMA>
<OMS name="mapsto" cd="sts"/>
<OMV name="VectorOfVector"/>
<OMV name="Matrix"/>
</OMA>
</OMOBJ>
</Signature>
<Signature name="upper_Hessenberg">
<OMOBJ>
<OMA>
<OMS name="mapsto" cd="sts"/>
<OMV name="VectorOfVector"/>
<OMV name="Matrix"/>
</OMA>
</OMOBJ>
</Signature>
<Signature name="lower_Hessenberg">
<OMOBJ>
<OMA>
<OMS name="mapsto" cd="sts"/>
<OMV name="VectorOfVector"/>
<OMV name="Matrix"/>
</OMA>
</OMOBJ>
</Signature>
<Signature name="tridiagonal">
<OMOBJ>
<OMA>
<OMS name="mapsto" cd="sts"/>
<OMV name="VectorOfVector"/>
<OMV name="Matrix"/>
</OMA>
</OMOBJ>
</Signature>
</CDSignatures>
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