<a name=r38_0600>
<title>matrix_augment</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>MATRIX_AUGMENT</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
Matrix augment, matrix stack:
<P>
<P>
<P> <H3>
syntax: </H3>
<em>matrix_augment</em>{<matrix\_list>}
<P>
<P>
<P>
(If you are feeling lazy then the braces can be omitted.)
<P>
<P>
<matrix\_list> :- matrices.
<P>
<P>
<em>matrix_augment</em>sticks the matrices in <matrix\_list>
together horizontally.
<P>
<P>
<em>matrix_stack</em>sticks the matrices in <matrix\_list>
together vertically.
<P>
<P>
<P> <H3>
examples: </H3>
<p><pre><tt>
matrix_augment({A,A});
[1 2 3 1 2 3]
[ ]
[4 5 6 4 5 6]
[ ]
[7 8 9 7 8 9]
matrix_stack(A,A);
[1 2 3]
[ ]
[4 5 6]
[ ]
[7 8 9]
[ ]
[1 2 3]
[ ]
[4 5 6]
[ ]
[7 8 9]
</tt></pre><p>Related functions:
<a href=r38_0550.html#r38_0577>augment_columns</a>,
<a href=r38_0600.html#r38_0615>stack_rows</a>,
<a href=r38_0600.html#r38_0616>sub_matrix</a>.
<P>
<P>
<a name=r38_0601>
<title>matrixp</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>MATRIXP</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
<P>
<P>
<P> <H3>
syntax: </H3>
<em>matrixp</em>(<test\_input>)
<P>
<P>
<P>
<test\_input> :- anything you like.
<P>
<P>
<em>matrixp</em>is a boolean function that returns t if the input is a
matrix and nil otherwise.
<P>
<P>
<P> <H3>
examples: </H3>
<p><pre><tt>
matrixp A;
t
matrixp(doodlesackbanana);
nil
</tt></pre><p>Related functions:
<a href=r38_0600.html#r38_0614>squarep</a>,
<a href=r38_0600.html#r38_0621>symmetricp</a>.
<P>
<P>
<a name=r38_0602>
<title>matrix_stack</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>MATRIX_STACK</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
see:
<a href=r38_0600.html#r38_0600>matrix_augment</a>.
<P>
<P>
<a name=r38_0603>
<title>minor</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>MINOR</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
<P>
<P>
<P> <H3>
syntax: </H3>
<em>minor</em>(<matrix>,<r>,<c>)
<P>
<P>
<P>
<matrix> :- a
<a href=r38_0300.html#r38_0345>matrix</a>.
<r>,<c> :- positive integers.
<P>
<P>
<em>minor</em>computes the (<r>,<c>)'th minor of <matrix>.
This is created by removing the <r>'th row and the <c>'th
column from <matrix>.
<P>
<P>
<P> <H3>
examples: </H3>
<p><pre><tt>
minor(A,1,3);
[4 5]
[ ]
[7 8]
</tt></pre><p>Related functions:
<a href=r38_0600.html#r38_0609>remove_columns</a>,
<a href=r38_0600.html#r38_0610>remove_rows</a>.
<P>
<P>
<a name=r38_0604>
<title>mult_columns</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>MULT_COLUMNS</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
Mult columns, mult rows:
<P>
<P>
<P> <H3>
syntax: </H3>
<em>mult_columns</em>(<matrix>,<column\_list>,<expr>)
<P>
<P>
<P>
<matrix> :- a
<a href=r38_0300.html#r38_0345>matrix</a>.
<P>
<P>
<column\_list> :- a positive integer or a list of positive
integers.
<P>
<P>
<expr> :- an algebraic expression.
<P>
<P>
<em>mult_columns</em>returns a copy of <matrix> in which the
columns specified in <column\_list> have been multiplied by
<expr>.
<P>
<P>
<em>mult_rows</em>performs the same task on the rows of <matrix>.
<P>
<P>
<P> <H3>
examples: </H3>
<p><pre><tt>
mult_columns(A,{1,3},x);
[ x 2 3*x]
[ ]
[4*x 5 6*x]
[ ]
[7*x 8 9*x]
mult_rows(A,2,10);
[1 2 3 ]
[ ]
[40 50 60]
[ ]
[7 8 9 ]
</tt></pre><p>Related functions:
<a href=r38_0550.html#r38_0575>add_to_columns</a>,
<a href=r38_0550.html#r38_0576>add_to_rows</a>.
<P>
<P>
<a name=r38_0605>
<title>mult_rows</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>MULT_ROWS</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
see:
<a href=r38_0600.html#r38_0604>mult_columns</a>.
<P>
<P>
<a name=r38_0606>
<title>pivot</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>PIVOT</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
<P>
<P>
<P> <H3>
syntax: </H3>
<em>pivot</em>(<matrix>,<r>,<c>)
<P>
<P>
<P>
<matrix> :- a matrix.
<P>
<P>
<r>,<c> :- positive integers such that <matrix>(<r>,
<c>) neq 0.
<P>
<P>
<em>pivot</em>pivots <matrix> about it's (<r>,<c>)'th
entry.
<P>
<P>
To do this, multiples of the <r>'th row are added to every other
row in the matrix.
<P>
<P>
This means that the <c>'th column will be 0 except for the
(<r>,<c>)'th entry.
<P>
<P>
<P> <H3>
examples: </H3>
<p><pre><tt>
pivot(A,2,3);
[ - 1 ]
[-1 ------ 0]
[ 2 ]
[ ]
[4 5 6]
[ ]
[ 1 ]
[1 --- 0]
[ 2 ]
</tt></pre><p>Related functions:
<a href=r38_0600.html#r38_0612>rows_pivot</a>.
<P>
<P>
<a name=r38_0607>
<title>pseudo_inverse</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>PSEUDO_INVERSE</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
<P>
<P>
<P> <H3>
syntax: </H3>
<em>pseudo_inverse</em>(<matrix>)
<P>
<P>
<P>
<matrix> :- a
<a href=r38_0300.html#r38_0345>matrix</a>.
<P>
<P>
<em>pseudo_inverse</em>, also known as the Moore-Penrose inverse,
computes the pseudo inverse of <matrix>.
<P>
<P>
Given the singular value decomposition of <matrix>, i.e:
A = U*P*V^T, then the pseudo inverse A^-1 is defined by
A^-1 = V^T*P^-1*U.
<P>
<P>
Thus <matrix> * pseudo_inverse(A) = Id.
(Id is the identity matrix).
<P>
<P>
<P> <H3>
examples: </H3>
<p><pre><tt>
R := mat((1,2,3,4),(9,8,7,6));
[1 2 3 4]
r := [ ]
[9 8 7 6]
on rounded;
pseudo_inverse(R);
[ - 0.199999999996 0.100000000013 ]
[ ]
[ - 0.0499999999988 0.0500000000037 ]
[ ]
[ 0.0999999999982 - 5.57825497203e-12]
[ ]
[ 0.249999999995 - 0.0500000000148 ]
</tt></pre><p>Related functions:
<a href=r38_0600.html#r38_0617>svd</a>.
<P>
<P>
<a name=r38_0608>
<title>random_matrix</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>RANDOM_MATRIX</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
<P>
<P>
<P> <H3>
syntax: </H3>
<em>random_matrix</em>(<r>,<c>,<limit>)
<P>
<P>
<P>
<r>,<c>,<limit> :- positive integers.
<P>
<P>
<em>random_matrix</em>creates an <r> by <c> matrix with random
entries in the range -limit <entry <limit.
<P>
<P>
Switches:
<P>
<P>
<em>imaginary</em>:- if on then matrix entries are x+i*y where -limit <x,y
<<limit>.
<P>
<P>
<em>not_negative</em>:- if on then 0 <entry <<limit>. In the imagina
ry
case we have 0 <x,y <<limit>.
<P>
<P>
<em>only_integer</em>:- if on then each entry is an integer. In the imaginary
case x and y are integers.
<P>
<P>
<em>symmetric</em>:- if on then the matrix is symmetric.
<P>
<P>
<em>upper_matrix</em>:- if on then the matrix is upper triangular.
<P>
<P>
<em>lower_matrix</em>:- if on then the matrix is lower triangular.
<P>
<P>
<P> <H3>
examples: </H3>
<p><pre><tt>
on rounded;
random_matrix(3,3,10);
[ - 8.11911717343 - 5.71677292768 0.620580830035 ]
[ ]
[ - 0.032596262422 7.1655452861 5.86742633837 ]
[ ]
[ - 9.37155438255 - 7.55636708637 - 8.88618627557]
on only_integer, not_negative, upper_matrix, imaginary;
random_matrix(4,4,10);
[70*i + 15 28*i + 8 2*i + 79 27*i + 44]
[ ]
[ 0 46*i + 95 9*i + 63 95*i + 50]
[ ]
[ 0 0 31*i + 75 14*i + 65]
[ ]
[ 0 0 0 5*i + 52 ]
</tt></pre><p>
<a name=r38_0609>
<title>remove_columns</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>REMOVE_COLUMNS</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
Remove columns, remove rows:
<P>
<P>
<P> <H3>
syntax: </H3>
<em>remove_columns</em>(<matrix>,<column\_list>)
<P>
<P>
<P>
<matrix> :- a
<a href=r38_0300.html#r38_0345>matrix</a>.
<column\_list> :- either a positive integer or a list of positive
integers.
<P>
<P>
<em>remove_columns</em>removes the columns specified in
<column\_list> from <matrix>.
<P>
<P>
<em>remove_rows</em>performs the same task on the rows of <matrix>.
<P>
<P>
<P> <H3>
examples: </H3>
<p><pre><tt>
remove_columns(A,2);
[1 3]
[ ]
[4 6]
[ ]
[7 9]
remove_rows(A,{1,3});
[4 5 6]
</tt></pre><p>Related functions:
<a href=r38_0600.html#r38_0603>minor</a>.
<P>
<P>
<a name=r38_0610>
<title>remove_rows</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>REMOVE_ROWS</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
see:
<a href=r38_0600.html#r38_0609>remove_columns</a>.
<P>
<P>
<a name=r38_0611>
<title>row_dim</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>ROW_DIM</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
see:
<a href=r38_0550.html#r38_0584>column_dim</a>.
<P>
<P>
<a name=r38_0612>
<title>rows_pivot</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>ROWS_PIVOT</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
<P>
<P>
<P> <H3>
syntax: </H3>
<em>rows_pivot</em>(<matrix>,<r>,<c>,{<row\_list>})
<P>
<P>
<P>
<matrix> :- a namerefmatrix.
<P>
<P>
<r>,<c> :- positive integers such that <matrix>(<r>,
<c>) neq 0.
<P>
<P>
<row\_list> :- positive integer or a list of positive integers.
<P>
<P>
<em>rows_pivot</em>performs the same task as <em>pivot</em> but applies
the pivot only to the rows specified in <row\_list>.
<P>
<P>
<P> <H3>
examples: </H3>
<p><pre><tt>
N := mat((1,2,3),(4,5,6),(7,8,9),(1,2,3),(4,5,6));
[1 2 3]
[ ]
[4 5 6]
[ ]
n := [7 8 9]
[ ]
[1 2 3]
[ ]
[4 5 6]
rows_pivot(N,2,3,{4,5});
[1 2 3]
[ ]
[4 5 6]
[ ]
[7 8 9]
[ ]
[ - 1 ]
[-1 ------ 0]
[ 2 ]
[ ]
[0 0 0]
</tt></pre><p>Related functions:
<a href=r38_0600.html#r38_0606>pivot</a>.
<P>
<P>
<a name=r38_0613>
<title>simplex</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>SIMPLEX</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
<P>
<P>
<P> <H3>
syntax: </H3>
<em>simplex</em>(<max/min>,<objective function>,
{<linear inequalities>})
<P>
<P>
<P>
<max/min> :- either max or min (signifying maximize and
minimize).
<P>
<P>
<objective function> :- the function you are maximizing or
minimizing.
<P>
<P>
<linear inequalities> :- the constraint inequalities. Each one must
be of the form sum of variables (
<=,=,>=) number.
<P>
<P>
<em>simplex</em>applies the revised simplex algorithm to find the
optimal(either maximum or minimum) value of the
<objective function> under the linear inequality constraints.
<P>
<P>
It returns {optimal value,{ values of variables at this optimal}}.
<P>
<P>
The algorithm implies that all the variables are non-negative.
<P>
<P>
<P> <H3>
examples: </H3>
<p><pre><tt>
simplex(max,x+y,{x>=10,y>=20,x+y<=25});
***** Error in simplex: Problem has no feasible solution
simplex(max,10x+5y+5.5z,{5x+3z<=200,x+0.1y+0.5z<=12,
0.1x+0.2y+0.3z<=9, 30x+10y+50z<=1500});
{525.0,{x=40.0,y=25.0,z=0}}
</tt></pre><p>
<a name=r38_0614>
<title>squarep</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>SQUAREP</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
<P>
<P>
<P> <H3>
syntax: </H3>
<em>squarep</em>(<matrix>)
<P>
<P>
<P>
<matrix> :- a
<a href=r38_0300.html#r38_0345>matrix</a>.
<P>
<P>
<em>squarep</em>is a predicate that returns t if the <matrix> is
square and nil otherwise.
<P>
<P>
<P> <H3>
examples: </H3>
<p><pre><tt>
squarep(mat((1,3,5)));
nil
squarep(A);
t
</tt></pre><p>Related functions:
<a href=r38_0600.html#r38_0601>matrixp</a>,
<a href=r38_0600.html#r38_0621>symmetricp</a>.
<P>
<P>
<a name=r38_0615>
<title>stack_rows</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>STACK_ROWS</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
see:
<a href=r38_0550.html#r38_0577>augment_columns</a>.
<P>
<P>
<a name=r38_0616>
<title>sub_matrix</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>SUB_MATRIX</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
<P>
<P>
<P> <H3>
syntax: </H3>
<em>sub_matrix</em>(<matrix>,<row\_list>,<column\_list>)
<P>
<P>
<P>
<matrix> :- a matrix.
<row\_list>, <column\_list> :- either a positive integer or a
list of positive integers.
<P>
<P>
namesub_matrix produces the matrix consisting of the intersection of
the rows specified in <row\_list> and the columns specified in
<column\_list>.
<P>
<P>
<P> <H3>
examples: </H3>
<p><pre><tt>
sub_matrix(A,{1,3},{2,3});
[2 3]
[ ]
[8 9]
</tt></pre><p>Related functions:
<a href=r38_0550.html#r38_0577>augment_columns</a>,
<a href=r38_0600.html#r38_0615>stack_rows</a>.
<P>
<P>
<a name=r38_0617>
<title>svd</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>SVD</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
<P>
<P>
Singular value decomposition:
<P>
<P>
<P> <H3>
syntax: </H3>
<em>svd</em>(<matrix>)
<P>
<P>
<P>
<matrix> :- a
<a href=r38_0300.html#r38_0345>matrix</a> containing only numeric entries.
<P>
<P>
<em>svd</em>computes the singular value decomposition of <matrix>.
<P>
<P>
It returns
<P>
<P>
{U,P,V}
<P>
<P>
where A = U*P*V^T
<P>
<P>
and P = diag(sigma(1) ... sigma(n)).
<P>
<P>
sigma(i) for i= 1 ... n are the singular values of
<matrix>.
<P>
<P>
n is the column dimension of <matrix>.
<P>
<P>
The singular values of <matrix> are the non-negative square roots
of the eigenvalues of A^T*A.
<P>
<P>
U and V are such that U*U^T = V*V^T = V^T*V = Id.
Id is the identity matrix.
<P>
<P>
<P> <H3>
examples: </H3>
<p><pre><tt>
Q := mat((1,3),(-4,3));
[1 3]
q := [ ]
[-4 3]
on rounded;
svd(Q);
{
[ 0.289784137735 0.957092029805]
[ ]
[ - 0.957092029805 0.289784137735]
,
[5.1491628629 0 ]
[ ]
[ 0 2.9130948854]
,
[ - 0.687215403194 0.726453707825 ]
[ ]
[ - 0.726453707825 - 0.687215403194]
}
</tt></pre><p>
<a name=r38_0618>
<title>swap_columns</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>SWAP_COLUMNS</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
Swap columns, swap rows:
<P>
<P>
<P> <H3>
syntax: </H3>
<em>swap_columns</em>(<matrix>,<c1>,<c2>)
<P>
<P>
<P>
<matrix> :- a
<a href=r38_0300.html#r38_0345>matrix</a>.
<P>
<P>
<c1>,<c1> :- positive integers.
<P>
<P>
<em>swap_columns</em>swaps column <c1> of <matrix> with
column <c2>.
<P>
<P>
<em>swap_rows</em>performs the same task on two rows of <matrix>.
<P>
<P>
<P> <H3>
examples: </H3>
<p><pre><tt>
swap_columns(A,2,3);
[1 3 2]
[ ]
[4 6 5]
[ ]
[7 9 8]
swap_rows(A,1,3);
[7 8 9]
[ ]
[4 5 6]
[ ]
[1 2 3]
</tt></pre><p>Related functions:
<a href=r38_0600.html#r38_0619>swap_entries</a>.
<P>
<P>
<a name=r38_0619>
<title>swap_entries</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>SWAP_ENTRIES</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
<P>
<P>
<P> <H3>
syntax: </H3>
<em>swap_entries</em>(<matrix>,{<r1>,<c1>},{<r2>,
<c2>})
<P>
<P>
<P>
<matrix> :- a
<a href=r38_0300.html#r38_0345>matrix</a>.
<P>
<P>
<r1>,<c1>,<r2>,<c2> :- positive integers.
<P>
<P>
<em>swap_entries</em>swaps <matrix>(<r1>,<c1>) with
<matrix>(<r2>,<c2>).
<P>
<P>
<P> <H3>
examples: </H3>
<p><pre><tt>
swap_entries(A,{1,1},{3,3});
[9 2 3]
[ ]
[4 5 6]
[ ]
[7 8 1]
</tt></pre><p>Related functions:
<a href=r38_0600.html#r38_0618>swap_columns</a>,
<a href=r38_0600.html#r38_0620>swap_rows</a>.
<P>
<P>
<a name=r38_0620>
<title>swap_rows</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>SWAP_ROWS</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
see:
<a href=r38_0600.html#r38_0618>swap_columns</a>.
<P>
<P>
<a name=r38_0621>
<title>symmetricp</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>SYMMETRICP</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
<P>
<P>
<P> <H3>
syntax: </H3>
<em>symmetricp</em>(<matrix>)
<P>
<P>
<P>
<matrix> :- a
<a href=r38_0300.html#r38_0345>matrix</a>.
<P>
<P>
<em>symmetricp</em>is a predicate that returns t if the matrix is symmetric
and nil otherwise.
<P>
<P>
<P> <H3>
examples: </H3>
<p><pre><tt>
symmetricp(make_identity(11));
t
symmetricp(A);
nil
</tt></pre><p>Related functions:
<a href=r38_0600.html#r38_0601>matrixp</a>,
<a href=r38_0600.html#r38_0614>squarep</a>.
<P>
<P>
<a name=r38_0622>
<title>toeplitz</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>TOEPLITZ</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
<P>
<P>
<P> <H3>
syntax: </H3>
<em>toeplitz</em>(<expr\_list>)
<P>
<P>
<P>
(If you are feeling lazy then the braces can be omitted.)
<P>
<P>
<expr\_list> :- list of algebraic expressions.
<P>
<P>
<em>toeplitz</em>creates the toeplitz matrix from the <expr\_list>.
<P>
<P>
This is a square symmetric matrix in which the first expression is
placed on the diagonal and the i'th expression is placed on the (i-1)'th
sub and super diagonals.
<P>
<P>
It has dimension n where n is the number of expressions.
<P>
<P>
<P> <H3>
examples: </H3>
<p><pre><tt>
toeplitz({w,x,y,z});
[w x y z]
[ ]
[x w x y]
[ ]
[y x w x]
[ ]
[z y x w]
</tt></pre><p>
<a name=r38_0623>
<title>vandermonde</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>VANDERMONDE</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
<P>
<P>
<P> <H3>
syntax: </H3>
<em>vandermonde</em>({<expr\_list>})
<P>
<P>
<P>
(If you are feeling lazy then the braces can be omitted.)
<P>
<P>
<expr\_list> :- list of algebraic expressions.
<P>
<P>
<em>vandermonde</em>creates the vandermonde matrix from the
<expr\_list>.
<P>
<P>
This is the square matrix in which the (i,j)'th entry is
<expr\_list>(i)^(j-1).
<P>
<P>
It has dimension n where n is the number of expressions.
<P>
<P>
<P> <H3>
examples: </H3>
<p><pre><tt>
vandermonde({x,2*y,3*z});
[ 2 ]
[1 x x ]
[ ]
[ 2]
[1 2*y 4*y ]
[ ]
[ 2]
[1 3*z 9*z ]
</tt></pre><p>
<a name=r38_0624>
<title>Linear Algebra package</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>Linear Algebra package</b><menu>
<li><a href=r38_0550.html#r38_0571>Linear Algebra package introduction</a><P>
<li><a href=r38_0550.html#r38_0572>fast_la switch</a><P>
<li><a href=r38_0550.html#r38_0573>add_columns operator</a><P>
<li><a href=r38_0550.html#r38_0574>add_rows operator</a><P>
<li><a href=r38_0550.html#r38_0575>add_to_columns operator</a><P>
<li><a href=r38_0550.html#r38_0576>add_to_rows operator</a><P>
<li><a href=r38_0550.html#r38_0577>augment_columns operator</a><P>
<li><a href=r38_0550.html#r38_0578>band_matrix operator</a><P>
<li><a href=r38_0550.html#r38_0579>block_matrix operator</a><P>
<li><a href=r38_0550.html#r38_0580>char_matrix operator</a><P>
<li><a href=r38_0550.html#r38_0581>char_poly operator</a><P>
<li><a href=r38_0550.html#r38_0582>cholesky operator</a><P>
<li><a href=r38_0550.html#r38_0583>coeff_matrix operator</a><P>
<li><a href=r38_0550.html#r38_0584>column_dim operator</a><P>
<li><a href=r38_0550.html#r38_0585>companion operator</a><P>
<li><a href=r38_0550.html#r38_0586>copy_into operator</a><P>
<li><a href=r38_0550.html#r38_0587>diagonal operator</a><P>
<li><a href=r38_0550.html#r38_0588>extend operator</a><P>
<li><a href=r38_0550.html#r38_0589>find_companion operator</a><P>
<li><a href=r38_0550.html#r38_0590>get_columns operator</a><P>
<li><a href=r38_0550.html#r38_0591>get_rows operator</a><P>
<li><a href=r38_0550.html#r38_0592>gram_schmidt operator</a><P>
<li><a href=r38_0550.html#r38_0593>hermitian_tp operator</a><P>
<li><a href=r38_0550.html#r38_0594>hessian operator</a><P>
<li><a href=r38_0550.html#r38_0595>hilbert operator</a><P>
<li><a href=r38_0550.html#r38_0596>jacobian operator</a><P>
<li><a href=r38_0550.html#r38_0597>jordan_block operator</a><P>
<li><a href=r38_0550.html#r38_0598>lu_decom operator</a><P>
<li><a href=r38_0550.html#r38_0599>make_identity operator</a><P>
<li><a href=r38_0600.html#r38_0600>matrix_augment operator</a><P>
<li><a href=r38_0600.html#r38_0601>matrixp operator</a><P>
<li><a href=r38_0600.html#r38_0602>matrix_stack operator</a><P>
<li><a href=r38_0600.html#r38_0603>minor operator</a><P>
<li><a href=r38_0600.html#r38_0604>mult_columns operator</a><P>
<li><a href=r38_0600.html#r38_0605>mult_rows operator</a><P>
<li><a href=r38_0600.html#r38_0606>pivot operator</a><P>
<li><a href=r38_0600.html#r38_0607>pseudo_inverse operator</a><P>
<li><a href=r38_0600.html#r38_0608>random_matrix operator</a><P>
<li><a href=r38_0600.html#r38_0609>remove_columns operator</a><P>
<li><a href=r38_0600.html#r38_0610>remove_rows operator</a><P>
<li><a href=r38_0600.html#r38_0611>row_dim operator</a><P>
<li><a href=r38_0600.html#r38_0612>rows_pivot operator</a><P>
<li><a href=r38_0600.html#r38_0613>simplex operator</a><P>
<li><a href=r38_0600.html#r38_0614>squarep operator</a><P>
<li><a href=r38_0600.html#r38_0615>stack_rows operator</a><P>
<li><a href=r38_0600.html#r38_0616>sub_matrix operator</a><P>
<li><a href=r38_0600.html#r38_0617>svd operator</a><P>
<li><a href=r38_0600.html#r38_0618>swap_columns operator</a><P>
<li><a href=r38_0600.html#r38_0619>swap_entries operator</a><P>
<li><a href=r38_0600.html#r38_0620>swap_rows operator</a><P>
<li><a href=r38_0600.html#r38_0621>symmetricp operator</a><P>
<li><a href=r38_0600.html#r38_0622>toeplitz operator</a><P>
<li><a href=r38_0600.html#r38_0623>vandermonde operator</a><P>
</menu>
<a name=r38_0625>
<title>Smithex</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>SMITHEX</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
The operator <em>smithex</em> computes the Smith normal form S of a
<a href=r38_0300.html#r38_0345>matrix</a> A (say). It returns {S,P,P^-1} where P
*S*P^-1 = A.
<P>
<P>
<P> <H3>
syntax: </H3>
<em>smithex</em>(<matrix>,<variable>)
<P>
<P>
<matrix> :- a rectangular
<a href=r38_0300.html#r38_0345>matrix</a> of univariate polynomials in
<variable>.
<variable> :- the variable.
<P>
<P>
<P>
<P> <H3>
examples: </H3>
<p><pre><tt>
a := mat((x,x+1),(0,3*x^2));
[x x + 1]
[ ]
a := [ 2 ]
[0 3*x ]
smithex(a,x);
[1 0 ] [1 0] [x x + 1]
{ [ ], [ ], [ ] }
[ 3] [ 2 ] [ ]
[0 x ] [3*x 1] [-3 -3 ]
</tt></pre><p>
<a name=r38_0626>
<title>Smithex_int</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>SMITHEX\_INT</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
The operator <em>smithex_int</em> performs the same task as <em>smithex</em>
but on matrices containing only integer entries. Namely,
<em>smithex_int</em> returns {S,P,P^-1} where S is the smith normal
form of the input
<a href=r38_0300.html#r38_0345>matrix</a> (A say), and P*S*P^-1 = A.
<P>
<P>
<P> <H3>
syntax: </H3>
<em>smithex_int</em>(<matrix>)
<P>
<P>
<matrix> :- a rectangular
<a href=r38_0300.html#r38_0345>matrix</a> of integer entries.
<P>
<P>
<P>
<P> <H3>
examples: </H3>
<p><pre><tt>
a := mat((9,-36,30),(-36,192,-180),(30,-180,180));
[ 9 -36 30 ]
[ ]
a := [-36 192 -180]
[ ]
[30 -180 180 ]
smithex_int(a);
[3 0 0 ] [-17 -5 -4 ] [1 -24 30 ]
[ ] [ ] [ ]
{ [0 12 0 ], [64 19 15 ], [-1 25 -30] }
[ ] [ ] [ ]
[0 0 60] [-50 -15 -12] [0 -1 1 ]
</tt></pre><p>
<a name=r38_0627>
<title>Frobenius</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>FROBENIUS</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
The operator <em>frobenius</em> computes the <em>frobenius</em> normal form F of
a
<a href=r38_0300.html#r38_0345>matrix</a> (A say). It returns {F,P,P^-1} where P
*F*P^-1 = A.
<P>
<P>
<P> <H3>
syntax: </H3>
<em>frobenius</em>(<matrix>)
<P>
<P>
<matrix> :- a square
<a href=r38_0300.html#r38_0345>matrix</a>.
<P>
<P>
<P>
Field Extensions:
<P>
<P>
By default, calculations are performed in the rational numbers. To
extend this field the
<a href=r38_0600.html#r38_0635>arnum</a> package can be used. The package must
first be loaded by load_package arnum;. The field can now be extended
by using the defpoly command. For example, defpoly sqrt2**2-2; will
extend the field to include the square root of 2 (now defined by sqrt2).
<P>
<P>
Modular Arithmetic:
<P>
<P>
<em>Frobenius</em>can also be calculated in a modular base. To do this
first type on modular;. Then setmod p; (where p is a prime) will set
the modular base of calculation to p. By further typing on balanced_mod
the answer will appear using a symmetric modular representation. See
<a href=r38_0600.html#r38_0628>ratjordan</a> for an example.
<P>
<P>
<P> <H3>
examples: </H3>
<p><pre><tt>
a := mat((x,x^2),(3,5*x));
[ 2 ]
[x x ]
a := [ ]
[3 5*x]
frobenius(a);
[ 2] [1 x] [ - x ]
{ [0 - 2*x ], [ ], [1 -----] }
[ ] [0 3] [ 3 ]
[1 6*x ] [ ]
[ 1 ]
[0 --- ]
[ 3 ]
load_package arnum;
defpoly sqrt2**2-2;
a := mat((sqrt2,5),(7*sqrt2,sqrt2));
[ sqrt2 5 ]
a := [ ]
[7*sqrt2 sqrt2]
frobenius(a);
[0 35*sqrt2 - 2] [1 sqrt2 ] [ 1 ]
{ [ ], [ ], [1 - --- ] }
[1 2*sqrt2 ] [1 7*sqrt2] [ 7 ]
[ ]
[ 1 ]
[0 ----*sqrt2]
[ 14 ]
</tt></pre><p>
<a name=r38_0628>
<title>Ratjordan</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>RATJORDAN</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
The operator <em>ratjordan</em> computes the rational Jordan normal form R
of a
<a href=r38_0300.html#r38_0345>matrix</a> (A say). It returns {R,P,P^-1} where P
*R*P^-1 = A.
<P>
<P>
<P> <H3>
syntax: </H3>
<em>ratjordan</em>(<matrix>)
<P>
<P>
<matrix> :- a square
<a href=r38_0300.html#r38_0345>matrix</a>.
<P>
<P>
<P>
Field Extensions:
<P>
<P>
By default, calculations are performed in the rational numbers. To
extend this field the <em>arnum</em> package can be used. The package must
first be loaded by load_package arnum;. The field can now be extended
by using the defpoly command. For example, defpoly sqrt2**2-2; will
extend the field to include the square root of 2 (now defined by sqrt2).
See
<a href=r38_0600.html#r38_0627>frobenius</a> for an example.
<P>
<P>
Modular Arithmetic:
<P>
<P>
<em>ratjordan</em>can also be calculated in a modular base. To do this
first type on modular;. Then setmod p; (where p is a prime) will set
the modular base of calculation to p. By further typing on balanced_mod
the answer will appear using a symmetric modular representation.
<P>
<P>
<P> <H3>
examples: </H3>
<p><pre><tt>
a := mat((5,4*x),(2,x^2));
[5 4*x]
[ ]
a := [ 2 ]
[2 x ]
ratjordan(a);
[0 x*( - 5*x + 8)] [1 5] [ -5 ]
{ [ ], [ ], [1 -----] }
[ 2 ] [0 2] [ 2 ]
[1 x + 5 ] [ ]
[ 1 ]
[0 -----]
[ 2 ]
on modular;
setmod 23;
a := mat((12,34),(56,78));
[12 11]
a := [ ]
[10 9 ]
ratjordan(a);
[15 0] [16 8] [1 21]
{ [ ], [ ], [ ] }
[0 6] [19 4] [1 4 ]
on balanced_mod;
ratjordan(a);
[- 8 0] [ - 7 8] [1 - 2]
{ [ ], [ ], [ ] }
[ 0 6] [ - 4 4] [1 4 ]
</tt></pre><p>
<a name=r38_0629>
<title>Jordansymbolic</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>JORDANSYMBOLIC</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
The operator <em>jordansymbolic</em> computes the Jordan normal form J
of a
<a href=r38_0300.html#r38_0345>matrix</a> (A say). It returns {J,L,P,P^-1} where
P*J*P^-1 = A. L = {ll,mm} where mm is a name and ll is a list of
irreducible factors of p(mm).
<P>
<P>
<P> <H3>
syntax: </H3>
<em>jordansymbolic</em>(<matrix>)
<P>
<P>
<matrix> :- a square
<a href=r38_0300.html#r38_0345>matrix</a>.
<P>
<P>
<P>
Field Extensions:
<P>
<P>
By default, calculations are performed in the rational numbers. To
extend this field the
<a href=r38_0600.html#r38_0635>arnum</a> package can be used. The package must
first be loaded by load_package arnum;. The field can now be extended
by using the defpoly command. For example, defpoly sqrt2**2-2; will
extend the field to include the square root of 2 (now defined by sqrt2).
See
<a href=r38_0600.html#r38_0627>frobenius</a> for an example.
<P>
<P>
Modular Arithmetic:
<P>
<P>
<em>jordansymbolic</em>can also be calculated in a modular base. To do this
first type on modular;. Then setmod p; (where p is a prime) will set
the modular base of calculation to p. By further typing on balanced_mod
the answer will appear using a symmetric modular representation. See
<a href=r38_0600.html#r38_0628>ratjordan</a> for an example.
<P>
<P>
<P> <H3>
examples: </H3>
<p><pre><tt>
a := mat((1,y),(2,5*y));
[1 y ]
a := [ ]
[2 5*y]
jordansymbolic(a);
{
[lambda11 0 ]
[ ]
[ 0 lambda12]
,
2
lambda - 5*lambda*y - lambda + 3*y,lambda,
[lambda11 - 5*y lambda12 - 5*y]
[ ]
[ 2 2 ]
,
[ 2*lambda11 - 5*y - 1 5*lambda11*y - lambda11 - y + 1 ]
[---------------------- ---------------------------------]
[ 2 2 ]
[ 25*y - 2*y + 1 2*(25*y - 2*y + 1) ]
[ ]
[ 2*lambda12 - 5*y - 1 5*lambda12*y - lambda12 - y + 1 ]
[---------------------- ---------------------------------]
[ 2 2 ]
[ 25*y - 2*y + 1 2*(25*y - 2*y + 1) ]
}
</tt></pre><p>
<a name=r38_0630>
<title>Jordan</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>JORDAN</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>operator</b><P>
<P>
The operator <em>jordan</em> computes the Jordan normal form J
of a
<a href=r38_0300.html#r38_0345>matrix</a> (A say). It returns {J,P,P^-1} where P
*J*P^-1 = A.
<P>
<P>
<P> <H3>
syntax: </H3>
<em>jordan</em>(<matrix>)
<P>
<P>
<matrix> :- a square
<a href=r38_0300.html#r38_0345>matrix</a>.
<P>
<P>
<P>
Field Extensions:
By default, calculations are performed in the rational numbers. To
extend this field the <em>arnum</em> package can be used. The package must
first be loaded by load_package arnum;. The field can now be extended
by using the defpoly command. For example, defpoly sqrt2**2-2; will
extend the field to include the square root of 2 (now defined by sqrt2).
See
<a href=r38_0600.html#r38_0627>frobenius</a> for an example.
<P>
<P>
Modular Arithmetic:
<em>Jordan</em> can also be calculated in a modular base. To do this
first type on modular;. Then setmod p; (where p is a prime) will set
the modular base of calculation to p. By further typing on balanced_mod
the answer will appear using a symmetric modular representation. See
<a href=r38_0600.html#r38_0628>ratjordan</a> for an example.
<P>
<P>
<P> <H3>
examples: </H3>
<p><pre><tt>
a := mat((1,x),(0,x));
[1 x]
a := [ ]
[0 x]
jordan(a);
{
[1 0]
[ ]
[0 x]
,
[ 1 x ]
[------- --------------]
[ x - 1 2 ]
[ x - 2*x + 1 ]
[ ]
[ 1 ]
[ 0 ------- ]
[ x - 1 ]
,
[x - 1 - x ]
[ ]
[ 0 x - 1]
}
</tt></pre><p>
<a name=r38_0631>
<title>Matrix Normal Forms</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>Matrix Normal Forms</b><menu>
<li><a href=r38_0600.html#r38_0625>Smithex operator</a><P>
<li><a href=r38_0600.html#r38_0626>Smithex\_int operator</a><P>
<li><a href=r38_0600.html#r38_0627>Frobenius operator</a><P>
<li><a href=r38_0600.html#r38_0628>Ratjordan operator</a><P>
<li><a href=r38_0600.html#r38_0629>Jordansymbolic operator</a><P>
<li><a href=r38_0600.html#r38_0630>Jordan operator</a><P>
</menu>
<a name=r38_0632>
<title>Miscellaneous_Packages</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>MISCELLANEOUS PACKAGES</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>introduction</b><P>
<P>
REDUCE includes a large number of packages that have been contributed by
users from various fields. Some of these, together with their relevant
commands, switches and so on (e.g., the NUMERIC package), have
been described elsewhere. This section describes those packages for which
no separate help material exists. Each has its own switches, commands,
and operators, and some redefine special characters to aid in their
notation. However, the brief descriptions given here do not include all
such information. Readers are referred to the general package
documentation in this case, which can be found, along with the source
code, under the subdirectories <em>doc</em> and <em>src</em> in the
<em>reduce</em> directory. The
<a href=r38_0100.html#r38_0127>load_package</a> command is used to
load the files you wish into your system. There will be a short delay
while the package is loaded. A package cannot be unloaded. Once it
is in your system, it stays there until you end the session. Each package
also has a test file, which you will find under its name in the
<em>$reduce/xmpl</em> directory.
<P>
<P>
Finally, it should be mentioned that such user-contributed packages are
unsupported; any questions or problems should be directed to their
authors.
<P>
<P>
<a name=r38_0633>
<title>ALGINT_package</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>ALGINT</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>package</b><P>
<P>
<P>
<P>
Author: James H. Davenport
<P>
<P>
The <em>algint</em> package provides indefinite integration of square roots.
This package, which is an extension of the basic integration package
distributed with REDUCE, will analytically integrate a wide range of
expressions involving square roots. The
<a href=r38_0250.html#r38_0265>algint</a> switch provides for
the use of the facilities given by the package, and is automatically turned
on when the package is loaded. If you want to return to the standard
integration algorithms, turn
<a href=r38_0250.html#r38_0265>algint</a> off. An error message is given
if you try to turn the
<a href=r38_0250.html#r38_0265>algint</a> switch on when its package is not
loaded.
<P>
<P>
<a name=r38_0634>
<title>APPLYSYM</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>APPLYSYM</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>package</b><P>
<P>
<P>
<P>
Author: Thomas Wolf
<P>
<P>
This package provides programs APPLYSYM, QUASILINPDE and DETRAFO for
computing with infinitesimal symmetries of differential equations.
<P>
<P>
<a name=r38_0635>
<title>ARNUM</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>ARNUM</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>package</b><P>
<P>
<P>
<P>
Author: Eberhard Schruefer
<P>
<P>
This package provides facilities for handling algebraic numbers as polynomial
coefficients in REDUCE calculations. It includes facilities for introducing
indeterminates to represent algebraic numbers, for calculating splitting
fields, and for factoring and finding greatest common divisors in such
domains.
<P>
<P>
<a name=r38_0636>
<title>ASSIST</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>ASSIST</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>package</b><P>
<P>
<P>
<P>
Author: Hubert Caprasse
<P>
<P>
ASSIST contains a large number of additional general purpose functions
that allow a user to better adapt REDUCE to various calculational
strategies and to make the programming task more straightforward and more
efficient.
<P>
<P>
<a name=r38_0637>
<title>AVECTOR</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>AVECTOR</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>package</b><P>
<P>
<P>
<P>
Author: David Harper
<P>
<P>
This package provides REDUCE with the ability to perform vector algebra
using the same notation as scalar algebra. The basic algebraic operations
are supported, as are differentiation and integration of vectors with
respect to scalar variables, cross product and dot product, component
manipulation and application of scalar functions (e.g. cosine) to a vector
to yield a vector result.
<P>
<P>
<a name=r38_0638>
<title>BOOLEAN</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>BOOLEAN</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>package</b><P>
<P>
<P>
<P>
Author: Herbert Melenk
<P>
<P>
This package supports the computation with boolean expressions in the
propositional calculus. The data objects are composed from algebraic
expressions connected by the infix boolean operators and, or,
implies, equiv, and the unary prefix operator not.
Boolean allows you to simplify expressions built from these
operators, and to test properties like equivalence, subset property etc.
<P>
<P>
<a name=r38_0639>
<title>CALI</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>CALI</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>package</b><P>
<P>
<P>
<P>
Author: Hans-Gert Gr"abe
<P>
<P>
This package contains algorithms for computations in commutative algebra
closely related to the Groebner algorithm for ideals and modules. Its
heart is a new implementation of the Groebner algorithm that also allows
for the computation of syzygies. This implementation is also applicable to
submodules of free modules with generators represented as rows of a matrix.
<P>
<P>
<a name=r38_0640>
<title>CAMAL</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>CAMAL</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>package</b><P>
<P>
<P>
<P>
Author: John P. Fitch
<P>
<P>
This package implements in REDUCE the Fourier transform procedures of the
CAMAL package for celestial mechanics.
<P>
<P>
<a name=r38_0641>
<title>CHANGEVR</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>CHANGEVR</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>package</b><P>
<P>
Author: G. Ucoluk
<P>
<P>
This package provides facilities for changing the independent variables in
a differential equation. It is basically the application of the chain rule.
<P>
<P>
<a name=r38_0642>
<title>COMPACT</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>COMPACT</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>package</b><P>
<P>
<P>
<P>
Author: Anthony C. Hearn
<P>
<P>
COMPACT is a package of functions for the reduction of a polynomial in the
presence of side relations. COMPACT applies the side relations to the
polynomial so that an equivalent expression results with as few terms as
possible. For example, the evaluation of
<P>
<P>
<p><pre><tt>
compact(s*(1-sin x^2)+c*(1-cos x^2)+sin x^2+cos x^2,
{cos x^2+sin x^2=1});
</tt></pre><p>yields the result
<p><pre><tt>
2 2
SIN(X) *C + COS(X) *S + 1
</tt></pre><p><P>
<P>
The first argument to the operator <em>compact</em> is the expression
and the second is a list of side relations that can be
equations or simple expressions (implicitly equated to zero). The
kernels in the side relations may also be free variables with the
same meaning as in rules, e.g.
<p><pre><tt>
sin_cos_identity := {cos ~w^2+sin ~w^2=1}$
compact(u,in_cos_identity);
</tt></pre><p><P>
<P>
Also the full rule syntax with the replacement operator is allowed here.
<P>
<P>
<a name=r38_0643>
<title>CRACK</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>CRACK</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>package</b><P>
<P>
<P>
<P>
Authors: Andreas Brand, Thomas Wolf
<P>
<P>
CRACK is a package for solving overdetermined systems of partial or
ordinary differential equations (PDEs, ODEs). Examples of programs which
make use of CRACK for investigating ODEs (finding symmetries, first
integrals, an equivalent Lagrangian or a ``differential factorization'') are
included.
<P>
<P>
<a name=r38_0644>
<title>CVIT</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>CVIT</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>package</b><P>
<P>
<P>
<P>
Authors: V.Ilyin, A.Kryukov, A.Rodionov, A.Taranov
<P>
<P>
This package provides an alternative method for computing traces of Dirac
gamma matrices, based on an algorithm by Cvitanovich that treats gamma
matrices as 3-j symbols.
<P>
<P>
<a name=r38_0645>
<title>DEFINT</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>DEFINT</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>package</b><P>
<P>
<P>
<P>
Authors: Kerry Gaskell, Stanley M. Kameny, Winfried Neun
<P>
<P>
This package finds the definite integral of an expression in a stated
interval. It uses several techniques, including an innovative approach
based on the Meijer G-function, and contour integration.
<P>
<P>
<a name=r38_0646>
<title>DESIR</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>DESIR</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>package</b><P>
<P>
<P>
<P>
Authors: C. Dicrescenzo, F. Richard-Jung, E. Tournier
<P>
<P>
This package enables the basis of formal solutions to be computed for an
ordinary homogeneous differential equation with polynomial coefficients
over Q of any order, in the neighborhood of zero (regular or irregular
singular point, or ordinary point).
<P>
<P>
<a name=r38_0647>
<title>DFPART</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>DFPART</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>package</b><P>
<P>
<P>
<P>
Author: Herbert Melenk
<P>
<P>
This package supports computations with total and partial derivatives of
formal function objects. Such computations can be useful in the context
of differential equations or power series expansions.
<P>
<P>
<a name=r38_0648>
<title>DUMMY</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>DUMMY</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>package</b><P>
<P>
<P>
<P>
Author: Alain Dresse
<P>
<P>
This package allows a user to find the canonical form of expressions
involving dummy variables. In that way, the simplification of
polynomial expressions can be fully done. The indeterminates are general
operator objects endowed with as few properties as possible. In that way
the package may be used in a large spectrum of applications.
<P>
<P>
<a name=r38_0649>
<title>EXCALC</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>EXCALC</b> _ _ _ _ _ _ _ _ _ _ _ _ <b>package</b><P>
<P>
<P>
<P>
Author: Eberhard Schruefer
<P>
<P>
The <em>excalc</em> package is designed for easy use by all who are familiar
with the calculus of Modern Differential Geometry. The program is currently
able to handle scalar-valued exterior forms, vectors and operations between
them, as well as non-scalar valued forms (indexed forms). It is thus an ideal
tool for studying differential equations, doing calculations in general
relativity and field theories, or doing simple things such as calculating the
Laplacian of a tensor field for an arbitrary given frame.
<P>
<P>