reduce-historical/r38/lisp/csl/r38.doc/r38_0550.html

<a name=r38_0550>

<title>Axes_names</title></a>
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<b>AXES NAMES</b><P>
<P>
 
Inside REDUCE the choice of variable names for a graph is completely 
free. For referring to the GNUPLOT axes the names 
X and Y for 2 dimensions, X,Y and Z for 3 dimensions are used 
in the usual schoolbook sense independent from the variables of 
the REDUCE expression. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt></tt></pre><p>
<a name=r38_0551>

<title>Pointset</title></a>
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<b>POINTSET</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>type</b><P>
<P>
 
 <P>
<P>
A curve can be give as a set of precomputed points (a polygon) 
in 2 or 3 dimensions. Such a point set is a 
<a href=r38_0050.html#r38_0053>list</a> 
of points, where each point is a 
<a href=r38_0050.html#r38_0053>list</a> 2 (or 3) 
numbers. These numbers are interpreted as <em>(x,y)</em> 
(or <em>x,y,z</em>) coordinates. All points of one set must have 
the same dimension. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt></tt></pre><p>Also a surface in 3d can be given by precomputed point
s, 
but only on a logically orthogonal mesh: the surface is defined 
by a list of curves (in 3d) which must have a uniform length. 
GNUPLOT then will draw an orthogonal mesh by first drawing the 
given lines, and second connecting the 1st point of the 1st curve 
with the 1st point of the 2nd curve, that one with the 1st point 
of the 3rd curve and so on for all curves and for all indexes. 
<P>
<P>

<a name=r38_0552>

<title>PLOT</title></a>
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<b>PLOT</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
 <P>
<P>
The command <em>plot</em> is the main entry for drawing a 
picture from inside REDUCE. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>plot</em>(&lt;spec&gt;,&lt;spec&gt;,...) 
<P>
<P>
<P>
where &lt;spec&gt; is a &lt;function&gt;, a &lt;range&gt; or an &lt;option&gt;. 

<P>
<P>
&lt;function&gt;: 
<P>
<P>
- an expression depending 
on one unknown (e.g. <em>sin(x)</em> or two unknowns (e.g. 
<em>sin(x+y)</em>, 
<P>
<P>
- an equation with a function on its right-hand 
side and a single name on its left-hand side (e.g. 
<em>z=sin(x+y)</em> where the name on the left-hand side specifies 
the dependent variable. 
<P>
<P>
- a list of functions: 
if in 2 dimensions the picture should have more than one 
curve the expressions can be given as list (e.g. <em>{sin(x),cos(x)}</em>). 
<P>
<P>
- an equation with zero left or right hand side describing 
 an implicit curve in two dimensions (e.g. <em>x**3+x*y**3-9x=0</em>). 
<P>
<P>
- a point set: the graph can be given 
as point set in 2 dimensions or a 
<a href=r38_0550.html#r38_0551>pointset</a> or pointset list 
in 3 dimensions. 
<P>
<P>
&lt;range&gt;: 
<P>
<P>
Each dependent and independent variable can be limited 
to an interval by an equation where the left-hand side specifies 
the variable and the right-hand side defines the 
<a href=r38_0400.html#r38_0423>interval</a>, 
e.g. <em>x=( -3 .. 5)</em>. 
<P>
<P>
If omitted the independent variables 
range from -10 to 10 and the dependent variable is limited only 
by the precision of the IEEE floating point arithmetic. 
<P>
<P>
&lt;option&gt;: 
<P>
<P>
An option can be an equation equating a variable 
and a value (in general a string), or a keyword(GNUPLOT switch). 
These have to be included in the gnuplot command arguments directly. 
Strings have to be enclosed in 
string quotes (see 
<a href=r38_0001.html#r38_0003>string</a>). Available options are: 
<P>
<P>

<a href=r38_0550.html#r38_0554>title</a>: assign a heading (default: empty) 
<P>
<P>

<a href=r38_0550.html#r38_0555>xlabel</a>: set label for the x axis 
<P>
<P>

<a href=r38_0550.html#r38_0556>ylabel</a>: set label for the y axis 
<P>
<P>

<a href=r38_0550.html#r38_0557>zlabel</a>: set label for the z axis 
<P>
<P>

<a href=r38_0550.html#r38_0558>terminal</a>: select an output device 
<P>
<P>

<a href=r38_0550.html#r38_0559>size</a>: rescale the picture 
<P>
<P>

<a href=r38_0550.html#r38_0560>view</a>: set a viewpoint 
<P>
<P>
<em>(no)</em>
<a href=r38_0550.html#r38_0561>contour</a>: 3d: add contour lines 
<P>
<P>
<em>(no)</em>
<a href=r38_0550.html#r38_0562>surface</a>: 3d: draw surface (default: yes) 
<P>
<P>
<em>(no)</em>
<a href=r38_0550.html#r38_0563>hidden3d</a>: 3d: remove hidden lines (default: n
o) 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
plot(cos x);

plot(s=sin phi,phi=(-3 .. 3));

plot(sin phi,cos phi,phi=(-3 .. 3));

plot (cos sqrt(x**2 + y**2),x=(-3 .. 3),y=(-3 .. 3),hidden3d);

plot {{0,0},{0,1},{1,1},{0,0},{1,0},{0,1},{0.5,1.5},{1,1},{1,0}};



on rounded;

w:=for j:=1:200 collect {1/j*sin j,1/j*cos j,j/200}$

plot w; 
</tt></pre><p>Additional control of the <em>plot</em> operation: 

<a href=r38_0550.html#r38_0565>plotrefine</a>, 

<a href=r38_0550.html#r38_0566>plot_xmesh</a>, 
<a href=r38_0550.html#r38_0567>plot_ymesh</a>, 
<a href=r38_0550.html#r38_0569>trplot</a>, 

<a href=r38_0550.html#r38_0564>plotkeep</a>, 
<a href=r38_0550.html#r38_0568>show_grid</a>. 
<P>
<P>

<a name=r38_0553>

<title>PLOTRESET</title></a>
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<b>PLOTRESET</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>command</b><P>
<P>
 
The command <em>plotreset</em> closes the current GNUPLOT windows. 
The next call to 
<a href=r38_0550.html#r38_0552>plot</a> will create a new one. <em>plotreset
</em> 
can also be used to reset the system status after technical problems. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>plotreset</em>; 
<P>
<P>
<P>

<a name=r38_0554>

<title>title</title></a>
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<b>TITLE</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>variable</b><P>
<P>
 
 <P>
<P>

<a href=r38_0550.html#r38_0552>plot</a>option: 
Assign a title to the GNUPLOT graph. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>title</em>= &lt;string&gt; 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
title=&quot;annual revenue in 1993&quot;
</tt></pre><p>
<a name=r38_0555>

<title>xlabel</title></a>
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<b>XLABEL</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>variable</b><P>
<P>
 
 <P>
<P>

<a href=r38_0550.html#r38_0552>plot</a>option: 
Assign a name to to the x axis (see 
<a href=r38_0550.html#r38_0550>axes names</a>). 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>xlabel</em>= &lt;string&gt; 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
xlabel=&quot;month&quot;
</tt></pre><p>
<a name=r38_0556>

<title>ylabel</title></a>
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<b>YLABEL</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>variable</b><P>
<P>
 
 <P>
<P>

<a href=r38_0550.html#r38_0552>plot</a>option: 
Assign a name to to the x axis (see 
<a href=r38_0550.html#r38_0550>axes names</a>). 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>ylabel</em>= &lt;string&gt; 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
ylabel=&quot;million forint&quot;
</tt></pre><p>
<a name=r38_0557>

<title>zlabel</title></a>
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<b>ZLABEL</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>variable</b><P>
<P>
 
 <P>
<P>

<a href=r38_0550.html#r38_0552>plot</a>option: 
Assign a name to to the z axis (see 
<a href=r38_0550.html#r38_0550>axes names</a>). 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>zlabel</em>= &lt;string&gt; 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
zlabel=&quot;local weight&quot;
</tt></pre><p>
<a name=r38_0558>

<title>terminal</title></a>
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<b>TERMINAL</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>variable</b><P>
<P>
 
 <P>
<P>

<a href=r38_0550.html#r38_0552>plot</a>option: 
Select a different output device. The possible values here 
depend highly on the facilities installed for your GNUPLOT 
software. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>terminal</em>= &lt;string&gt; 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
terminal=&quot;x11&quot;
</tt></pre><p>
<a name=r38_0559>

<title>size</title></a>
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<b>SIZE</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>variable</b><P>
<P>
 
 <P>
<P>

<a href=r38_0550.html#r38_0552>plot</a>option: 
Rescale the graph (not the window!) in x and y direction. 
Default is 1.0 (no rescaling). 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>size</em>= &quot;&lt;sx&gt;,&lt;sy&gt;&quot; 
<P>
<P>
<P>
where &lt;sx&gt;,&lt;sy&gt; are floating point number not too 
far from 1.0. 
 <P> <H3> 
examples: </H3>
<p><pre><tt>
size=&quot;0.7,1&quot;
</tt></pre><p><P>
<P>

<a name=r38_0560>

<title>view</title></a>
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<b>VIEW</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>variable</b><P>
<P>
 
 <P>
<P>

<a href=r38_0550.html#r38_0552>plot</a>option: 
Set a new viewpoint by turning the object around the x and then 
around the z axis (see 
<a href=r38_0550.html#r38_0550>axes names</a>). 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>view</em>= &quot;&lt;sx&gt;,&lt;sz&gt;&quot; 
<P>
<P>
<P>
where &lt;sx&gt;,&lt;sz&gt; are floating point number representing 
angles in degrees. 
 <P> <H3> 
examples: </H3>
<p><pre><tt>
view=&quot;30,130&quot;
</tt></pre><p><P>
<P>

<a name=r38_0561>

<title>contour</title></a>
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<b>CONTOUR</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>switch</b><P>
<P>
 
 <P>
<P>

<a href=r38_0550.html#r38_0552>plot</a>option: 
If <em>contour</em> is member of the options for a 3d 
<a href=r38_0550.html#r38_0552>plot</a> 
contour lines are projected to the z=0 plane 
(see 
<a href=r38_0550.html#r38_0550>axes names</a>). The absence of contour lines 
can be selected explicitly by including <em>nocontour</em>. Default 
is <em>nocontour</em>. 
<P>
<P>

<a name=r38_0562>

<title>surface</title></a>
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<b>SURFACE</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>switch</b><P>
<P>
 
 <P>
<P>

<a href=r38_0550.html#r38_0552>plot</a>option: 
If <em>surface</em> is member of the options for a 3d 
<a href=r38_0550.html#r38_0552>plot</a> 
the surface is drawn. The absence of the surface plotting 
can be selected by including <em>nosurface</em>, e.g. if 
only the 
<a href=r38_0550.html#r38_0561>contour</a> should be visualized. Default is <em>
surface</em>. 
<P>
<P>

<a name=r38_0563>

<title>hidden3d</title></a>
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<b>HIDDEN3D</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>switch</b><P>
<P>
 
 <P>
<P>

<a href=r38_0550.html#r38_0552>plot</a>option: 
If <em>hidden3d</em> is member of the options for a 3d 
<a href=r38_0550.html#r38_0552>plot</a> 
hidden lines are removed from the picture. Otherwise a 
surface is drawn as transparent object. Default is 
<em>nohidden3d</em>. Selecting <em>hidden3d</em> increases the 
computing time substantially. 
<P>
<P>

<a name=r38_0564>

<title>PLOTKEEP</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
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<b>PLOTKEEP</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>switch</b><P>
<P>
 
 <P>
<P>
Normally all intermediate data sets are deleted after terminating 
a plot session. If the switch <em>plotkeep</em> is set 
<a href=r38_0200.html#r38_0210>on</a>, 
the data sets are kept for eventual post processing independent 
of REDUCE. 
<P>
<P>

<a name=r38_0565>

<title>PLOTREFINE</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
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<b>PLOTREFINE</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>switch</b><P>
<P>
 
 <P>
<P>
In general 
<a href=r38_0550.html#r38_0552>plot</a> tries to generate smooth pictures by eva
luating 
the functions at interior points until the distances are fine 
enough. This can require a lot of computing time if the 
single function evaluation is expensive. The refinement is 
controlled by the switch <em>plotrefine</em> which is 
<a href=r38_0200.html#r38_0210>on</a> 
by default. When you turn it 
<a href=r38_0200.html#r38_0209>off</a> the functions will 
be evaluated only at the basic points (see 
<a href=r38_0550.html#r38_0566>plot_xmesh</a>, 

<a href=r38_0550.html#r38_0567>plot_ymesh</a>). 
<P>
<P>

<a name=r38_0566>

<title>plot_xmesh</title></a>
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E"></p>
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<b>PLOT_XMESH</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>variable</b><P>
<P>
 
 <P>
<P>
The integer value of the global variable <em>plot_xmesh</em> 
defines the number of initial function evaluations in x 
direction (see 
<a href=r38_0550.html#r38_0550>axes names</a>) for 
<a href=r38_0550.html#r38_0552>plot</a>. For 2d graphs additional 
points will be used as long as 
<a href=r38_0550.html#r38_0565>plotrefine</a> is <em>on</em>. 
For 3d graphs this number defines also the number of mesh lines 
orthogonal to the x axis. 
<P>
<P>

<a name=r38_0567>

<title>plot_ymesh</title></a>
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E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>PLOT_YMESH</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>variable</b><P>
<P>
 
 <P>
<P>
The integer value of the global variable <em>plot_ymesh</em> 
defines for 3d 
<a href=r38_0550.html#r38_0552>plot</a> calls the number of function evaluations
 in y 
direction (see 
<a href=r38_0550.html#r38_0550>axes names</a>) and the number of mesh lines 
orthogonal to the y axis. 
<P>
<P>

<a name=r38_0568>

<title>SHOW_GRID</title></a>
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E"></p>
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<b>SHOW_GRID</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>switch</b><P>
<P>
 
 <P>
<P>
The grid for localizing an implicitly defined curve in 
<a href=r38_0550.html#r38_0552>plot</a> 
consists of triangles. These are computed initially equally distributed 
over the x-y plane controlled by 
<a href=r38_0550.html#r38_0566>plot_xmesh</a>. The grid is 
refined adaptively in several levels. The final grid can be visualized 
by setting on the switch <em>show_grid</em>. 
<P>
<P>

<a name=r38_0569>

<title>TRPLOT</title></a>
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E"></p>
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<b>TRPLOT</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>switch</b><P>
<P>
 
 <P>
<P>
In general the interaction between REDUCE and GNUPLOT is performed 
as silently as possible. However, sometimes it might be useful 
to see the GNUPLOT commands generated by REDUCE, e.g. for a 
postprocessing of generated data sets independent of REDUCE. 
When the switch <em>trplot</em> is set on all GNUPLOT commands will 
be printed to the standard output additionally. 
<P>
<P>

<a name=r38_0570>

<title>Gnuplot 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>Gnuplot package</b><menu>
<li><a href=r38_0500.html#r38_0549>GNUPLOT and REDUCE introduction</a><P>
<li><a href=r38_0550.html#r38_0550>Axes names concept</a><P>
<li><a href=r38_0550.html#r38_0551>Pointset type</a><P>
<li><a href=r38_0550.html#r38_0552>PLOT command</a><P>
<li><a href=r38_0550.html#r38_0553>PLOTRESET command</a><P>
<li><a href=r38_0550.html#r38_0554>title variable</a><P>
<li><a href=r38_0550.html#r38_0555>xlabel variable</a><P>
<li><a href=r38_0550.html#r38_0556>ylabel variable</a><P>
<li><a href=r38_0550.html#r38_0557>zlabel variable</a><P>
<li><a href=r38_0550.html#r38_0558>terminal variable</a><P>
<li><a href=r38_0550.html#r38_0559>size variable</a><P>
<li><a href=r38_0550.html#r38_0560>view variable</a><P>
<li><a href=r38_0550.html#r38_0561>contour switch</a><P>
<li><a href=r38_0550.html#r38_0562>surface switch</a><P>
<li><a href=r38_0550.html#r38_0563>hidden3d switch</a><P>
<li><a href=r38_0550.html#r38_0564>PLOTKEEP switch</a><P>
<li><a href=r38_0550.html#r38_0565>PLOTREFINE switch</a><P>
<li><a href=r38_0550.html#r38_0566>plot_xmesh variable</a><P>
<li><a href=r38_0550.html#r38_0567>plot_ymesh variable</a><P>
<li><a href=r38_0550.html#r38_0568>SHOW_GRID switch</a><P>
<li><a href=r38_0550.html#r38_0569>TRPLOT switch</a><P>
</menu>
<a name=r38_0571>

<title>Linear_Algebra_package</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
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<b>LINEAR ALGEBRA PACKAGE</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>introduction</b><P>
<P>
 
<P>
<P>
This section briefly describes what's available in the Linear Algebra 
package. 
<P>
<P>
Note on examples: In the examples throughout this 
document, the matrix A will be 
<p><pre><tt>
     [1  2  3]
     [4  5  6]
     [7  8  9].
</tt></pre><p><P>
<P>
The functions can be divided into four categories: 
<P>
<P>
Basic matrix handling 
 <P>
<P>

<a href=r38_0550.html#r38_0573>add_columns</a>, 
<P>
<P>

<a href=r38_0550.html#r38_0574>add_rows</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0575>add_to_columns</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0576>add_to_rows</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0577>augment_columns</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0581>char_poly</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0584>column_dim</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0586>copy_into</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0587>diagonal</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0588>extend</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0589>find_companion</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0590>get_columns</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0591>get_rows</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0593>hermitian_tp</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0600>matrix_augment</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0602>matrix_stack</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0603>minor</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0604>mult_columns</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0605>mult_rows</a>, 
<P>
<P>

<a href=r38_0600.html#r38_0606>pivot</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0609>remove_columns</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0610>remove_rows</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0611>row_dim</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0612>rows_pivot</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0615>stack_rows</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0616>sub_matrix</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0618>swap_columns</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0619>swap_entries</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0620>swap_rows</a>. 
<P>
<P>
Constructors -- functions that create matrices 
 <P>
<P>

<a href=r38_0550.html#r38_0578>band_matrix</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0579>block_matrix</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0580>char_matrix</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0583>coeff_matrix</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0585>companion</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0594>hessian</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0595>hilbert</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0596>jacobian</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0597>jordan_block</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0599>make_identity</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0608>random_matrix</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0622>toeplitz</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0623>vandermonde</a>. 
<P>
<P>
High level algorithms 
 <P>
<P>

<a href=r38_0550.html#r38_0581>char_poly</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0582>cholesky</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0592>gram_schmidt</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0598>lu_decom</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0607>pseudo_inverse</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0613>simplex</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0617>svd</a>. 
<P>
<P>
Normal Forms 
<P>
<P>
There is a separate package, NORMFORM, for computing 
the following matrix normal forms in REDUCE: 
<P>
<P>

<a href=r38_0600.html#r38_0625>smithex</a>, 
<P>
<P>

<a href=r38_0600.html#r38_0626>smithex_int</a>, 
<P>
<P>

<a href=r38_0600.html#r38_0627>frobenius</a>, 
<P>
<P>

<a href=r38_0600.html#r38_0628>ratjordan</a>, 
<P>
<P>

<a href=r38_0600.html#r38_0629>jordansymbolic</a>, 
<P>
<P>

<a href=r38_0600.html#r38_0630>jordan</a>. 
<P>
<P>
Predicates 
<P>
<P>

<a href=r38_0600.html#r38_0601>matrixp</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0614>squarep</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0621>symmetricp</a>. 
<P>
<P>

<a name=r38_0572>

<title>fast_la</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
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<b>FAST_LA</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>switch</b><P>
<P>
 
<P>
<P>
By turning the <em>fast_la</em> switch on, the speed of the following 
functions will be increased: 
<P>
<P>

<a href=r38_0550.html#r38_0573>add_columns</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0574>add_rows</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0577>augment_columns</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0584>column_dim</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0586>copy_into</a>, 
 <P>
<P>

<a href=r38_0550.html#r38_0599>make_identity</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0600>matrix_augment</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0602>matrix_stack</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0603>minor</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0604>mult_columns</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0605>mult_rows</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0606>pivot</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0609>remove_columns</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0610>remove_rows</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0612>rows_pivot</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0614>squarep</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0615>stack_rows</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0616>sub_matrix</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0618>swap_columns</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0619>swap_entries</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0620>swap_rows</a>, 
 <P>
<P>

<a href=r38_0600.html#r38_0621>symmetricp</a>. 
 <P>
<P>
The increase in speed will be negligible unless you are making a 
significant number (i.e. thousands) of calls. When using this switch, 
error checking is minimized. This means that illegal input may give 
strange error messages. Beware. 
<P>
<P>

<a name=r38_0573>

<title>add_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>ADD_COLUMNS</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
Add columns, add rows: 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>add_columns</em>(&lt;matrix&gt;,&lt;c1&gt;,&lt;c2&gt;,&lt;expr&gt;) 
<P>
<P>
<P>
&lt;matrix&gt; :- a 
<a href=r38_0300.html#r38_0345>matrix</a>. 
<P>
<P>
&lt;c1&gt;,&lt;c2&gt; :- positive integers. 
<P>
<P>
&lt;expr&gt; :- a scalar expression. 
<P>
<P>
The Operator <em>add_columns</em> replaces column &lt;\meta{c2&gt;} of 
&lt;matrix&gt; by &lt;expr&gt; * column(&lt;matrix&gt;,&lt;c1&gt;) + 
column(&lt;matrix&gt;,&lt;c2&gt;). 
<P>
<P>
<em>add_rows</em>performs the equivalent task on the rows of 
&lt;matrix&gt;. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>

add_columns(A,1,2,x); 

  [1   x + 2   3]
  [             ]
  [4  4*x + 5  6]
  [             ]
  [7  7*x + 8  9]



add_rows(A,2,3,5); 

  [1   2   3 ]
  [          ]
  [4   5   6 ]
  [          ]
  [27  33  39]

</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>, 
<a href=r38_0600.html#r38_0604>mult_columns</a>, 

<a href=r38_0600.html#r38_0605>mult_rows</a>. 
<P>
<P>

<a name=r38_0574>

<title>add_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>ADD_ROWS</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 see: 
<a href=r38_0550.html#r38_0573>add_columns</a>. 
<P>
<P>

<a name=r38_0575>

<title>add_to_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>ADD_TO_COLUMNS</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
Add to columns, add to rows: 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>add_to_columns</em>(&lt;matrix&gt;,&lt;column\_list&gt;,&lt;expr&gt;) 
<P>
<P>
<P>
&lt;matrix&gt; :- a matrix. 
<P>
<P>
&lt;column\_list&gt; :- a positive integer or a list of positive 
 integers. 
<P>
<P>
&lt;expr&gt; :- a scalar expression. 
<P>
<P>
<em>add_to_columns</em>adds &lt;expr&gt; to each column specified in 
&lt;column\_list&gt; of &lt;matrix&gt;. 
<P>
<P>
<em>add_to_rows</em>performs the equivalent task on the rows of 
&lt;matrix&gt;. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>

add_to_columns(A,{1,2},10); 

  [11  12  3]
  [         ]
  [14  15  6]
  [         ]
  [17  18  9]



add_to_rows(A,2,-x) 

   
  [   1         2         3    ]
  [                            ]
  [ - x + 4   - x + 5   - x + 6]
  [                            ]
  [   7         8         9    ]

</tt></pre><p>Related functions: 

<a href=r38_0550.html#r38_0573>add_columns</a>, 
<a href=r38_0550.html#r38_0574>add_rows</a>, 
<a href=r38_0600.html#r38_0605>mult_rows</a>, 

<a href=r38_0600.html#r38_0604>mult_columns</a>. 
<P>
<P>

<a name=r38_0576>

<title>add_to_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>ADD_TO_ROWS</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 see: 
<a href=r38_0550.html#r38_0575>add_to_columns</a>. 
<P>
<P>

<a name=r38_0577>

<title>augment_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>AUGMENT_COLUMNS</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
Augment columns, stack rows: 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>augment_columns</em>(&lt;matrix&gt;,&lt;column\_list&gt;) 
<P>
<P>
<P>
&lt;matrix&gt; :- a matrix. 
<P>
<P>
&lt;column\_list&gt; :- either a positive integer or a list of positive 
 integers. 
<P>
<P>
<em>augment_columns</em>gets hold of the columns of &lt;matrix&gt; 
specified in <em>column_list</em> and sticks them together. 
<P>
<P>
<em>stack_rows</em>performs the same task on rows of &lt;matrix&gt;. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>

augment_columns(A,{1,2}) 

   
  [1  2]
  [    ]
  [4  5]
  [    ]
  [7  8]



stack_rows(A,{1,3}) 

  [1  2  3]
  [       ]
  [7  8  9]

</tt></pre><p>Related functions: 

<a href=r38_0550.html#r38_0590>get_columns</a>, 
<a href=r38_0550.html#r38_0591>get_rows</a>, 
<a href=r38_0600.html#r38_0616>sub_matrix</a>. 
<P>
<P>

<a name=r38_0578>

<title>band_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>BAND_MATRIX</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>band_matrix</em>(&lt;expr\_list&gt;,&lt;square\_size&gt;) 
<P>
<P>
<P>
&lt;expr\_list&gt; :- either a single scalar expression or a list of 
 an odd number of scalar expressions. 
<P>
<P>
&lt;square\_size&gt; :- a positive integer. 
<P>
<P>
<em>band_matrix</em>creates a square matrix of dimension 
&lt;square\_size&gt;. The diagonal consists of the middle expression 
of the &lt;expr\_list&gt;. The expressions to the left of this fill 
the required number of sub_diagonals and the expressions to the right 
the super_diagonals. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>

band_matrix({x,y,z},6) 

  [y  z  0  0  0  0]
  [                ]
  [x  y  z  0  0  0]
  [                ]
  [0  x  y  z  0  0]
  [                ]
  [0  0  x  y  z  0]
  [                ]
  [0  0  0  x  y  z]
  [                ]
  [0  0  0  0  x  y]

</tt></pre><p>Related functions: 
<a href=r38_0550.html#r38_0587>diagonal</a>. 
<P>
<P>

<a name=r38_0579>

<title>block_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>BLOCK_MATRIX</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>block_matrix</em>(&lt;r&gt;,&lt;c&gt;,&lt;matrix\_list&gt;) 
<P>
<P>
<P>
&lt;r&gt;,&lt;c&gt; :- positive integers. 
<P>
<P>
&lt;matrix\_list&gt; :- a list of matrices. 
<P>
<P>
<em>block_matrix</em>creates a matrix that consists of &lt;r&gt; by 
&lt;c&gt; matrices filled from the &lt;matrix\_list&gt; row wise. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
B := make_identity(2); 

       [1  0]
  b := [    ]
       [0  1]



C := mat((5),(5)); 

       [5]
  c := [ ]
       [5]



D := mat((22,33),(44,55)); 

       [22  33]
  d := [      ]
       [44  55]



block_matrix(2,3,{B,C,D,D,C,B}); 


  [1   0   5  22  33]
  [                 ]
  [0   1   5  44  55]
  [                 ]
  [22  33  5  1   0 ]
  [                 ]
  [44  55  5  0   1 ]

</tt></pre><p>
<a name=r38_0580>

<title>char_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>CHAR_MATRIX</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>char_matrix</em>(&lt;matrix&gt;,&lt;lambda&gt;) 
<P>
<P>
<P>
&lt;matrix&gt; :- a square matrix. 
&lt;lambda&gt; :- a symbol or algebraic expression. 
<P>
<P>
&lt;char\_matrix&gt; creates the characteristic matrix C of 
&lt;matrix&gt;. 
<P>
<P>
This is C = &lt;lambda&gt; * Id - A. 
Id is the identity matrix. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>

char_matrix(A,x); 

  [x - 1   -2     -3  ]
  [                   ]
  [ -4    x - 5   -6  ]
  [                   ]
  [ -7     -8    x - 9]

</tt></pre><p>Related functions: 
<a href=r38_0550.html#r38_0581>char_poly</a>. 
<P>
<P>

<a name=r38_0581>

<title>char_poly</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>CHAR_POLY</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>char_poly</em>(&lt;matrix&gt;,&lt;lambda&gt;) 
<P>
<P>
<P>
&lt;matrix&gt; :- a square matrix. 
<P>
<P>
&lt;lambda&gt; :- a symbol or algebraic expression. 
<P>
<P>
<em>char_poly</em>finds the characteristic polynomial of &lt;matrix&gt;. 
This is the determinant of &lt;lambda&gt; * Id - A. 
Id is the identity matrix. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
char_poly(A,x); 

   3     2
  x -15*x -18*x

</tt></pre><p>Related functions: 
<a href=r38_0550.html#r38_0580>char_matrix</a>. 
<P>
<P>

<a name=r38_0582>

<title>cholesky</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>CHOLESKY</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>cholesky</em>(&lt;matrix&gt;) 
<P>
<P>
<P>
&lt;matrix&gt; :- a positive definite matrix containing numeric entries. 
<P>
<P>
<em>cholesky</em>computes the cholesky decomposition of &lt;matrix&gt;. 
<P>
<P>
It returns {L,U} where L is a lower matrix, U is an upper matrix, 
A = LU, and U = L^T. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
F := mat((1,1,0),(1,3,1),(0,1,1)); 


       [1  1  0]
       [       ]
  f := [1  3  1]
       [       ]
       [0  1  1]



on rounded; 

cholesky(F); 

  {
   [1        0               0       ]
   [                                 ]
   [1  1.41421356237         0       ]
   [                                 ]
   [0  0.707106781187  0.707106781187]
   ,
   [1        1              0       ]
   [                                ]
   [0  1.41421356237  0.707106781187]
   [                                ]
   [0        0        0.707106781187]
  }

</tt></pre><p>Related functions: 
<a href=r38_0550.html#r38_0598>lu_decom</a>. 
<P>
<P>

<a name=r38_0583>

<title>coeff_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>COEFF_MATRIX</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>coeff_matrix</em>({&lt;lineq\_list&gt;}) 
<P>
<P>
<P>
(If you are feeling lazy then the braces can be omitted.) 
<P>
<P>
&lt;lineq\_list&gt; :- linear equations. Can be of the form equation = number 
or just equation. 
<P>
<P>
<em>coeff_matrix</em>creates the coefficient matrix C of the linear 
equations. 
<P>
<P>
It returns {C,X,B} such that CX = B. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>

coeff_matrix({x+y+4*z=10,y+x-z=20,x+y+4}); 


  {
   [4   1  1]
   [        ]
   [-1  1  1]
   [        ]
   [0   1  1]
   ,
   [z]
   [ ]
   [y]
   [ ]
   [x]
   ,
   [10]
   [  ]
   [20]
   [  ]
   [-4]
  }

</tt></pre><p>
<a name=r38_0584>

<title>column_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>COLUMN_DIM</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
Column dimension, row dimension: 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>column_dim</em>(&lt;matrix&gt;) 
<P>
<P>
<P>
&lt;matrix&gt; :- a matrix. 
<P>
<P>
<em>column_dim</em>finds the column dimension of &lt;matrix&gt;. 
<P>
<P>
<em>row_dim</em>finds the row dimension of &lt;matrix&gt;. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>

column_dim(A); 

  3 


row_dim(A); 

  3 

</tt></pre><p>
<a name=r38_0585>

<title>companion</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>COMPANION</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>companion</em>(&lt;poly&gt;,&lt;x&gt;) 
<P>
<P>
<P>
&lt;poly&gt; :- a monic univariate polynomial in &lt;x&gt;. 
<P>
<P>
&lt;x&gt; :- the variable. 
<P>
<P>
<em>companion</em>creates the companion matrix C of &lt;poly&gt;. 
<P>
<P>
This is the square matrix of dimension n, where n is the degree of 
&lt;poly&gt; w.r.t. &lt;x&gt;. 
<P>
<P>
The entries of C are: 
<P>
<P>
C(i,n) = -coeffn(&lt;poly&gt;,&lt;x&gt;,i-1) for i = 1 
 ... n, C(i,i-1) = 1 for i = 2 ... n and 
 the rest are 0. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>

companion(x^4+17*x^3-9*x^2+11,x); 


  [0  0  0  -11]
  [            ]
  [1  0  0   0 ]
  [            ]
  [0  1  0   9 ]
  [            ]
  [0  0  1  -17]

</tt></pre><p>Related functions: 

<a href=r38_0550.html#r38_0589>find_companion</a>. 
<P>
<P>

<a name=r38_0586>

<title>copy_into</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>COPY_INTO</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>copy_into</em>(&lt;A&gt;,&lt;B&gt;,&lt;r&gt;,&lt;c&gt;) 
<P>
<P>
<P>
&lt;A&gt;,&lt;B&gt; :- matrices. 
<P>
<P>
&lt;r&gt;,&lt;c&gt; :- positive integers. 
<P>
<P>
<em>copy_into</em>copies matrix &lt;matrix&gt; into &lt;B&gt; with 
&lt;matrix&gt;(1,1) at &lt;B&gt;(&lt;r&gt;,&lt;c&gt;). 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt> 

G := mat((0,0,0,0,0),(0,0,0,0,0),(0,0,0,0,0),(0,0,0,0,0),(0,0,0,0,0)); 


       [0  0  0  0  0]
       [             ]
       [0  0  0  0  0]
       [             ]
  g := [0  0  0  0  0]
       [             ]
       [0  0  0  0  0]
       [             ]
       [0  0  0  0  0]



copy_into(A,G,1,2); 

  [0  1  2  3  0]
  [             ]
  [0  4  5  6  0]
  [             ]
  [0  7  8  9  0]
  [             ]
  [0  0  0  0  0]
  [             ]
  [0  0  0  0  0]

</tt></pre><p>Related functions: 

<a href=r38_0550.html#r38_0577>augment_columns</a>, 
<a href=r38_0550.html#r38_0588>extend</a>, 
<a href=r38_0600.html#r38_0600>matrix_augment</a>, 

<a href=r38_0600.html#r38_0602>matrix_stack</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_0587>

<title>diagonal</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>DIAGONAL</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>diagonal</em>({&lt;mat\_list&gt;}) 
<P>
<P>
<P>
(If you are feeling lazy then the braces can be omitted.) 
<P>
<P>
&lt;mat\_list&gt; :- each can be either a scalar expression or a 
square 
<a href=r38_0300.html#r38_0345>matrix</a>. 
<P>
<P>
<em>diagonal</em>creates a matrix that contains the input on the 
diagonal. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>

H := mat((66,77),(88,99)); 

       [66  77]
  h := [      ]
       [88  99]



diagonal({A,x,H}); 

  [1  2  3  0  0   0 ]
  [                  ]
  [4  5  6  0  0   0 ]
  [                  ]
  [7  8  9  0  0   0 ]
  [                  ]
  [0  0  0  x  0   0 ]
  [                  ]
  [0  0  0  0  66  77]
  [                  ]
  [0  0  0  0  88  99]

</tt></pre><p>Related functions: 

<a href=r38_0550.html#r38_0597>jordan_block</a>. 
<P>
<P>

<a name=r38_0588>

<title>extend</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>EXTEND</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>extend</em>(&lt;matrix&gt;,&lt;r&gt;,&lt;c&gt;,&lt;expr&gt;) 
<P>
<P>
<P>
&lt;matrix&gt; :- a 
<a href=r38_0300.html#r38_0345>matrix</a>. 
<P>
<P>
&lt;r&gt;,&lt;c&gt; :- positive integers. 
<P>
<P>
&lt;expr&gt; :- algebraic expression or symbol. 
<P>
<P>
<em>extend</em>returns a copy of &lt;matrix&gt; that has been extended by 
&lt;r&gt; rows and &lt;c&gt; columns. The new entries are made equal to 
&lt;expr&gt;. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>

extend(A,1,2,x); 

  [1  2  3  x  x]
  [             ]
  [4  5  6  x  x]
  [             ]
  [7  8  9  x  x]
  [             ]
  [x  x  x  x  x]

</tt></pre><p>Related functions: 

<a href=r38_0550.html#r38_0586>copy_into</a>, 
<a href=r38_0600.html#r38_0600>matrix_augment</a>, 
<a href=r38_0600.html#r38_0602>matrix_stack</a>, 

<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_0589>

<title>find_companion</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>FIND_COMPANION</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>find_companion</em>(&lt;matrix&gt;,&lt;x&gt;) 
<P>
<P>
<P>
&lt;matrix&gt; :- a 
<a href=r38_0300.html#r38_0345>matrix</a>. 
<P>
<P>
&lt;x&gt; :- the variable. 
<P>
<P>
Given a companion matrix, <em>find_companion</em> finds the polynomial 
from which it was made. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>

C := companion(x^4+17*x^3-9*x^2+11,x); 


       [0  0  0  -11]
       [            ]
       [1  0  0   0 ]
  c := [            ]
       [0  1  0   9 ]
       [            ]
       [0  0  1  -17]



find_companion(C,x); 

   4     3    2
  x +17*x -9*x +11

</tt></pre><p>Related functions: 

<a href=r38_0550.html#r38_0585>companion</a>. 
<P>
<P>

<a name=r38_0590>

<title>get_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>GET_COLUMNS</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
Get columns, get rows: 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>get_columns</em>(&lt;matrix&gt;,&lt;column\_list&gt;) 
<P>
<P>
<P>
&lt;matrix&gt; :- a 
<a href=r38_0300.html#r38_0345>matrix</a>. 
<P>
<P>
&lt;c&gt; :- either a positive integer or a list of positive 
 integers. 
<P>
<P>
<em>get_columns</em>removes the columns of &lt;matrix&gt; specified in 
&lt;column\_list&gt; and returns them as a list of column matrices. 
<P>
<P>
<em>get_rows</em>performs the same task on the rows of &lt;matrix&gt;. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>

get_columns(A,{1,3}); 

  {
   [1]
   [ ]
   [4]
   [ ]
   [7]
   ,
   [3]
   [ ]
   [6]
   [ ]
   [9]
  }



get_rows(A,2); 

  {
   [4  5  6]
  }

</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_0591>

<title>get_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>GET_ROWS</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
see: 
<a href=r38_0550.html#r38_0590>get_columns</a>. 
<P>
<P>

<a name=r38_0592>

<title>gram_schmidt</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>GRAM_SCHMIDT</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>gram_schmidt</em>({&lt;vec\_list&gt;}) 
<P>
<P>
<P>
(If you are feeling lazy then the braces can be omitted.) 
<P>
<P>
&lt;vec\_list&gt; :- linearly independent vectors. Each vector must be 
written as a list, eg:{1,0,0}. 
<P>
<P>
<em>gram_schmidt</em>performs the gram_schmidt orthonormalization on 
the input vectors. 
<P>
<P>
It returns a list of orthogonal normalized vectors. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>

gram_schmidt({{1,0,0},{1,1,0},{1,1,1}}); 


  {{1,0,0},{0,1,0},{0,0,1}} 



gram_schmidt({{1,2},{3,4}}); 


        1         2        2*sqrt(5)   -sqrt(5)
  {{ ------- , ------- },{ --------- , -------- }}
     sqrt(5)   sqrt(5)         5          5

</tt></pre><p>
<a name=r38_0593>

<title>hermitian_tp</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>HERMITIAN_TP</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em> hermitian_tp</em>(&lt;matrix&gt;) 
<P>
<P>
<P>
&lt;matrix&gt; :- a 
<a href=r38_0300.html#r38_0345>matrix</a>. 
<P>
<P>
<em>hermitian_tp</em>computes the hermitian transpose of &lt;matrix&gt;. 
<P>
<P>
This is a 
<a href=r38_0300.html#r38_0345>matrix</a> in which the (i,j)'th entry is the con
jugate 
of the (j,i)'th entry of &lt;matrix&gt;. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>

J := mat((i+1,i+2,i+3),(4,5,2),(1,i,0)); 


       [i + 1  i + 2  i + 3]
       [                   ]
  j := [  4      5      2  ]
       [                   ]
       [  1      i      0  ]



hermitian_tp(j); 

  [ - i + 1  4   1  ]
  [                 ]
  [ - i + 2  5   - i]
  [                 ]
  [ - i + 3  2   0  ]

</tt></pre><p>Related functions: 

<a href=r38_0300.html#r38_0348>tp</a>. 
<P>
<P>

<a name=r38_0594>

<title>hessian</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>HESSIAN</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>hessian</em>(&lt;expr&gt;,&lt;variable\_list&gt;) 
<P>
<P>
<P>
&lt;expr&gt; :- a scalar expression. 
<P>
<P>
&lt;variable\_list&gt; :- either a single variable or a list of 
 variables. 
<P>
<P>
<em>hessian</em>computes the hessian matrix of &lt;expr&gt; w.r.t. the 
variables in &lt;variable\_list&gt;. 
<P>
<P>
This is an n by n matrix where n is the number of variables and the 
(i,j)'th entry is 
<a href=r38_0100.html#r38_0148>df</a>(&lt;expr&gt;,&lt;variable\_list&gt;(i), 
&lt;variable\_list&gt;(j)). 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>

hessian(x*y*z+x^2,{w,x,y,z}); 

  [0  0  0  0]
  [          ]
  [0  2  z  y]
  [          ]
  [0  z  0  x]
  [          ]
  [0  y  x  0]

</tt></pre><p>Related functions: 
<a href=r38_0100.html#r38_0148>df</a>. 
<P>
<P>

<a name=r38_0595>

<title>hilbert</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>HILBERT</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>hilbert</em>(&lt;square\_size&gt;,&lt;expr&gt;) 
<P>
<P>
<P>
&lt;square\_size&gt; :- a positive integer. 
<P>
<P>
&lt;expr&gt; :- an algebraic expression. 
<P>
<P>
<em>hilbert</em>computes the square hilbert matrix of dimension 
&lt;square\_size&gt;. 
<P>
<P>
This is the symmetric matrix in which the (i,j)'th entry is 
1/(i+j-&lt;expr&gt;). 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>

hilbert(3,y+x); 

  [    - 1          - 1          - 1    ]
  [-----------  -----------  -----------]
  [ x + y - 2    x + y - 3    x + y - 4 ]
  [                                     ]
  [    - 1          - 1          - 1    ]
  [-----------  -----------  -----------]
  [ x + y - 3    x + y - 4    x + y - 5 ]
  [                                     ]
  [    - 1          - 1          - 1    ]
  [-----------  -----------  -----------]
  [ x + y - 4    x + y - 5    x + y - 6 ]

</tt></pre><p>
<a name=r38_0596>

<title>jacobian</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>JACOBIAN</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>jacobian</em>(&lt;expr\_list&gt;,&lt;variable\_list&gt;) 
<P>
<P>
<P>
&lt;expr\_list&gt; :- either a single algebraic expression or a list 
 of algebraic expressions. 
<P>
<P>
&lt;variable\_list&gt; :- either a single variable or a list of 
 variables. 
<P>
<P>
<em>jacobian</em>computes the jacobian matrix of &lt;expr\_list&gt; 
w.r.t. &lt;variable\_list&gt;. 
<P>
<P>
This is a matrix whose (i,j)'th entry is 
<a href=r38_0100.html#r38_0148>df</a>(&lt;expr\_list&gt; 
(i),&lt;variable\_list&gt;(j)). 
<P>
<P>
The matrix is n by m where n is the number of variables and m the number 
of expressions. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
 
jacobian({x^4,x*y^2,x*y*z^3},{w,x,y,z}); 


  [      3                 ]
  [0  4*x     0       0    ]
  [                        ]
  [     2                  ]
  [0   y    2*x*y     0    ]
  [                        ]
  [      3     3          2]
  [0  y*z   x*z    3*x*y*z ]

</tt></pre><p>Related functions: 

<a href=r38_0550.html#r38_0594>hessian</a>, 
<a href=r38_0100.html#r38_0148>df</a>. 
<P>
<P>

<a name=r38_0597>

<title>jordan_block</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_BLOCK</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>jordan_block</em>(&lt;expr&gt;,&lt;square\_size&gt;) 
<P>
<P>
<P>
&lt;expr&gt; :- an algebraic expression or symbol. 
<P>
<P>
&lt;square\_size&gt; :- a positive integer. 
<P>
<P>
<em>jordan_block</em>computes the square jordan block matrix J of 
dimension &lt;square\_size&gt;. 
<P>
<P>
The entries of J are: 
<P>
<P>
J(i,i) = &lt;expr&gt; for i=1 
 ... n, J(i,i+1) = 1 for i=1 
 ... n-1, and all other entries are 0. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>

jordan_block(x,5); 

  [x  1  0  0  0]
  [             ]
  [0  x  1  0  0]
  [             ]
  [0  0  x  1  0]
  [             ]
  [0  0  0  x  1]
  [             ]
  [0  0  0  0  x]

</tt></pre><p>Related functions: 
<a href=r38_0550.html#r38_0587>diagonal</a>, 
<a href=r38_0550.html#r38_0585>companion</a>. 
<P>
<P>

<a name=r38_0598>

<title>lu_decom</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>LU_DECOM</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>lu_decom</em>(&lt;matrix&gt;) 
<P>
<P>
<P>
&lt;matrix&gt; :- a 
<a href=r38_0300.html#r38_0345>matrix</a> containing either numeric entries 
 or imaginary entries with numeric coefficients. 
<P>
<P>
<em>lu_decom</em>performs LU decomposition on &lt;matrix&gt;, ie: it 
returns {L,U} where L is a lower diagonal 
<a href=r38_0300.html#r38_0345>matrix</a>, U an 
upper diagonal 
<a href=r38_0300.html#r38_0345>matrix</a> and A = LU. 
<P>
<P>
Caution: 
<P>
<P>
The algorithm used can swap the rows of &lt;matrix&gt; during the 
calculation. This means that LU does not equal &lt;matrix&gt; but a row 
equivalent of it. Due to this, <em>lu_decom</em> returns {L,U,vec}. 
The call <em>convert(meta{matrix</em>,vec)} will return the matrix that has 
been decomposed, i.e: LU = convert(&lt;matrix&gt;,vec). 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>

K := mat((1,3,5),(-4,3,7),(8,6,4)); 


       [1   3  5]
       [        ]
  k := [-4  3  7]
       [        ]
       [8   6  4]



on rounded;

lu :=  lu_decom(K); 

  lu := {
         [8    0      0  ]
         [               ]
         [-4  6.0     0  ]
         [               ]
         [1   2.25  1.125]
         ,
         [1  0.75  0.5]
         [            ]
         [0   1    1.5]
         [            ]
         [0   0     1 ]
         ,
         [3 2 3]}



first lu * second lu; 

  [8   6.0  4.0]
  [            ]
  [-4  3.0  7.0]
  [            ]
  [1   3.0  5.0]



convert(K,third lu); 

  P := mat((i+1,i+2,i+3),(4,5,2),(1,i,0));  _ _ _ 
       [i + 1  i + 2  i + 3]
       [                   ]
  p := [  4      5      2  ]
       [                   ]
       [  1      i      0  ]


lu :=  lu_decom(P); 

  lu := {
         [  1        0                      0                ]
         [                                                   ]
         [  4     - 4*i + 5                 0                ]
         [                                                   ]
         [i + 1      3       0.414634146341*i + 2.26829268293]
         ,
         [1  i                 0                ]
         [                                      ]
         [0  1  0.19512195122*i + 0.243902439024]
         [                                      ]
         [0  0                 1                ]
         ,
         [3 2 3]}



first lu * second lu; 

  [  1      i       0   ]
  [                     ]
  [  4      5      2.0  ]
  [                     ]
  [i + 1  i + 2  i + 3.0]



convert(P,third lu); 

  [  1      i      0  ]
  [                   ]
  [  4      5      2  ]
  [                   ]
  [i + 1  i + 2  i + 3]

</tt></pre><p> 
<P>
<P>
Related functions: 
<a href=r38_0550.html#r38_0582>cholesky</a>. 
<P>
<P>

<a name=r38_0599>

<title>make_identity</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>MAKE_IDENTITY</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>make_identity</em>(&lt;square\_size&gt;) 
<P>
<P>
<P>
&lt;square\_size&gt; :- a positive integer. 
<P>
<P>
<em>make_identity</em>creates the identity matrix of dimension 
&lt;square\_size&gt;. 
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>

make_identity(4); 

  [1  0  0  0]
  [          ]
  [0  1  0  0]
  [          ]
  [0  0  1  0]
  [          ]
  [0  0  0  1]

</tt></pre><p>Related functions: 
<a href=r38_0550.html#r38_0587>diagonal</a>. 
<P>
<P>