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<H1><A NAME="SEC131" HREF="octave_toc.html#TOC131">Matrix Manipulation</A></H1>

<P>
There are a number of functions available for checking to see if the
elements of a matrix meet some condition, and for rearranging the
elements of a matrix.  For example, Octave can easily tell you if all
the elements of a matrix are finite, or are less than some specified
value.  Octave can also rotate the elements, extract the upper- or
lower-triangular parts, or sort the columns of a matrix.

</P>



<H2><A NAME="SEC132" HREF="octave_toc.html#TOC132">Finding Elements and Checking Conditions</A></H2>

<P>
The functions <CODE>any</CODE> and <CODE>all</CODE> are useful for determining
whether any or all of the elements of a matrix satisfy some condition.
The <CODE>find</CODE> function is also useful in determining which elements of
a matrix meet a specified condition.

</P>
<P>
<DL>
<DT><U>Built-in Function:</U>  <B>any</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX574"></A>
For a vector argument, return 1 if any element of the vector is
nonzero.

</P>
<P>
For a matrix argument, return a row vector of ones and
zeros with each element indicating whether any of the elements of the
corresponding column of the matrix are nonzero.  For example,

</P>

<PRE>
any (eye (2, 4))
     => [ 1, 1, 0, 0 ]
</PRE>

<P>
To see if any of the elements of a matrix are nonzero, you can use a
statement like

</P>

<PRE>
any (any (a))
</PRE>

</DL>

<P>
<DL>
<DT><U>Built-in Function:</U>  <B>all</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX575"></A>
The function <CODE>all</CODE> behaves like the function <CODE>any</CODE>, except
that it returns true only if all the elements of a vector, or all the
elements in a column of a matrix, are nonzero.
</DL>

</P>
<P>
Since the comparison operators (see section <A HREF="octave_9.html#SEC70">Comparison Operators</A>) return matrices
of ones and zeros, it is easy to test a matrix for many things, not just
whether the elements are nonzero.  For example, 

</P>

<PRE>
all (all (rand (5) &#60; 0.9))
     => 0
</PRE>

<P>
tests a random 5 by 5 matrix to see if all of its elements are less
than 0.9.

</P>
<P>
Note that in conditional contexts (like the test clause of <CODE>if</CODE> and
<CODE>while</CODE> statements) Octave treats the test as if you had typed
<CODE>all (all (condition))</CODE>.

</P>
<P>
<DL>
<DT><U>Function File:</U> [<VAR>err</VAR>, <VAR>y1</VAR>, ...] = <B>common_size</B> <I>(<VAR>x1</VAR>, ...)</I>
<DD><A NAME="IDX576"></A>
Determine if all input arguments are either scalar or of common
size.  If so, <VAR>err</VAR> is zero, and <VAR>yi</VAR> is a matrix of the
common size with all entries equal to <VAR>xi</VAR> if this is a scalar or
<VAR>xi</VAR> otherwise.  If the inputs cannot be brought to a common size,
errorcode is 1, and <VAR>yi</VAR> is <VAR>xi</VAR>.  For example,

</P>

<PRE>
[errorcode, a, b] = common_size ([1 2; 3 4], 5)
     => errorcode = 0
     => a = [ 1, 2; 3, 4 ]
     => b = [ 5, 5; 5, 5 ]
</PRE>

<P>
This is useful for implementing functions where arguments can either
be scalars or of common size.
</DL>

</P>
<P>
<DL>
<DT><U>Function File:</U>  <B>diff</B> <I>(<VAR>x</VAR>, <VAR>k</VAR>)</I>
<DD><A NAME="IDX577"></A>
If <VAR>x</VAR> is a vector of length <VAR>n</VAR>, <CODE>diff (<VAR>x</VAR>)</CODE> is the
vector of first differences
 <VAR>x</VAR>(2) - <VAR>x</VAR>(1), ..., <VAR>x</VAR>(n) - <VAR>x</VAR>(n-1).

</P>
<P>
If <VAR>x</VAR> is a matrix, <CODE>diff (<VAR>x</VAR>)</CODE> is the matrix of column
differences.

</P>
<P>
The second argument is optional.  If supplied, <CODE>diff (<VAR>x</VAR>,
<VAR>k</VAR>)</CODE>, where <VAR>k</VAR> is a nonnegative integer, returns the
<VAR>k</VAR>-th differences.
</DL>

</P>
<P>
<DL>
<DT><U>Mapping Function:</U>  <B>isinf</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX578"></A>
Return 1 for elements of <VAR>x</VAR> that are infinite and zero
otherwise. For example,

</P>

<PRE>
isinf ([13, Inf, NaN])
     => [ 0, 1, 0 ]
</PRE>

</DL>

<P>
<DL>
<DT><U>Mapping Function:</U>  <B>isnan</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX579"></A>
Return 1 for elements of <VAR>x</VAR> that are NaN values and zero
otherwise. For example,

</P>

<PRE>
isnan ([13, Inf, NaN])
     => [ 0, 0, 1 ]
</PRE>

</DL>

<P>
<DL>
<DT><U>Mapping Function:</U>  <B>finite</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX580"></A>
Return 1 for elements of <VAR>x</VAR> that are NaN values and zero
otherwise. For example,

</P>

<PRE>
finite ([13, Inf, NaN])
     => [ 1, 0, 0 ]
</PRE>

</DL>

<P>
<DL>
<DT><U>Loadable Function:</U>  <B>find</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX581"></A>
Return a vector of indices of nonzero elements of a matrix.  To obtain a
single index for each matrix element, Octave pretends that the columns
of a matrix form one long vector (like Fortran arrays are stored).  For
example,

</P>

<PRE>
find (eye (2))
     => [ 1; 4 ]
</PRE>

<P>
If two outputs are requested, <CODE>find</CODE> returns the row and column
indices of nonzero elements of a matrix.  For example,

</P>

<PRE>
[i, j] = find (2 * eye (2))
     => i = [ 1; 2 ]
     => j = [ 1; 2 ]
</PRE>

<P>
If three outputs are requested, <CODE>find</CODE> also returns a vector
containing the nonzero values.  For example,

</P>

<PRE>
[i, j, v] = find (3 * eye (2))
     => i = [ 1; 2 ]
     => j = [ 1; 2 ]
     => v = [ 3; 3 ]
</PRE>

</DL>
<P>
        


<H2><A NAME="SEC133" HREF="octave_toc.html#TOC133">Rearranging Matrices</A></H2>

<P>
<DL>
<DT><U>Function File:</U>  <B>fliplr</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX582"></A>
Return a copy of <VAR>x</VAR> with the order of the columns reversed.  For
example, 

</P>

<PRE>
fliplr ([1, 2; 3, 4])
     =>  2  1
         4  3
</PRE>

</DL>

<P>
<DL>
<DT><U>Function File:</U>  <B>flipud</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX583"></A>
Return a copy of <VAR>x</VAR> with the order of the rows reversed.  For
example,

</P>

<PRE>
flipud ([1, 2; 3, 4])
     =>  3  4
         1  2
</PRE>

</DL>

<P>
<DL>
<DT><U>Function File:</U>  <B>rot90</B> <I>(<VAR>x</VAR>, <VAR>n</VAR>)</I>
<DD><A NAME="IDX584"></A>
Return a copy of <VAR>x</VAR> with the elements rotated counterclockwise in
90-degree increments.  The second argument is optional, and specifies
how many 90-degree rotations are to be applied (the default value is 1).
Negative values of <VAR>n</VAR> rotate the matrix in a clockwise direction.
For example,

</P>

<PRE>
rot90 ([1, 2; 3, 4], -1)
     =>  3  1
         4  2
</PRE>

<P>
rotates the given matrix clockwise by 90 degrees.  The following are all
equivalent statements:

</P>

<PRE>
rot90 ([1, 2; 3, 4], -1)
==
rot90 ([1, 2; 3, 4], 3)
==
rot90 ([1, 2; 3, 4], 7)
</PRE>

</DL>

<P>
<DL>
<DT><U>Function File:</U>  <B>reshape</B> <I>(<VAR>a</VAR>, <VAR>m</VAR>, <VAR>n</VAR>)</I>
<DD><A NAME="IDX585"></A>
Return a matrix with <VAR>m</VAR> rows and <VAR>n</VAR> columns whose elements are
taken from the matrix <VAR>a</VAR>.  To decide how to order the elements,
Octave pretends that the elements of a matrix are stored in column-major
order (like Fortran arrays are stored).

</P>
<P>
For example,

</P>

<PRE>
reshape ([1, 2, 3, 4], 2, 2)
     =>  1  3
         2  4
</PRE>

<P>
If the variable <CODE>do_fortran_indexing</CODE> is nonzero, the
<CODE>reshape</CODE> function is equivalent to

</P>

<PRE>
retval = zeros (m, n);
retval (:) = a;
</PRE>

<P>
but it is somewhat less cryptic to use <CODE>reshape</CODE> instead of the
colon operator.  Note that the total number of elements in the original
matrix must match the total number of elements in the new matrix.
</DL>

</P>
<P>
<DL>
<DT><U>Function File:</U>  <B>shift</B> <I>(<VAR>x</VAR>, <VAR>b</VAR>)</I>
<DD><A NAME="IDX586"></A>
If <VAR>x</VAR> is a vector, perform a circular shift of length <VAR>b</VAR> of
the elements of <VAR>x</VAR>.

</P>
<P>
If <VAR>x</VAR> is a matrix, do the same for each column of <VAR>x</VAR>.
</DL>

</P>
<P>
<DL>
<DT><U>Loadable Function:</U> [<VAR>s</VAR>, <VAR>i</VAR>] = <B>sort</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX587"></A>
Return a copy of <VAR>x</VAR> with the elements elements arranged in
increasing order.  For matrices, <CODE>sort</CODE> orders the elements in each
column.

</P>
<P>
For example,

</P>

<PRE>
sort ([1, 2; 2, 3; 3, 1])
     =>  1  1
         2  2
         3  3
</PRE>

<P>
The <CODE>sort</CODE> function may also be used to produce a matrix
containing the original row indices of the elements in the sorted
matrix.  For example,

</P>

<PRE>
[s, i] = sort ([1, 2; 2, 3; 3, 1])
     => s = 1  1
            2  2
            3  3
     => i = 1  3
            2  1
            3  2
</PRE>

</DL>

<P>
Since the <CODE>sort</CODE> function does not allow sort keys to be specified,
it can't be used to order the rows of a matrix according to the values
of the elements in various columns<A NAME="DOCF6" HREF="octave_foot.html#FOOT6">(6)</A>
in a single call.  Using the second output, however, it is possible to
sort all rows based on the values in a given column.  Here's an example
that sorts the rows of a matrix based on the values in the second
column.

</P>

<PRE>
a = [1, 2; 2, 3; 3, 1];
[s, i] = sort (a (:, 2));
a (i, :)
     =>  3  1
         1  2
         2  3
</PRE>

<P>
<DL>
<DT><U>Function File:</U>  <B>tril</B> <I>(<VAR>a</VAR>, <VAR>k</VAR>)</I>
<DD><A NAME="IDX588"></A>
<DT><U>Function File:</U>  <B>triu</B> <I>(<VAR>a</VAR>, <VAR>k</VAR>)</I>
<DD><A NAME="IDX589"></A>
Return a new matrix formed by extracting extract the lower (<CODE>tril</CODE>)
or upper (<CODE>triu</CODE>) triangular part of the matrix <VAR>a</VAR>, and
setting all other elements to zero.  The second argument is optional,
and specifies how many diagonals above or below the main diagonal should
also be set to zero.

</P>
<P>
The default value of <VAR>k</VAR> is zero, so that <CODE>triu</CODE> and
<CODE>tril</CODE> normally include the main diagonal as part of the result
matrix.

</P>
<P>
If the value of <VAR>k</VAR> is negative, additional elements above (for
<CODE>tril</CODE>) or below (for <CODE>triu</CODE>) the main diagonal are also
selected.

</P>
<P>
The absolute value of <VAR>k</VAR> must not be greater than the number of
sub- or super-diagonals.

</P>
<P>
For example,

</P>

<PRE>
tril (ones (3), -1)
     =>  0  0  0
         1  0  0
         1  1  0
</PRE>

<P>
and

</P>

<PRE>
tril (ones (3), 1)
     =>  1  1  0
         1  1  1
         1  1  1
</PRE>

</DL>

<P>
<DL>
<DT><U>Function File:</U>  <B>vec</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX590"></A>
Return the vector obtained by stacking the columns of the matrix <VAR>x</VAR>
one above the other.
</DL>

</P>
<P>
<DL>
<DT><U>Function File:</U>  <B>vech</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX591"></A>
Return the vector obtained by eliminating all supradiagonal elements of
the square matrix <VAR>x</VAR> and stacking the result one column above the
other.
</DL>

</P>


<H2><A NAME="SEC134" HREF="octave_toc.html#TOC134">Special Utility Matrices</A></H2>

<P>
<DL>
<DT><U>Built-in Function:</U>  <B>eye</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX592"></A>
<DT><U>Built-in Function:</U>  <B>eye</B> <I>(<VAR>n</VAR>, <VAR>m</VAR>)</I>
<DD><A NAME="IDX593"></A>
Return an identity matrix.  If invoked with a single scalar argument,
<CODE>eye</CODE> returns a square matrix with the dimension specified.  If you
supply two scalar arguments, <CODE>eye</CODE> takes them to be the number of
rows and columns.  If given a vector with two elements, <CODE>eye</CODE> uses
the values of the elements as the number of rows and columns,
respectively.  For example,

</P>

<PRE>
eye (3)
     =>  1  0  0
         0  1  0
         0  0  1
</PRE>

<P>
The following expressions all produce the same result:

</P>

<PRE>
eye (2)
==
eye (2, 2)
==
eye (size ([1, 2; 3, 4])
</PRE>

<P>
For compatibility with MATLAB, calling <CODE>eye</CODE> with no arguments
is equivalent to calling it with an argument of 1.
</DL>

</P>
<P>
<DL>
<DT><U>Built-in Function:</U>  <B>ones</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX594"></A>
<DT><U>Built-in Function:</U>  <B>ones</B> <I>(<VAR>n</VAR>, <VAR>m</VAR>)</I>
<DD><A NAME="IDX595"></A>
Return a matrix whose elements are all 1.  The arguments are handled
the same as the arguments for <CODE>eye</CODE>.

</P>
<P>
If you need to create a matrix whose values are all the same, you should
use an expression like

</P>

<PRE>
val_matrix = val * ones (n, m)
</PRE>

</DL>

<P>
<DL>
<DT><U>Built-in Function:</U>  <B>zeros</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX596"></A>
<DT><U>Built-in Function:</U>  <B>zeros</B> <I>(<VAR>n</VAR>, <VAR>m</VAR>)</I>
<DD><A NAME="IDX597"></A>
Return a matrix whose elements are all 0.  The arguments are handled
the same as the arguments for <CODE>eye</CODE>.
</DL>

</P>
<P>
<DL>
<DT><U>Loadable Function:</U>  <B>rand</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX598"></A>
<DT><U>Loadable Function:</U>  <B>rand</B> <I>(<VAR>n</VAR>, <VAR>m</VAR>)</I>
<DD><A NAME="IDX599"></A>
<DT><U>Loadable Function:</U>  <B>rand</B> <I>(<CODE>"seed"</CODE>, <VAR>x</VAR>)</I>
<DD><A NAME="IDX600"></A>
Return a matrix with random elements uniformly distributed on the
interval (0, 1).  The arguments are handled the same as the arguments
for <CODE>eye</CODE>.  In
addition, you can set the seed for the random number generator using the
form

</P>

<PRE>
rand ("seed", <VAR>x</VAR>)
</PRE>

<P>
where <VAR>x</VAR> is a scalar value.  If called as

</P>

<PRE>
rand ("seed")
</PRE>

<P>
<CODE>rand</CODE> returns the current value of the seed.
</DL>

</P>
<P>
<DL>
<DT><U>Loadable Function:</U>  <B>randn</B> <I>(<VAR>x</VAR>)</I>
<DD><A NAME="IDX601"></A>
<DT><U>Loadable Function:</U>  <B>randn</B> <I>(<VAR>n</VAR>, <VAR>m</VAR>)</I>
<DD><A NAME="IDX602"></A>
<DT><U>Loadable Function:</U>  <B>randn</B> <I>(<CODE>"seed"</CODE>, <VAR>x</VAR>)</I>
<DD><A NAME="IDX603"></A>
Return a matrix with normally distributed random elements.  The
arguments are handled the same as the arguments for <CODE>eye</CODE>.  In
addition, you can set the seed for the random number generator using the
form

</P>

<PRE>
randn ("seed", <VAR>x</VAR>)
</PRE>

<P>
where <VAR>x</VAR> is a scalar value.  If called as

</P>

<PRE>
randn ("seed")
</PRE>

<P>
<CODE>randn</CODE> returns the current value of the seed.
</DL>

</P>
<P>
The <CODE>rand</CODE> and <CODE>randn</CODE> functions use separate generators.
This ensures that

</P>

<PRE>
rand ("seed", 13);
randn ("seed", 13);
u = rand (100, 1);
n = randn (100, 1);
</PRE>

<P>
and

</P>

<PRE>
rand ("seed", 13);
randn ("seed", 13);
u = zeros (100, 1);
n = zeros (100, 1);
for i = 1:100
  u(i) = rand ();
  n(i) = randn ();
end
</PRE>

<P>
produce equivalent results.

</P>
<P>
Normally, <CODE>rand</CODE> and <CODE>randn</CODE> obtain their initial
seeds from the system clock, so that the sequence of random numbers is
not the same each time you run Octave.  If you really do need for to
reproduce a sequence of numbers exactly, you can set the seed to a
specific value.

</P>
<P>
If it is invoked without arguments, <CODE>rand</CODE> and <CODE>randn</CODE> return a
single element of a random sequence.

</P>
<P>
The <CODE>rand</CODE> and <CODE>randn</CODE> functions use Fortran code from
RANLIB, a library of fortran routines for random number generation,
compiled by Barry W. Brown and James Lovato of the Department of
Biomathematics at The University of Texas, M.D. Anderson Cancer Center,
Houston, TX 77030.

</P>
<P>
<DL>
<DT><U>Built-in Function:</U>  <B>diag</B> <I>(<VAR>v</VAR>, <VAR>k</VAR>)</I>
<DD><A NAME="IDX604"></A>
Return a diagonal matrix with vector <VAR>v</VAR> on diagonal <VAR>k</VAR>.  The
second argument is optional.  If it is positive, the vector is placed on
the <VAR>k</VAR>-th super-diagonal.  If it is negative, it is placed on the
<VAR>-k</VAR>-th sub-diagonal.  The default value of <VAR>k</VAR> is 0, and the
vector is placed on the main diagonal.  For example,

</P>

<PRE>
diag ([1, 2, 3], 1)
     =>  0  1  0  0
         0  0  2  0
         0  0  0  3
         0  0  0  0
</PRE>

</DL>

<P>
The functions <CODE>linspace</CODE> and <CODE>logspace</CODE> make it very easy to
create vectors with evenly or logarithmically spaced elements.
See section <A HREF="octave_5.html#SEC51">Ranges</A>.

</P>
<P>
<DL>
<DT><U>Function File:</U>  <B>linspace</B> <I>(<VAR>base</VAR>, <VAR>limit</VAR>, <VAR>n</VAR>)</I>
<DD><A NAME="IDX605"></A>
Return a row vector with <VAR>n</VAR> linearly spaced elements between
<VAR>base</VAR> and <VAR>limit</VAR>.  The number of elements, <VAR>n</VAR>, must be
greater than 1.  The <VAR>base</VAR> and <VAR>limit</VAR> are always included in
the range.  If <VAR>base</VAR> is greater than <VAR>limit</VAR>, the elements are
stored in decreasing order.  If the number of points is not specified, a
value of 100 is used.

</P>
<P>
The <CODE>linspace</CODE> function always returns a row vector, regardless of
the value of <CODE>prefer_column_vectors</CODE>.
</DL>

</P>
<P>
<DL>
<DT><U>Function File:</U>  <B>logspace</B> <I>(<VAR>base</VAR>, <VAR>limit</VAR>, <VAR>n</VAR>)</I>
<DD><A NAME="IDX606"></A>
Similar to <CODE>linspace</CODE> except that the values are logarithmically
spaced from
10^base to 10^limit.

</P>
<P>
If <VAR>limit</VAR> is equal to
pi,
the points are between
10^base and pi,
<EM>not</EM>
10^base and 10^pi,
in order to  be compatible with the corresponding MATLAB function.
</DL>

</P>
<P>
<DL>
<DT><U>Built-in Variable:</U> <B>treat_neg_dim_as_zero</B>
<DD><A NAME="IDX607"></A>
If the value of <CODE>treat_neg_dim_as_zero</CODE> is nonzero, expressions
like

</P>

<PRE>
eye (-1)
</PRE>

<P>
produce an empty matrix (i.e., row and column dimensions are zero).
Otherwise, an error message is printed and control is returned to the
top level.  The default value is 0.
</DL>

</P>


<H2><A NAME="SEC135" HREF="octave_toc.html#TOC135">Famous Matrices</A></H2>

<P>
The following functions return famous matrix forms.

</P>
<P>
<DL>
<DT><U>Function File:</U>  <B>hankel</B> <I>(<VAR>c</VAR>, <VAR>r</VAR>)</I>
<DD><A NAME="IDX608"></A>
Return the Hankel matrix constructed given the first column <VAR>c</VAR>, and
(optionally) the last row <VAR>r</VAR>.  If the last element of <VAR>c</VAR> is
not the same as the first element of <VAR>r</VAR>, the last element of
<VAR>c</VAR> is used.  If the second argument is omitted, the last row is
taken to be the same as the first column.

</P>
<P>
A Hankel matrix formed from an m-vector <VAR>c</VAR>, and an n-vector
<VAR>r</VAR>, has the elements

</P>

<PRE>
H (i, j) = c (i+j-1),  i+j-1 &#60;= m;
H (i, j) = r (i+j-m),  otherwise
</PRE>

</DL>

<P>
<DL>
<DT><U>Function File:</U>  <B>hilb</B> <I>(<VAR>n</VAR>)</I>
<DD><A NAME="IDX609"></A>
Return the Hilbert matrix of order <VAR>n</VAR>.  The
i, j
element of a Hilbert matrix is defined as

</P>

<PRE>
H (i, j) = 1 / (i + j - 1)
</PRE>

</DL>

<P>
<DL>
<DT><U>Function File:</U>  <B>invhilb</B> <I>(<VAR>n</VAR>)</I>
<DD><A NAME="IDX610"></A>
Return the inverse of a Hilbert matrix of order <VAR>n</VAR>.  This is exact.
Compare with the numerical calculation of <CODE>inverse (hilb (n))</CODE>,
which suffers from the ill-conditioning of the Hilbert matrix, and the
finite precision of your computer's floating point arithmetic.
</DL>

</P>
<P>
<DL>
<DT><U>Function File:</U>  <B>sylvester_matrix</B> <I>(<VAR>k</VAR>)</I>
<DD><A NAME="IDX611"></A>
Return the Sylvester matrix of order
n = 2^k.
</DL>

</P>
<P>
<DL>
<DT><U>Function File:</U>  <B>toeplitz</B> <I>(<VAR>c</VAR>, <VAR>r</VAR>)</I>
<DD><A NAME="IDX612"></A>
Return the Toeplitz matrix constructed given the first column <VAR>c</VAR>,
and (optionally) the first row <VAR>r</VAR>.  If the first element of <VAR>c</VAR>
is not the same as the first element of <VAR>r</VAR>, the first element of
<VAR>c</VAR> is used.  If the second argument is omitted, the first row is
taken to be the same as the first column.

</P>
<P>
A square Toeplitz matrix has the form

</P>

<PRE>
c(0)  r(1)   r(2)  ...  r(n)
c(1)  c(0)   r(1)      r(n-1)
c(2)  c(1)   c(0)      r(n-2)
 .                       .
 .                       .
 .                       .

c(n) c(n-1) c(n-2) ...  c(0)
</PRE>

</DL>

<P>
<DL>
<DT><U>Function File:</U>  <B>vander</B> <I>(<VAR>c</VAR>)</I>
<DD><A NAME="IDX613"></A>
Return the Vandermonde matrix whose next to last column is <VAR>c</VAR>.

</P>
<P>
A Vandermonde matrix has the form

</P>

<PRE>
c(0)^n ... c(0)^2  c(0)  1
c(1)^n ... c(1)^2  c(1)  1
 .           .      .    .
 .           .      .    .
 .           .      .    .
                 
c(n)^n ... c(n)^2  c(n)  1
</PRE>

</DL>

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