@c Copyright (C) 1996, 1997, 1999, 2000, 2002, 2007, 2008, 2009 John W. Eaton
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@c 
@c You should have received a copy of the GNU General Public License
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@node Polynomial Manipulations
@chapter Polynomial Manipulations

In Octave, a polynomial is represented by its coefficients (arranged
in descending order).  For example, a vector @var{c} of length
@math{N+1} corresponds to the following polynomial of order
@iftex
@tex
 $N$
$$
 p (x) = c_1 x^N + \ldots + c_N x + c_{N+1}.
$$
@end tex
@end iftex
@ifinfo
 @var{N}

@example
p(x) = @var{c}(1) x^@var{N} + ... + @var{c}(@var{N}) x + @var{c}(@var{N}+1).
@end example
@end ifinfo

@menu
* Evaluating Polynomials::
* Finding Roots::
* Products of Polynomials::
* Derivatives and Integrals::
* Polynomial Interpolation::
* Miscellaneous Functions::
@end menu

@node Evaluating Polynomials
@section Evaluating Polynomials

The value of a polynomial represented by the vector @var{c} can be evaluated
at the point @var{x} very easily, as the following example shows:

@example
N = length(c)-1;
val = dot( x.^(N:-1:0), c );
@end example

@noindent
While the above example shows how easy it is to compute the value of a
polynomial, it isn't the most stable algorithm.  With larger polynomials
you should use more elegant algorithms, such as Horner's Method, which
is exactly what the Octave function @code{polyval} does.

In the case where @var{x} is a square matrix, the polynomial given by
@var{c} is still well-defined.  As when @var{x} is a scalar the obvious
implementation is easily expressed in Octave, but also in this case
more elegant algorithms perform better.  The @code{polyvalm} function
provides such an algorithm.

@DOCSTRING(polyval)

@DOCSTRING(polyvalm)

@node Finding Roots
@section Finding Roots

Octave can find the roots of a given polynomial.  This is done by computing
the companion matrix of the polynomial (see the @code{compan} function
for a definition), and then finding its eigenvalues.

@DOCSTRING(roots)

@DOCSTRING(compan)

@DOCSTRING(mpoles)

@node Products of Polynomials
@section Products of Polynomials

@DOCSTRING(conv)

@DOCSTRING(convn)

@DOCSTRING(deconv)

@DOCSTRING(conv2)

@DOCSTRING(polygcd)

@DOCSTRING(residue)

@node Derivatives and Integrals
@section Derivatives and Integrals

Octave comes with functions for computing the derivative and the integral
of a polynomial.  The functions @code{polyderiv} and @code{polyint}
both return new polynomials describing the result.  As an example we'll
compute the definite integral of @math{p(x) = x^2 + 1} from 0 to 3.

@example
c = [1, 0, 1];
integral = polyint(c);
area = polyval(integral, 3) - polyval(integral, 0)
@result{} 12
@end example

@DOCSTRING(polyderiv)

@DOCSTRING(polyder)

@DOCSTRING(polyinteg)

@DOCSTRING(polyint)

@node Polynomial Interpolation
@section Polynomial Interpolation

Octave comes with good support for various kinds of interpolation,
most of which are described in @ref{Interpolation}.  One simple alternative
to the functions described in the aforementioned chapter, is to fit
a single polynomial to some given data points.  To avoid a highly
fluctuating polynomial, one most often wants to fit a low-order polynomial
to data.  This usually means that it is necessary to fit the polynomial
in a least-squares sense, which is what the @code{polyfit} function does.

@DOCSTRING(polyfit)

In situations where a single polynomial isn't good enough, a solution
is to use several polynomials pieced together.  The function @code{mkpp}
creates a piece-wise polynomial, @code{ppval} evaluates the function 
created by @code{mkpp}, and @code{unmkpp} returns detailed information
about the function.

The following example shows how to combine two linear functions and a
quadratic into one function.  Each of these functions is expressed
on adjoined intervals.

@example
x = [-2, -1, 1, 2];
p = [ 0,  1, 0;
      1, -2, 1;
      0, -1, 1 ];
pp = mkpp(x, p);
xi = linspace(-2, 2, 50);
yi = ppval(pp, xi);
plot(xi, yi);
@end example

@DOCSTRING(ppval)

@DOCSTRING(mkpp)

@DOCSTRING(unmkpp)

@node Miscellaneous Functions
@section Miscellaneous Functions

@DOCSTRING(poly)

@DOCSTRING(polyout)

@DOCSTRING(polyreduce)