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@c Copyright (C) 1996, 1997, 1999, 2000, 2002, 2007 John W. Eaton
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@c This file is part of Octave.
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@c Octave is free software; you can redistribute it and/or modify it
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@c under the terms of the GNU General Public License as published by the
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@c Free Software Foundation; either version 3 of the License, or (at
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@c your option) any later version.
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@c Octave is distributed in the hope that it will be useful, but WITHOUT
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@c ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
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@c FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
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@c You should have received a copy of the GNU General Public License
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@c along with Octave; see the file COPYING. If not, see
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@c <http://www.gnu.org/licenses/>.
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@node Polynomial Manipulations
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@chapter Polynomial Manipulations
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In Octave, a polynomial is represented by its coefficients (arranged
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in descending order). For example, a vector @var{c} of length
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@math{N+1} corresponds to the following polynomial of order
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p (x) = c_1 x^N + \ldots + c_N x + c_{N+1}.
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p(x) = @var{c}(1) x^@var{N} + ... + @var{c}(@var{N}) x + @var{c}(@var{N}+1).
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* Evaluating Polynomials::
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* Products of Polynomials::
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* Derivatives and Integrals::
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* Polynomial Interpolation::
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* Miscellaneous Functions::
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@node Evaluating Polynomials
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@section Evaluating Polynomials
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The value of a polynomial represented by the vector @var{c} can be evaluated
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at the point @var{x} very easily, as the following example shows.
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val = dot( x.^(N:-1:0), c );
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While the above example shows how easy it is to compute the value of a
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polynomial, it isn't the most stable algorithm. With larger polynomials
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you should use more elegant algorithms, such as Horner's Method, which
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is exactly what the Octave function @code{polyval} does.
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In the case where @var{x} is a square matrix, the polynomial given by
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@var{c} is still well-defined. As when @var{x} is a scalar the obvious
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implementation is easily expressed in Octave, but also in this case
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more elegant algorithms perform better. The @code{polyvalm} function
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provides such an algorithm.
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@section Finding Roots
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Octave can find the roots of a given polynomial. This is done by computing
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the companion matrix of the polynomial (see the @code{compan} function
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for a definition), and then finding its eigenvalues.
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@node Products of Polynomials
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@section Products of Polynomials
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@node Derivatives and Integrals
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@section Derivatives and Integrals
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Octave comes with functions for computing the derivative and the integral
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of a polynomial. The functions @code{polyderiv} and @code{polyint}
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both return new polynomials describing the result. As an example we'll
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compute the definite integral of @math{p(x) = x^2 + 1} from 0 to 3.
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integral = polyint(c);
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area = polyval(integral, 3) - polyval(integral, 0)
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@DOCSTRING(polyderiv)
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@node Polynomial Interpolation
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@section Polynomial Interpolation
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Octave comes with good support for various kinds of interpolation,
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most of which are described in @ref{Interpolation}. One simple alternative
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to the functions described in the aforementioned chapter, is to fit
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a single polynomial to some given data points. To avoid a highly
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fluctuating polynomial, one most often wants to fit a low-order polynomial
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to data. This usually means that it is necessary to fit the polynomial
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in a least-squares sense, which is what the @code{polyfit} function does.
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In situations where a single polynomial isn't good enough, a solution
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is to use several polynomials pieced together. The function @code{mkpp}
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creates a piece-wise polynomial, @code{ppval} evaluates the function
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created by @code{mkpp}, and @code{unmkpp} returns detailed information
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The following example shows how to combine two linear functions and a
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quadratic into one function. Each of these functions is expressed
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on adjoined intervals.
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xi = linspace(-2, 2, 50);
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@node Miscellaneous Functions
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@section Miscellaneous Functions
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@DOCSTRING(polyreduce)
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doc/interpreter/poly.txi
