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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
@tex
 $N$
$$
 p (x) = c_1 x^N + \ldots + c_N x + c_{N+1}.
$$
@end tex
@ifnottex
 @var{N}

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

@end ifnottex

@menu
* Evaluating Polynomials::
* Finding Roots::
* Products of Polynomials::
* Derivatives / Integrals / Transforms::
* 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
@group
N = length (c) - 1;
val = dot (x.^(N:-1:0), c);
@end group
@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 @nospell{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.

@c polyval scripts/polynomial/polyval.m
@anchor{XREFpolyval}
@deftypefn  {} {@var{y} =} polyval (@var{p}, @var{x})
@deftypefnx {} {@var{y} =} polyval (@var{p}, @var{x}, [], @var{mu})
@deftypefnx {} {[@var{y}, @var{dy}] =} polyval (@var{p}, @var{x}, @var{s})
@deftypefnx {} {[@var{y}, @var{dy}] =} polyval (@var{p}, @var{x}, @var{s}, @var{mu})

Evaluate the polynomial @var{p} at the specified values of @var{x}.

If @var{x} is a vector or matrix, the polynomial is evaluated for each of
the elements of @var{x}.

When @var{mu} is present, evaluate the polynomial for
(@var{x}-@var{mu}(1))/@var{mu}(2).

In addition to evaluating the polynomial, the second output represents the
prediction interval, @var{y} +/- @var{dy}, which contains at least 50% of
the future predictions.  To calculate the prediction interval, the
structured variable @var{s}, originating from @code{polyfit}, must be
supplied.

@seealso{@ref{XREFpolyvalm,,polyvalm}, @ref{XREFpolyaffine,,polyaffine}, @ref{XREFpolyfit,,polyfit}, @ref{XREFroots,,roots}, @ref{XREFpoly,,poly}}
@end deftypefn


@c polyvalm scripts/polynomial/polyvalm.m
@anchor{XREFpolyvalm}
@deftypefn {} {} polyvalm (@var{c}, @var{x})
Evaluate a polynomial in the matrix sense.

@code{polyvalm (@var{c}, @var{x})} will evaluate the polynomial in the
matrix sense, i.e., matrix multiplication is used instead of element by
element multiplication as used in @code{polyval}.

The argument @var{x} must be a square matrix.
@seealso{@ref{XREFpolyval,,polyval}, @ref{XREFroots,,roots}, @ref{XREFpoly,,poly}}
@end deftypefn


@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.

@c roots scripts/polynomial/roots.m
@anchor{XREFroots}
@deftypefn {} {} roots (@var{c})

Compute the roots of the polynomial @var{c}.

For a vector @var{c} with @math{N} components, return the roots of the
polynomial
@tex
$$
c_1 x^{N-1} + \cdots + c_{N-1} x + c_N.
$$
@end tex
@ifnottex

@example
c(1) * x^(N-1) + @dots{} + c(N-1) * x + c(N)
@end example

@end ifnottex

As an example, the following code finds the roots of the quadratic
polynomial
@tex
$$ p(x) = x^2 - 5. $$
@end tex
@ifnottex

@example
p(x) = x^2 - 5.
@end example

@end ifnottex

@example
@group
c = [1, 0, -5];
roots (c)
@result{}  2.2361
@result{} -2.2361
@end group
@end example

Note that the true result is
@tex
$\pm \sqrt{5}$
@end tex
@ifnottex
@math{+/- sqrt(5)}
@end ifnottex
which is roughly
@tex
$\pm 2.2361$.
@end tex
@ifnottex
@math{+/- 2.2361}.
@end ifnottex
@seealso{@ref{XREFpoly,,poly}, @ref{XREFcompan,,compan}, @ref{XREFfzero,,fzero}}
@end deftypefn


@c polyeig scripts/polynomial/polyeig.m
@anchor{XREFpolyeig}
@deftypefn  {} {@var{z} =} polyeig (@var{C0}, @var{C1}, @dots{}, @var{Cl})
@deftypefnx {} {[@var{v}, @var{z}] =} polyeig (@var{C0}, @var{C1}, @dots{}, @var{Cl})

Solve the polynomial eigenvalue problem of degree @var{l}.

Given an @var{n}x@var{n} matrix polynomial

@code{@var{C}(@var{s}) = @var{C0} + @var{C1} @var{s} + @dots{} + @var{Cl}
@var{s}^@var{l}}

@code{polyeig} solves the eigenvalue problem

@code{(@var{C0} + @var{C1} @var{z} + @dots{} + @var{Cl} z^@var{l}) @var{v} =
0}.

Note that the eigenvalues @var{z} are the zeros of the matrix polynomial.
@var{z} is a row vector with @code{@var{n}*@var{l}} elements.  @var{v} is a
matrix (@var{n} x @var{n}*@var{l}) with columns that correspond to the
eigenvectors.

@seealso{@ref{XREFeig,,eig}, @ref{XREFeigs,,eigs}, @ref{XREFcompan,,compan}}
@end deftypefn


@c compan scripts/polynomial/compan.m
@anchor{XREFcompan}
@deftypefn {} {} compan (@var{c})
Compute the companion matrix corresponding to polynomial coefficient vector
@var{c}.

The companion matrix is
@tex
$$
A = \left[\matrix{
 -c_2/c_1 & -c_3/c_1 & \cdots & -c_N/c_1 & -c_{N+1}/c_1\cr
     1    &     0    & \cdots &     0    &         0   \cr
     0    &     1    & \cdots &     0    &         0   \cr
  \vdots  &   \vdots & \ddots &  \vdots  &      \vdots \cr
     0    &     0    & \cdots &     1    &         0}\right].
$$
@end tex
@ifnottex
@c Set example in small font to prevent overfull line

@smallexample
@group
     _                                                        _
    |  -c(2)/c(1)   -c(3)/c(1)  @dots{}  -c(N)/c(1)  -c(N+1)/c(1)  |
    |       1            0      @dots{}       0             0      |
    |       0            1      @dots{}       0             0      |
A = |       .            .      .         .             .      |
    |       .            .       .        .             .      |
    |       .            .        .       .             .      |
    |_      0            0      @dots{}       1             0     _|
@end group
@end smallexample

@end ifnottex
The eigenvalues of the companion matrix are equal to the roots of the
polynomial.
@seealso{@ref{XREFroots,,roots}, @ref{XREFpoly,,poly}, @ref{XREFeig,,eig}}
@end deftypefn


@c mpoles scripts/polynomial/mpoles.m
@anchor{XREFmpoles}
@deftypefn  {} {[@var{multp}, @var{idxp}] =} mpoles (@var{p})
@deftypefnx {} {[@var{multp}, @var{idxp}] =} mpoles (@var{p}, @var{tol})
@deftypefnx {} {[@var{multp}, @var{idxp}] =} mpoles (@var{p}, @var{tol}, @var{reorder})
Identify unique poles in @var{p} and their associated multiplicity.

The output is ordered from largest pole to smallest pole.

If the relative difference of two poles is less than @var{tol} then they are
considered to be multiples.  The default value for @var{tol} is 0.001.

If the optional parameter @var{reorder} is zero, poles are not sorted.

The output @var{multp} is a vector specifying the multiplicity of the poles.
@code{@var{multp}(n)} refers to the multiplicity of the Nth pole
@code{@var{p}(@var{idxp}(n))}.

For example:

@example
@group
p = [2 3 1 1 2];
[m, n] = mpoles (p)
   @result{} m = [1; 1; 2; 1; 2]
   @result{} n = [2; 5; 1; 4; 3]
   @result{} p(n) = [3, 2, 2, 1, 1]
@end group
@end example

@seealso{@ref{XREFresidue,,residue}, @ref{XREFpoly,,poly}, @ref{XREFroots,,roots}, @ref{XREFconv,,conv}, @ref{XREFdeconv,,deconv}}
@end deftypefn


@node Products of Polynomials
@section Products of Polynomials

@c conv scripts/polynomial/conv.m
@anchor{XREFconv}
@deftypefn  {} {} conv (@var{a}, @var{b})
@deftypefnx {} {} conv (@var{a}, @var{b}, @var{shape})
Convolve two vectors @var{a} and @var{b}.

When @var{a} and @var{b} are the coefficient vectors of two polynomials, the
convolution represents the coefficient vector of the product polynomial.

The size of the result is determined by the optional @var{shape} argument
which takes the following values

@table @asis
@item @var{shape} = @qcode{"full"}
Return the full convolution.  (default)
The result is a vector with length equal to
@code{length (@var{a}) + length (@var{b}) - 1}.

@item @var{shape} = @qcode{"same"}
Return the central part of the convolution with the same size as @var{a}.

@item @var{shape} = @qcode{"valid"}
Return only the parts which do not include zero-padded edges.
The size of the result is
@code{max (size (@var{a}) - size (@var{b}) + 1, 0)}.
@end table

@seealso{@ref{XREFdeconv,,deconv}, @ref{XREFconv2,,conv2}, @ref{XREFconvn,,convn}, @ref{XREFfftconv,,fftconv}}
@end deftypefn


@c convn libinterp/corefcn/conv2.cc
@anchor{XREFconvn}
@deftypefn  {} {@var{C} =} convn (@var{A}, @var{B})
@deftypefnx {} {@var{C} =} convn (@var{A}, @var{B}, @var{shape})
Return the n-D convolution of @var{A} and @var{B}.

The size of the result is determined by the optional @var{shape} argument
which takes the following values

@table @asis
@item @var{shape} = @qcode{"full"}
Return the full convolution.  (default)

@item @var{shape} = @qcode{"same"}
Return central part of the convolution with the same size as @var{A}.
The central part of the convolution begins at the indices
@code{floor ([size(@var{B})/2] + 1)}.

@item @var{shape} = @qcode{"valid"}
Return only the parts which do not include zero-padded edges.
The size of the result is @code{max (size (A) - size (B) + 1, 0)}.
@end table

@seealso{@ref{XREFconv2,,conv2}, @ref{XREFconv,,conv}}
@end deftypefn


@c deconv scripts/polynomial/deconv.m
@anchor{XREFdeconv}
@deftypefn  {} {@var{b} =} deconv (@var{y}, @var{a})
@deftypefnx {} {[@var{b}, @var{r}] =} deconv (@var{y}, @var{a})
Deconvolve two vectors (polynomial division).

@code{[@var{b}, @var{r}] = deconv (@var{y}, @var{a})} solves for @var{b} and
@var{r} such that @code{@var{y} = conv (@var{a}, @var{b}) + @var{r}}.

If @var{y} and @var{a} are polynomial coefficient vectors, @var{b} will
contain the coefficients of the polynomial quotient and @var{r} will be
a remainder polynomial of lowest order.
@seealso{@ref{XREFconv,,conv}, @ref{XREFresidue,,residue}}
@end deftypefn


@c conv2 libinterp/corefcn/conv2.cc
@anchor{XREFconv2}
@deftypefn  {} {} conv2 (@var{A}, @var{B})
@deftypefnx {} {} conv2 (@var{v1}, @var{v2}, @var{m})
@deftypefnx {} {} conv2 (@dots{}, @var{shape})
Return the 2-D convolution of @var{A} and @var{B}.

The size of the result is determined by the optional @var{shape} argument
which takes the following values

@table @asis
@item @var{shape} = @qcode{"full"}
Return the full convolution.  (default)

@item @var{shape} = @qcode{"same"}
Return the central part of the convolution with the same size as @var{A}.
The central part of the convolution begins at the indices
@code{floor ([size(@var{B})/2] + 1)}.

@item @var{shape} = @qcode{"valid"}
Return only the parts which do not include zero-padded edges.
The size of the result is @code{max (size (A) - size (B) + 1, 0)}.
@end table

When the third argument is a matrix, return the convolution of the matrix
@var{m} by the vector @var{v1} in the column direction and by the vector
@var{v2} in the row direction.
@seealso{@ref{XREFconv,,conv}, @ref{XREFconvn,,convn}}
@end deftypefn


@c polygcd scripts/polynomial/polygcd.m
@anchor{XREFpolygcd}
@deftypefn  {} {@var{q} =} polygcd (@var{b}, @var{a})
@deftypefnx {} {@var{q} =} polygcd (@var{b}, @var{a}, @var{tol})

Find the greatest common divisor of two polynomials.

This is equivalent to the polynomial found by multiplying together all the
common roots.  Together with deconv, you can reduce a ratio of two
polynomials.

The tolerance @var{tol} defaults to @code{sqrt (eps)}.

@strong{Caution:} This is a numerically unstable algorithm and should not
be used on large polynomials.

Example code:

@example
@group
polygcd (poly (1:8), poly (3:12)) - poly (3:8)
@result{} [ 0, 0, 0, 0, 0, 0, 0 ]
deconv (poly (1:8), polygcd (poly (1:8), poly (3:12))) - poly (1:2)
@result{} [ 0, 0, 0 ]
@end group
@end example
@seealso{@ref{XREFpoly,,poly}, @ref{XREFroots,,roots}, @ref{XREFconv,,conv}, @ref{XREFdeconv,,deconv}, @ref{XREFresidue,,residue}}
@end deftypefn


@c residue scripts/polynomial/residue.m
@anchor{XREFresidue}
@deftypefn  {} {[@var{r}, @var{p}, @var{k}, @var{e}] =} residue (@var{b}, @var{a})
@deftypefnx {} {[@var{b}, @var{a}] =} residue (@var{r}, @var{p}, @var{k})
@deftypefnx {} {[@var{b}, @var{a}] =} residue (@var{r}, @var{p}, @var{k}, @var{e})
The first calling form computes the partial fraction expansion for the
quotient of the polynomials, @var{b} and @var{a}.

The quotient is defined as
@tex
$$
{B(s)\over A(s)} = \sum_{m=1}^M {r_m\over (s-p_m)^e_m}
  + \sum_{i=1}^N k_i s^{N-i}.
$$
@end tex
@ifnottex

@example
@group
B(s)    M       r(m)        N
---- = SUM ------------- + SUM k(i)*s^(N-i)
A(s)   m=1 (s-p(m))^e(m)   i=1
@end group
@end example

@end ifnottex
@noindent
where @math{M} is the number of poles (the length of the @var{r}, @var{p},
and @var{e}), the @var{k} vector is a polynomial of order @math{N-1}
representing the direct contribution, and the @var{e} vector specifies the
multiplicity of the m-th residue's pole.

For example,

@example
@group
b = [1, 1, 1];
a = [1, -5, 8, -4];
[r, p, k, e] = residue (b, a)
   @result{} r = [-2; 7; 3]
   @result{} p = [2; 2; 1]
   @result{} k = [](0x0)
   @result{} e = [1; 2; 1]
@end group
@end example

@noindent
which represents the following partial fraction expansion
@tex
$$
{s^2+s+1\over s^3-5s^2+8s-4} = {-2\over s-2} + {7\over (s-2)^2} + {3\over s-1}
$$
@end tex
@ifnottex

@example
@group
        s^2 + s + 1       -2        7        3
   ------------------- = ----- + ------- + -----
   s^3 - 5s^2 + 8s - 4   (s-2)   (s-2)^2   (s-1)
@end group
@end example

@end ifnottex

The second calling form performs the inverse operation and computes the
reconstituted quotient of polynomials, @var{b}(s)/@var{a}(s), from the
partial fraction expansion; represented by the residues, poles, and a direct
polynomial specified by @var{r}, @var{p} and @var{k}, and the pole
multiplicity @var{e}.

If the multiplicity, @var{e}, is not explicitly specified the multiplicity
is determined by the function @code{mpoles}.

For example:

@example
@group
r = [-2; 7; 3];
p = [2; 2; 1];
k = [1, 0];
[b, a] = residue (r, p, k)
   @result{} b = [1, -5, 9, -3, 1]
   @result{} a = [1, -5, 8, -4]

where mpoles is used to determine e = [1; 2; 1]
@end group
@end example

Alternatively the multiplicity may be defined explicitly, for example,

@example
@group
r = [7; 3; -2];
p = [2; 1; 2];
k = [1, 0];
e = [2; 1; 1];
[b, a] = residue (r, p, k, e)
   @result{} b = [1, -5, 9, -3, 1]
   @result{} a = [1, -5, 8, -4]
@end group
@end example

@noindent
which represents the following partial fraction expansion
@tex
$$
{-2\over s-2} + {7\over (s-2)^2} + {3\over s-1} + s = {s^4-5s^3+9s^2-3s+1\over s^3-5s^2+8s-4}
$$
@end tex
@ifnottex

@example
@group
 -2        7        3         s^4 - 5s^3 + 9s^2 - 3s + 1
----- + ------- + ----- + s = --------------------------
(s-2)   (s-2)^2   (s-1)          s^3 - 5s^2 + 8s - 4
@end group
@end example

@end ifnottex
@seealso{@ref{XREFmpoles,,mpoles}, @ref{XREFpoly,,poly}, @ref{XREFroots,,roots}, @ref{XREFconv,,conv}, @ref{XREFdeconv,,deconv}}
@end deftypefn


@node Derivatives / Integrals / Transforms
@section Derivatives / Integrals / Transforms

Octave comes with functions for computing the derivative and the integral
of a polynomial.  The functions @code{polyder} 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
@group
c = [1, 0, 1];
integral = polyint (c);
area = polyval (integral, 3) - polyval (integral, 0)
@result{} 12
@end group
@end example

@c polyder scripts/polynomial/polyder.m
@anchor{XREFpolyder}
@deftypefn  {} {} polyder (@var{p})
@deftypefnx {} {[@var{k}] =} polyder (@var{a}, @var{b})
@deftypefnx {} {[@var{q}, @var{d}] =} polyder (@var{b}, @var{a})
Return the coefficients of the derivative of the polynomial whose
coefficients are given by the vector @var{p}.

If a pair of polynomials is given, return the derivative of the product
@math{@var{a}*@var{b}}.

If two inputs and two outputs are given, return the derivative of the
polynomial quotient @math{@var{b}/@var{a}}.  The quotient numerator is
in @var{q} and the denominator in @var{d}.
@seealso{@ref{XREFpolyint,,polyint}, @ref{XREFpolyval,,polyval}, @ref{XREFpolyreduce,,polyreduce}}
@end deftypefn


@c polyint scripts/polynomial/polyint.m
@anchor{XREFpolyint}
@deftypefn  {} {} polyint (@var{p})
@deftypefnx {} {} polyint (@var{p}, @var{k})
Return the coefficients of the integral of the polynomial whose
coefficients are represented by the vector @var{p}.

The variable @var{k} is the constant of integration, which by default is
set to zero.
@seealso{@ref{XREFpolyder,,polyder}, @ref{XREFpolyval,,polyval}}
@end deftypefn


@c polyaffine scripts/polynomial/polyaffine.m
@anchor{XREFpolyaffine}
@deftypefn {} {} polyaffine (@var{f}, @var{mu})
Return the coefficients of the polynomial vector @var{f} after an affine
transformation.

If @var{f} is the vector representing the polynomial f(x), then
@code{@var{g} = polyaffine (@var{f}, @var{mu})} is the vector representing:

@example
g(x) = f( (x - @var{mu}(1)) / @var{mu}(2) )
@end example

@seealso{@ref{XREFpolyval,,polyval}, @ref{XREFpolyfit,,polyfit}}
@end deftypefn


@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, or a piecewise polynomial (spline) 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 just
is what the @code{polyfit} function does.

@c polyfit scripts/polynomial/polyfit.m
@anchor{XREFpolyfit}
@deftypefn  {} {@var{p} =} polyfit (@var{x}, @var{y}, @var{n})
@deftypefnx {} {[@var{p}, @var{s}] =} polyfit (@var{x}, @var{y}, @var{n})
@deftypefnx {} {[@var{p}, @var{s}, @var{mu}] =} polyfit (@var{x}, @var{y}, @var{n})
Return the coefficients of a polynomial @var{p}(@var{x}) of degree
@var{n} that minimizes the least-squares-error of the fit to the points
@code{[@var{x}, @var{y}]}.

If @var{n} is a logical vector, it is used as a mask to selectively force
the corresponding polynomial coefficients to be used or ignored.

The polynomial coefficients are returned in a row vector.

The optional output @var{s} is a structure containing the following fields:

@table @samp
@item R
Triangular factor R from the QR@tie{}decomposition.

@item X
The @nospell{Vandermonde} matrix used to compute the polynomial
coefficients.

@item C
The unscaled covariance matrix, formally equal to the inverse of
@var{x'}*@var{x}, but computed in a way minimizing roundoff error
propagation.

@item df
The degrees of freedom.

@item normr
The norm of the residuals.

@item yf
The values of the polynomial for each value of @var{x}.
@end table

The second output may be used by @code{polyval} to calculate the
statistical error limits of the predicted values.  In particular, the
standard deviation of @var{p} coefficients is given by

@code{sqrt (diag (s.C)/s.df)*s.normr}.

When the third output, @var{mu}, is present the coefficients, @var{p}, are
associated with a polynomial in

@code{@var{xhat} = (@var{x} - @var{mu}(1)) / @var{mu}(2)} @*
where @var{mu}(1) = mean (@var{x}), and @var{mu}(2) = std (@var{x}).

This linear transformation of @var{x} improves the numerical stability of
the fit.
@seealso{@ref{XREFpolyval,,polyval}, @ref{XREFpolyaffine,,polyaffine}, @ref{XREFroots,,roots}, @ref{XREFvander,,vander}, @ref{XREFzscore,,zscore}}
@end deftypefn


In situations where a single polynomial isn't good enough, a solution
is to use several polynomials pieced together.  The function
@code{splinefit} fits a piecewise polynomial (spline) to a set of
data.

@c splinefit scripts/polynomial/splinefit.m
@anchor{XREFsplinefit}
@deftypefn  {} {@var{pp} =} splinefit (@var{x}, @var{y}, @var{breaks})
@deftypefnx {} {@var{pp} =} splinefit (@var{x}, @var{y}, @var{p})
@deftypefnx {} {@var{pp} =} splinefit (@dots{}, "periodic", @var{periodic})
@deftypefnx {} {@var{pp} =} splinefit (@dots{}, "robust", @var{robust})
@deftypefnx {} {@var{pp} =} splinefit (@dots{}, "beta", @var{beta})
@deftypefnx {} {@var{pp} =} splinefit (@dots{}, "order", @var{order})
@deftypefnx {} {@var{pp} =} splinefit (@dots{}, "constraints", @var{constraints})

Fit a piecewise cubic spline with breaks (knots) @var{breaks} to the
noisy data, @var{x} and @var{y}.

@var{x} is a vector, and @var{y} is a vector or N-D array.  If @var{y} is an
N-D array, then @var{x}(j) is matched to @var{y}(:,@dots{},:,j).

@var{p} is a positive integer defining the number of intervals along
@var{x}, and @var{p}+1 is the number of breaks.  The number of points in
each interval differ by no more than 1.

The optional property @var{periodic} is a logical value which specifies
whether a periodic boundary condition is applied to the spline.  The
length of the period is @code{max (@var{breaks}) - min (@var{breaks})}.
The default value is @code{false}.

The optional property @var{robust} is a logical value which specifies
if robust fitting is to be applied to reduce the influence of outlying
data points.  Three iterations of weighted least squares are performed.
Weights are computed from previous residuals.  The sensitivity of outlier
identification is controlled by the property @var{beta}.  The value of
@var{beta} is restricted to the range, 0 < @var{beta} < 1.  The default
value is @var{beta} = 1/2.  Values close to 0 give all data equal
weighting.  Increasing values of @var{beta} reduce the influence of
outlying data.  Values close to unity may cause instability or rank
deficiency.

The fitted spline is returned as a piecewise polynomial, @var{pp}, and
may be evaluated using @code{ppval}.

The splines are constructed of polynomials with degree @var{order}.
The default is a cubic, @var{order}=3.  A spline with P pieces has
P+@var{order} degrees of freedom.  With periodic boundary conditions
the degrees of freedom are reduced to P.

The optional property, @var{constraints}, is a structure specifying linear
constraints on the fit.  The structure has three fields, @qcode{"xc"},
@qcode{"yc"}, and @qcode{"cc"}.

@table @asis
@item @qcode{"xc"}
Vector of the x-locations of the constraints.

@item @qcode{"yc"}
Constraining values at the locations @var{xc}.
The default is an array of zeros.

@item @qcode{"cc"}
Coefficients (matrix).  The default is an array of ones.  The number of
rows is limited to the order of the piecewise polynomials, @var{order}.
@end table

Constraints are linear combinations of derivatives of order 0 to
@var{order}-1 according to

@example
@group
@tex
$cc(1,j) \cdot y(xc(j)) + cc(2,j) \cdot y\prime(xc(j)) + ... = yc(:,\dots,:,j)$.
@end tex
@ifnottex
cc(1,j) * y(xc(j)) + cc(2,j) * y'(xc(j)) + ... = yc(:,...,:,j).
@end ifnottex
@end group
@end example

@seealso{@ref{XREFinterp1,,interp1}, @ref{XREFunmkpp,,unmkpp}, @ref{XREFppval,,ppval}, @ref{XREFspline,,spline}, @ref{XREFpchip,,pchip}, @ref{XREFppder,,ppder}, @ref{XREFppint,,ppint}, @ref{XREFppjumps,,ppjumps}}
@end deftypefn


The number of @var{breaks} (or knots) used to construct the piecewise
polynomial is a significant factor in suppressing the noise present in
the input data, @var{x} and @var{y}.  This is demonstrated by the example
below.

@example
@group
x = 2 * pi * rand (1, 200);
y = sin (x) + sin (2 * x) + 0.2 * randn (size (x));
## Uniform breaks
breaks = linspace (0, 2 * pi, 41); % 41 breaks, 40 pieces
pp1 = splinefit (x, y, breaks);
## Breaks interpolated from data
pp2 = splinefit (x, y, 10);  % 11 breaks, 10 pieces
## Plot
xx = linspace (0, 2 * pi, 400);
y1 = ppval (pp1, xx);
y2 = ppval (pp2, xx);
plot (x, y, ".", xx, [y1; y2])
axis tight
ylim auto
legend (@{"data", "41 breaks, 40 pieces", "11 breaks, 10 pieces"@})
@end group
@end example

@ifnotinfo
@noindent
The result of which can be seen in @ref{fig:splinefit1}.

@float Figure,fig:splinefit1
@center @image{splinefit1,4in}
@caption{Comparison of a fitting a piecewise polynomial with 41 breaks to one
with 11 breaks.  The fit with the large number of breaks exhibits a fast ripple
that is not present in the underlying function.}
@end float
@end ifnotinfo

The piecewise polynomial fit, provided by @code{splinefit}, has
continuous derivatives up to the @var{order}-1.  For example, a cubic fit
has continuous first and second derivatives.  This is demonstrated by
the code

@example
## Data (200 points)
x = 2 * pi * rand (1, 200);
y = sin (x) + sin (2 * x) + 0.1 * randn (size (x));
## Piecewise constant
pp1 = splinefit (x, y, 8, "order", 0);
## Piecewise linear
pp2 = splinefit (x, y, 8, "order", 1);
## Piecewise quadratic
pp3 = splinefit (x, y, 8, "order", 2);
## Piecewise cubic
pp4 = splinefit (x, y, 8, "order", 3);
## Piecewise quartic
pp5 = splinefit (x, y, 8, "order", 4);
## Plot
xx = linspace (0, 2 * pi, 400);
y1 = ppval (pp1, xx);
y2 = ppval (pp2, xx);
y3 = ppval (pp3, xx);
y4 = ppval (pp4, xx);
y5 = ppval (pp5, xx);
plot (x, y, ".", xx, [y1; y2; y3; y4; y5])
axis tight
ylim auto
legend (@{"data", "order 0", "order 1", "order 2", "order 3", "order 4"@})
@end example

@ifnotinfo
@noindent
The result of which can be seen in @ref{fig:splinefit2}.

@float Figure,fig:splinefit2
@center @image{splinefit2,4in}
@caption{Comparison of a piecewise constant, linear, quadratic, cubic, and
quartic polynomials with 8 breaks to noisy data.  The higher order solutions
more accurately represent the underlying function, but come with the
expense of computational complexity.}
@end float
@end ifnotinfo

When the underlying function to provide a fit to is periodic, @code{splinefit}
is able to apply the boundary conditions needed to manifest a periodic fit.
This is demonstrated by the code below.

@example
@group
## Data (100 points)
x = 2 * pi * [0, (rand (1, 98)), 1];
y = sin (x) - cos (2 * x) + 0.2 * randn (size (x));
## No constraints
pp1 = splinefit (x, y, 10, "order", 5);
## Periodic boundaries
pp2 = splinefit (x, y, 10, "order", 5, "periodic", true);
## Plot
xx = linspace (0, 2 * pi, 400);
y1 = ppval (pp1, xx);
y2 = ppval (pp2, xx);
plot (x, y, ".", xx, [y1; y2])
axis tight
ylim auto
legend (@{"data", "no constraints", "periodic"@})
@end group
@end example

@ifnotinfo
@noindent
The result of which can be seen in @ref{fig:splinefit3}.

@float Figure,fig:splinefit3
@center @image{splinefit3,4in}
@caption{Comparison of piecewise polynomial fits to a noisy periodic
function with, and without, periodic boundary conditions.}
@end float
@end ifnotinfo

More complex constraints may be added as well.  For example, the code below
illustrates a periodic fit with values that have been clamped at the endpoints,
and a second periodic fit which is hinged at the endpoints.

@example
## Data (200 points)
x = 2 * pi * rand (1, 200);
y = sin (2 * x) + 0.1 * randn (size (x));
## Breaks
breaks = linspace (0, 2 * pi, 10);
## Clamped endpoints, y = y' = 0
xc = [0, 0, 2*pi, 2*pi];
cc = [(eye (2)), (eye (2))];
con = struct ("xc", xc, "cc", cc);
pp1 = splinefit (x, y, breaks, "constraints", con);
## Hinged periodic endpoints, y = 0
con = struct ("xc", 0);
pp2 = splinefit (x, y, breaks, "constraints", con, "periodic", true);
## Plot
xx = linspace (0, 2 * pi, 400);
y1 = ppval (pp1, xx);
y2 = ppval (pp2, xx);
plot (x, y, ".", xx, [y1; y2])
axis tight
ylim auto
legend (@{"data", "clamped", "hinged periodic"@})
@end example

@ifnotinfo
@noindent
The result of which can be seen in @ref{fig:splinefit4}.

@float Figure,fig:splinefit4
@center @image{splinefit4,4in}
@caption{Comparison of two periodic piecewise cubic fits to a noisy periodic
signal.  One fit has its endpoints clamped and the second has its endpoints
hinged.}
@end float
@end ifnotinfo

The @code{splinefit} function also provides the convenience of a @var{robust}
fitting, where the effect of outlying data is reduced.  In the example below,
three different fits are provided.  Two with differing levels of outlier
suppression and a third illustrating the non-robust solution.

@example
## Data
x = linspace (0, 2*pi, 200);
y = sin (x) + sin (2 * x) + 0.05 * randn (size (x));
## Add outliers
x = [x, linspace(0,2*pi,60)];
y = [y, -ones(1,60)];
## Fit splines with hinged conditions
con = struct ("xc", [0, 2*pi]);
## Robust fitting, beta = 0.25
pp1 = splinefit (x, y, 8, "constraints", con, "beta", 0.25);
## Robust fitting, beta = 0.75
pp2 = splinefit (x, y, 8, "constraints", con, "beta", 0.75);
## No robust fitting
pp3 = splinefit (x, y, 8, "constraints", con);
## Plot
xx = linspace (0, 2*pi, 400);
y1 = ppval (pp1, xx);
y2 = ppval (pp2, xx);
y3 = ppval (pp3, xx);
plot (x, y, ".", xx, [y1; y2; y3])
legend (@{"data with outliers","robust, beta = 0.25", ...
         "robust, beta = 0.75", "no robust fitting"@})
axis tight
ylim auto
@end example

@ifnotinfo
@noindent
The result of which can be seen in @ref{fig:splinefit6}.

@float Figure,fig:splinefit6
@center @image{splinefit6,4in}
@caption{Comparison of two different levels of robust fitting (@var{beta} = 0.25 and 0.75) to noisy data combined with outlying data.  A conventional fit, without
robust fitting (@var{beta} = 0) is also included.}
@end float
@end ifnotinfo

A very specific form of polynomial interpretation is the Pad@'e approximant.
For control systems, a continuous-time delay can be modeled very simply with
the approximant.

@c padecoef scripts/polynomial/padecoef.m
@anchor{XREFpadecoef}
@deftypefn  {} {[@var{num}, @var{den}] =} padecoef (@var{T})
@deftypefnx {} {[@var{num}, @var{den}] =} padecoef (@var{T}, @var{N})
Compute the @var{N}th-order Pad@'e approximant of the continuous-time
delay @var{T} in transfer function form.

@tex
The Pad\'e approximant of $e^{-sT}$ is defined by the following equation
$$ e^{-sT} \approx {P_n(s) \over Q_n(s)} $$
where both $P_n(s)$ and $Q_n(s)$ are $N^{th}$-order rational functions
defined by the following expressions
$$ P_n(s)=\sum_{k=0}^N {(2N - k)!N!\over (2N)!k!(N - k)!}(-sT)^k $$
$$ Q_n(s) = P_n(-s) $$
@end tex
@ifnottex
The Pad@'e approximant of @nospell{@code{exp (-sT)}} is defined by the
following equation

@example
@group
             Pn(s)
exp (-sT) ~ -------
             Qn(s)
@end group
@end example

Where both @nospell{Pn(s) and Qn(s)} are @var{N}th-order rational functions
defined by the following expressions

@example
@group
         N    (2N - k)!N!        k
Pn(s) = SUM --------------- (-sT)
        k=0 (2N)!k!(N - k)!

Qn(s) = Pn(-s)
@end group
@end example

@end ifnottex

The inputs @var{T} and @var{N} must be non-negative numeric scalars.  If
@var{N} is unspecified it defaults to 1.

The output row vectors @var{num} and @var{den} contain the numerator and
denominator coefficients in descending powers of s.  Both are
@var{N}th-order polynomials.

For example:

@smallexample
@group
t = 0.1;
n = 4;
[num, den] = padecoef (t, n)
@result{} num =

      1.0000e-04  -2.0000e-02   1.8000e+00  -8.4000e+01   1.6800e+03

@result{} den =

      1.0000e-04   2.0000e-02   1.8000e+00   8.4000e+01   1.6800e+03
@end group
@end smallexample
@end deftypefn


The function, @code{ppval}, evaluates the piecewise polynomials, created
by @code{mkpp} or other means, and @code{unmkpp} returns detailed
information about the piecewise polynomial.

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
@group
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 group
@end example

@c mkpp scripts/polynomial/mkpp.m
@anchor{XREFmkpp}
@deftypefn  {} {@var{pp} =} mkpp (@var{breaks}, @var{coefs})
@deftypefnx {} {@var{pp} =} mkpp (@var{breaks}, @var{coefs}, @var{d})

Construct a piecewise polynomial (pp) structure from sample points
@var{breaks} and coefficients @var{coefs}.

@var{breaks} must be a vector of strictly increasing values.  The number of
intervals is given by @code{@var{ni} = length (@var{breaks}) - 1}.

When @var{m} is the polynomial order @var{coefs} must be of size:
@w{@var{ni}-by-(@var{m} + 1)}.

The i-th row of @var{coefs}, @code{@var{coefs}(@var{i},:)}, contains the
coefficients for the polynomial over the @var{i}-th interval, ordered from
highest (@var{m}) to lowest (@var{0}) degree.

@var{coefs} may also be a multi-dimensional array, specifying a
vector-valued or array-valued polynomial.  In that case the polynomial
order @var{m} is defined by the length of the last dimension of @var{coefs}.
The size of first dimension(s) are given by the scalar or vector @var{d}.
If @var{d} is not given it is set to @code{1}.  In this case
@code{@var{p}(@var{r}, @var{i}, :)} contains the coefficients for the
@var{r}-th polynomial defined on interval @var{i}.  In any case @var{coefs}
is reshaped to a 2-D matrix of size @code{[@var{ni}*prod(@var{d}) @var{m}]}.

Programming Note: @code{ppval} evaluates polynomials at
@code{@var{xi} - @var{breaks}(i)}, i.e., it subtracts the lower endpoint of
the current interval from @var{xi}.  This must be taken into account when
creating piecewise polynomials objects with @code{mkpp}.
@seealso{@ref{XREFunmkpp,,unmkpp}, @ref{XREFppval,,ppval}, @ref{XREFspline,,spline}, @ref{XREFpchip,,pchip}, @ref{XREFppder,,ppder}, @ref{XREFppint,,ppint}, @ref{XREFppjumps,,ppjumps}}
@end deftypefn


@c unmkpp scripts/polynomial/unmkpp.m
@anchor{XREFunmkpp}
@deftypefn {} {[@var{x}, @var{p}, @var{n}, @var{k}, @var{d}] =} unmkpp (@var{pp})

Extract the components of a piecewise polynomial structure @var{pp}.

This function is the inverse of @code{mkpp}: it extracts the inputs to
@code{mkpp} needed to create the piecewise polynomial structure @var{PP}.
The code below makes this relation explicit:

@example
@group
[breaks, coefs, numinter, order, dim] = unmkpp (pp);
pp2  = mkpp (breaks, coefs, dim);
@end group
@end example

The piecewise polynomial structure @code{pp2} obtained in this way, is
identical to the original @code{pp}.  The same can be obtained by directly
accessing the fields of the structure @code{pp}.

The components are:

@table @asis
@item @var{x}
Sample points or breaks.

@item @var{p}
Polynomial coefficients for points in sample interval.
@code{@var{p}(@var{i}, :)} contains the coefficients for the polynomial
over interval @var{i} ordered from highest to lowest degree.
If @code{@var{d} > 1}, then @var{p} is a matrix of size
@code{[@var{n}*prod(@var{d}) @var{m}]}, where the
@code{@var{i} + (1:@var{d})} rows are the coefficients of all the @var{d}
polynomials in the interval @var{i}.

@item @var{n}
Number of polynomial pieces or intervals,
@code{@var{n} = length (@var{x}) - 1}.

@item @var{k}
Order of the polynomial plus 1.

@item @var{d}
Number of polynomials defined for each interval.
@end table

@seealso{@ref{XREFmkpp,,mkpp}, @ref{XREFppval,,ppval}, @ref{XREFspline,,spline}, @ref{XREFpchip,,pchip}}
@end deftypefn


@c ppval scripts/polynomial/ppval.m
@anchor{XREFppval}
@deftypefn {} {@var{yi} =} ppval (@var{pp}, @var{xi})
Evaluate the piecewise polynomial structure @var{pp} at the points @var{xi}.

If @var{pp} describes a scalar polynomial function, the result is an array
of the same shape as @var{xi}.  Otherwise, the size of the result is
@code{[pp.dim, length(@var{xi})]} if @var{xi} is a vector, or
@code{[pp.dim, size(@var{xi})]} if it is a multi-dimensional array.
@seealso{@ref{XREFmkpp,,mkpp}, @ref{XREFunmkpp,,unmkpp}, @ref{XREFspline,,spline}, @ref{XREFpchip,,pchip}}
@end deftypefn


@c ppder scripts/polynomial/ppder.m
@anchor{XREFppder}
@deftypefn  {} {ppd =} ppder (pp)
@deftypefnx {} {ppd =} ppder (pp, m)
Compute the piecewise @var{m}-th derivative of a piecewise polynomial
struct @var{pp}.

If @var{m} is omitted the first derivative is calculated.
@seealso{@ref{XREFmkpp,,mkpp}, @ref{XREFppval,,ppval}, @ref{XREFppint,,ppint}}
@end deftypefn


@c ppint scripts/polynomial/ppint.m
@anchor{XREFppint}
@deftypefn  {} {@var{ppi} =} ppint (@var{pp})
@deftypefnx {} {@var{ppi} =} ppint (@var{pp}, @var{c})
Compute the integral of the piecewise polynomial struct @var{pp}.

@var{c}, if given, is the constant of integration.
@seealso{@ref{XREFmkpp,,mkpp}, @ref{XREFppval,,ppval}, @ref{XREFppder,,ppder}}
@end deftypefn


@c ppjumps scripts/polynomial/ppjumps.m
@anchor{XREFppjumps}
@deftypefn {} {@var{jumps} =} ppjumps (@var{pp})
Evaluate the boundary jumps of a piecewise polynomial.

If there are @math{n} intervals, and the dimensionality of @var{pp} is
@math{d}, the resulting array has dimensions @code{[d, n-1]}.
@seealso{@ref{XREFmkpp,,mkpp}}
@end deftypefn


@node Miscellaneous Functions
@section Miscellaneous Functions

@c poly scripts/polynomial/poly.m
@anchor{XREFpoly}
@deftypefn  {} {} poly (@var{A})
@deftypefnx {} {} poly (@var{x})
If @var{A} is a square @math{N}-by-@math{N} matrix, @code{poly (@var{A})}
is the row vector of the coefficients of @code{det (z * eye (N) - A)},
the characteristic polynomial of @var{A}.

For example, the following code finds the eigenvalues of @var{A} which are
the roots of @code{poly (@var{A})}.

@example
@group
roots (poly (eye (3)))
    @result{} 1.00001 + 0.00001i
       1.00001 - 0.00001i
       0.99999 + 0.00000i
@end group
@end example

In fact, all three eigenvalues are exactly 1 which emphasizes that for
numerical performance the @code{eig} function should be used to compute
eigenvalues.

If @var{x} is a vector, @code{poly (@var{x})} is a vector of the
coefficients of the polynomial whose roots are the elements of @var{x}.
That is, if @var{c} is a polynomial, then the elements of
@code{@var{d} = roots (poly (@var{c}))} are contained in @var{c}.  The
vectors @var{c} and @var{d} are not identical, however, due to sorting and
numerical errors.
@seealso{@ref{XREFroots,,roots}, @ref{XREFeig,,eig}}
@end deftypefn


@c polyout scripts/polynomial/polyout.m
@anchor{XREFpolyout}
@deftypefn  {} {} polyout (@var{c})
@deftypefnx {} {} polyout (@var{c}, @var{x})
@deftypefnx {} {@var{str} =} polyout (@dots{})
Display a formatted version of the polynomial @var{c}.

The formatted polynomial
@tex
$$ c(x) = c_1 x^n + \ldots + c_n x + c_{n+1} $$
@end tex
@ifnottex

@example
c(x) = c(1) * x^n + @dots{} + c(n) x + c(n+1)
@end example

@end ifnottex
is returned as a string or written to the screen if @code{nargout} is zero.

The second argument @var{x} specifies the variable name to use for each term
and defaults to the string @qcode{"s"}.
@seealso{@ref{XREFpolyreduce,,polyreduce}}
@end deftypefn


@c polyreduce scripts/polynomial/polyreduce.m
@anchor{XREFpolyreduce}
@deftypefn {} {} polyreduce (@var{c})
Reduce a polynomial coefficient vector to a minimum number of terms by
stripping off any leading zeros.
@seealso{@ref{XREFpolyout,,polyout}}
@end deftypefn

