poly.texi

开放gsl矩阵运算

@cindex polynomials, roots of 

This chapter describes functions for evaluating and solving polynomials.
There are routines for finding real and complex roots of quadratic and
cubic equations using analytic methods.  An iterative polynomial solver
is also available for finding the roots of general polynomials with real
coefficients (of any order).  The functions are declared in the header
file @code{gsl_poly.h}.

@menu
* Polynomial evaluation::       
* Quadratic equations::         
* Cubic equations::             
* General polynomial equations::  
* Roots of Polynomials Examples::  
* Roots of Polynomials References and Further Reading::  
@end menu

@node Polynomial evaluation
@section Polynomial evaluation
@cindex polynomial evaluation
@cindex evaluation of polynomials

@deftypefun double gsl_poly_eval (const double c[], const int @var{len}, const double @var{x})
This function evaluates the polynomial 
@c{$c[0] + c[1] x + c[2] x^2 + \dots + c[len-1] x^{len-1}$}
@math{c[0] + c[1] x + c[2] x^2 + \dots + c[len-1] x^@{len-1@}} using
Horner's method for stability.  The function is inlined when possible.
@end deftypefun


@node  Quadratic equations
@section Quadratic equations
@cindex quadratic equation, solving

@deftypefun int gsl_poly_solve_quadratic (double @var{a}, double @var{b}, double @var{c}, double *@var{x0}, double *@var{x1})
This function finds the real roots of the quadratic equation,

@tex
\beforedisplay
$$
a x^2 + b x + c = 0
$$
\afterdisplay
@end tex
@ifinfo
@example
a x^2 + b x + c = 0
@end example
@end ifinfo

@noindent
The number of real roots (either zero or two) is returned, and their
locations are stored in @var{x0} and @var{x1}.  If no real roots are
found then @var{x0} and @var{x1} are not modified.  When two real roots
are found they are stored in @var{x0} and @var{x1} in ascending
order.  The case of coincident roots is not considered special.  For
example @math{(x-1)^2=0} will have two roots, which happen to have
exactly equal values.

The number of roots found depends on the sign of the discriminant
@math{b^2 - 4 a c}.  This will be subject to rounding and cancellation
errors when computed in double precision, and will also be subject to
errors if the coefficients of the polynomial are inexact.  These errors
may cause a discrete change in the number of roots.  However, for
polynomials with small integer coefficients the discriminant can always
be computed exactly.

@end deftypefun

@deftypefun int gsl_poly_complex_solve_quadratic (double @var{a}, double @var{b}, double @var{c}, gsl_complex *@var{z0}, gsl_complex *@var{z1})

This function finds the complex roots of the quadratic equation,

@tex
\beforedisplay
$$
a z^2 + b z + c = 0
$$
\afterdisplay
@end tex
@ifinfo
@example
a z^2 + b z + c = 0
@end example
@end ifinfo

@noindent
The number of complex roots is returned (always two) and the locations
of the roots are stored in @var{z0} and @var{z1}.  The roots are returned
in ascending order, sorted first by their real components and then by
their imaginary components.

@end deftypefun


@node Cubic equations
@section Cubic equations
@cindex cubic equation, solving

@deftypefun int gsl_poly_solve_cubic (double @var{a}, double @var{b}, double @var{c}, double *@var{x0}, double *@var{x1}, double *@var{x2})

This function finds the real roots of the cubic equation,

@tex
\beforedisplay
$$
x^3 + a x^2 + b x + c = 0
$$
\afterdisplay
@end tex
@ifinfo
@example
x^3 + a x^2 + b x + c = 0
@end example
@end ifinfo

@noindent
with a leading coefficient of unity.  The number of real roots (either
one or three) is returned, and their locations are stored in @var{x0},
@var{x1} and @var{x2}.  If one real root is found then only @var{x0} is
modified.  When three real roots are found they are stored in @var{x0},
@var{x1} and @var{x2} in ascending order.  The case of coincident roots
is not considered special.  For example, the equation @math{(x-1)^3=0}
will have three roots with exactly equal values.

@end deftypefun

@deftypefun int gsl_poly_complex_solve_cubic (double @var{a}, double @var{b}, double @var{c}, gsl_complex *@var{z0}, gsl_complex *@var{z1}, gsl_complex *@var{z2})

This function finds the complex roots of the cubic equation,

@tex
\beforedisplay
$$
z^3 + a z^2 + b z + c = 0
$$
\afterdisplay
@end tex
@ifinfo
@example
z^3 + a z^2 + b z + c = 0
@end example
@end ifinfo

@noindent
The number of complex roots is returned (always three) and the locations
of the roots are stored in @var{z0}, @var{z1} and @var{z2}.  The roots
are returned in ascending order, sorted first by their real components
and then by their imaginary components.

@end deftypefun


@node General polynomial equations
@section General polynomial equations
@cindex general polynomial equations, solving

The roots of polynomial equations cannot be found analytically beyond
the special cases of the quadratic, cubic and quartic equation.  The
algorithm described in this section uses an iterative method to find the
approximate locations of roots of higher order polynomials.

@deftypefun {gsl_poly_complex_workspace *} gsl_poly_complex_workspace_alloc (size_t @var{n})
This function allocates space for a @code{gsl_poly_complex_workspace}
struct and a workspace suitable for solving a polynomial with @var{n}
coefficients using the routine @code{gsl_poly_complex_solve}.

The function returns a pointer to the newly allocated
@code{gsl_poly_complex_workspace} if no errors were detected, and a null
pointer in the case of error.
@end deftypefun

@deftypefun void gsl_poly_complex_workspace_free (gsl_poly_complex_workspace * @var{w})
This function frees all the memory associated with the workspace
@var{w}.
@end deftypefun

@deftypefun int gsl_poly_complex_solve (const double * @var{a}, size_t @var{n}, gsl_poly_complex_workspace * @var{w}, gsl_complex_packed_ptr @var{z})
This function computes the roots of the general polynomial 
@c{$P(x) = a_0 + a_1 x + a_2 x^2 + ... + a_{n-1} x^{n-1}$} 
@math{P(x) = a_0 + a_1 x + a_2 x^2 + ... + a_@{n-1@} x^@{n-1@}} using 
balanced-QR reduction of the companion matrix.  The parameter @var{n}
specifies the length of the coefficient array.  The coefficient of the
highest order term must be non-zero.  The function requires a workspace
@var{w} of the appropriate size.  The @math{n-1} roots are returned in
the packed complex array @var{z} of length @math{2(n-1)}, alternating
real and imaginary parts.

The function returns @code{GSL_SUCCESS} if all the roots are found and
@code{GSL_EFAILED} if the QR reduction does not converge.
@end deftypefun

@node Roots of Polynomials Examples
@section Examples

To demonstrate the use of the general polynomial solver we will take the
polynomial @math{P(x) = x^5 - 1} which has the following roots,

@tex
\beforedisplay
$$
1, e^{2\pi i /5}, e^{4\pi i /5}, e^{6\pi i /5}, e^{8\pi i /5}
$$
\afterdisplay
@end tex
@ifinfo
@example
1, e^@{2\pi i /5@}, e^@{4\pi i /5@}, e^@{6\pi i /5@}, e^@{8\pi i /5@}
@end example
@end ifinfo

@noindent
The following program will find these roots.

@example
#include <stdio.h>
#include <gsl/gsl_poly.h>

int
main (void)
@{
  int i;
  /* coefficient of P(x) =  -1 + x^5  */
  double a[6] = @{ -1, 0, 0, 0, 0, 1 @};  
  double z[10];

  gsl_poly_complex_workspace * w 
      = gsl_poly_complex_workspace_alloc (6);
  
  gsl_poly_complex_solve (a, 6, w, z);

  gsl_poly_complex_workspace_free (w);

  for (i = 0; i < 5; i++)
    @{
      printf("z%d = %+.18f %+.18f\n", 
             i, z[2*i], z[2*i+1]);
    @}

  return 0;
@}
@end example

@noindent
The output of the program is,

@example
bash$ ./a.out 
z0 = -0.809016994374947451 +0.587785252292473137
z1 = -0.809016994374947451 -0.587785252292473137
z2 = +0.309016994374947451 +0.951056516295153642
z3 = +0.309016994374947451 -0.951056516295153642
z4 = +1.000000000000000000 +0.000000000000000000
@end example
@noindent
which agrees with the analytic result, @math{z_n = \exp(2 \pi n i/5)}.

@node Roots of Polynomials References and Further Reading
@section References and Further Reading
@noindent
The balanced-QR method and its error analysis is described in the
following papers.

@itemize @asis
@item
R.S. Martin, G. Peters and J.H. Wilkinson, ``The QR Algorithm for Real
Hessenberg Matrices'', @cite{Numerische Mathematik}, 14 (1970), 219--231.

@item
B.N. Parlett and C. Reinsch, ``Balancing a Matrix for Calculation of
Eigenvalues and Eigenvectors'', @cite{Numerische Mathematik}, 13 (1969),
293--304.

@item
A. Edelman and H. Murakami, ``Polynomial roots from companion matrix
eigenvalues'', @cite{Mathematics of Computation}, Vol. 64 No. 210
(1995), 763--776.
@end itemize