#
# = Polynomials
# Contents:
# 1. {Polynomial Evaluation}[link:files/rdoc/poly_rdoc.html#1]
# 1. {Solving polynomial equations}[link:files/rdoc/poly_rdoc.html#2]
#    1. {Quadratic Equations}[link:files/rdoc/poly_rdoc.html#2.1]
#    1. {Cubic Equations}[link:files/rdoc/poly_rdoc.html#2.2]
#    1. {General Polynomial Equations}[link:files/rdoc/poly_rdoc.html#2.3]
# 1. {GSL::Poly Class}[link:files/rdoc/poly_rdoc.html#3]
#    1. {Constructors}[link:files/rdoc/poly_rdoc.html#3.1]
#    1. {Methods}[link:files/rdoc/poly_rdoc.html#3.2]
# 1. {Polynomial Fitting}[link:files/rdoc/poly_rdoc.html#4]
# 1. {Divided-difference representations}[link:files/rdoc/poly_rdoc.html#5]
# 1. {Extensions}[link:files/rdoc/poly_rdoc.html#6]
#    1. {Special Polynomials}[link:files/rdoc/poly_rdoc.html#6.1]
#    1. {Polynomial Operations}[link:files/rdoc/poly_rdoc.html#6.2]
#
# == {}[link:index.html"name="1] Polynomial Evaluation
# ---
# * GSL::Poly.eval(c, x)
#
#   Evaluates the polynomial <tt>c[0] + c[1]x + c[2]x^2 + ...</tt>. 
#   The polynomial coefficients <tt>c</tt> can be an <tt>Array</tt>, 
#   a <tt>GSL::Vector</tt>, or an <tt>NArray</tt>. The evaluation point <tt>x</tt>
#   is a <tt>Numeric</tt>, <tt>Array</tt>, <tt>GSL::Vector</tt> or <tt>NArray</tt>.
#   From GSL 1.11, <tt>x</tt> can be a complex number, and <tt>c</tt> can be a complex polynomial given by a <tt>GSL::Vector::Complex</tt> or an <tt>Array</tt>.
#
#   Ex)
#      >> require("gsl")
#      => true
#      >> GSL::Poly.eval([1, 2, 3], 2)
#      => 17.0
#      >> GSL::Poly.eval(GSL::Vector[1, 2, 3], 2)
#      => 17.0
#      >> GSL::Poly.eval(NArray[1.0, 2, 3], 2)
#      => 17.0
#      >> GSL::Poly.eval([1, 2, 3], [1, 2, 3])
#      => [6.0, 17.0, 34.0]
#      >> GSL::Poly.eval([1, 2, 3], GSL::Vector[1, 2, 3])
#      => GSL::Vector
#      [ 6.000e+00 1.700e+01 3.400e+01 ]
#      >> GSL::Poly.eval([1, 2, 3], NArray[1.0, 2, 3])
#      => NArray.float(3):
#      [ 6.0, 17.0, 34.0 ]
#
# ---
# * GSL::Poly.eval_derivs(c, x)
# * GSL::Poly.eval_derivs(c, x, lenres)
#
#   (GSL-1.13) Evaluate and return a polynomial and its derivatives. The output contains the values of d^k P/d x^k for the specified value of x starting with k = 0. The input polynomial <tt>c</tt> can be an <tt>Array</tt>, <tt>GSL::Poly</tt> or an <tt>NArray</tt>. If <tt>lenres</tt> is not given, <tt>lenres = LENGTH(c) + 1</tt> is used, therefore the last element of the output is 0.
#
# ---
# * GSL::Poly#eval_derivs(x)
# * GSL::Poly#eval_derivs(x, lenres)
#
#   (GSL-1.13) Evaluate and return a polynomial and its derivatives. The output contains the values of d^k P/d x^k for the specified value of x starting with k = 0. If <tt>lenres</tt> is not given, <tt>lenres = LENGTH(self) + 1</tt> is used, therefore the last element of the output is 0.
#    
#   Ex.)
#    >> ary = [1, 2, 3]
#    => [1, 2, 3]
#    >> GSL::Poly.eval_derivs(ary, 1)
#    => [6.0, 8.0, 6.0, 0.0]
#    >> na = NArray[1.0, 2, 3]
#    => NArray.float(3): 
#    [ 1.0, 2.0, 3.0 ]
#    >> GSL::Poly.eval_derivs(na, 1)
#    => NArray.float(4): 
#    [ 6.0, 8.0, 6.0, 0.0 ]
#    >> poly = GSL::Poly[1.0, 2, 3]
#    => GSL::Poly
#    [ 1.000e+00 2.000e+00 3.000e+00 ]
#    >> GSL::Poly.eval_derivs(poly, 1)
#    => GSL::Poly
#    [ 6.000e+00 8.000e+00 6.000e+00 0.000e+00 ]
#    >> poly.eval_derivs(1)
#    => GSL::Poly
#    [ 6.000e+00 8.000e+00 6.000e+00 0.000e+00 ]
#    >> poly.eval_derivs(1, 3)
#    => GSL::Poly
#    [ 6.000e+00 8.000e+00 6.000e+00 ]
#
# == {}[link:index.html"name="2] Solving polynomial equations
# === {}[link:index.html"name="2.1] Quadratic Equations
# ---
# * GSL::Poly::solve_quadratic(a, b, c)
# * GSL::Poly::solve_quadratic([a, b, c])
#
#   Find the real roots of the quadratic equation,
#     a x^2 + b x + c = 0
#   The coefficients are given by 3 numbers, or a Ruby array, 
#   or a <tt>GSL::Vector</tt> object. The roots are returned as a <tt>GSL::Vector</tt>.
#
#   * Ex: z^2 - 3z + 2 = 0
#       >> GSL::Poly::solve_quadratic(1, -3, 2)
#       => GSL::Vector: 
#       [ 1.000e+00 2.000e+00 ]
#
#
# ---
# * GSL::Poly::complex_solve_quadratic(a, b, c)
# * GSL::Poly::complex_solve_quadratic([a, b, c])
#
#   Find the complex roots of the quadratic equation,
#     a z^2 + b z + z = 0
#   The coefficients are given by 3 numbers or a Ruby array, or a 
#   <tt>GSL::Vector</tt>.
#   The roots are returned as a <tt>GSL::Vector::Complex</tt> of two elements.
#   
#   * Ex: z^2 - 3z + 2 = 0
#      >> require("gsl")
#      => true
#      >> GSL::Poly::complex_solve_quadratic(1, -3, 2)
#      [ [1.000e+00 0.000e+00] [2.000e+00 0.000e+00] ]   
#      => #<GSL::Vector::Complex:0x764014>
#      >> GSL::Poly::complex_solve_quadratic(1, -3, 2).real  <--- Real part
#      => GSL::Vector::View: 
#      [ 1.000e+00 2.000e+00 ]
#
# === {}[link:index.html"name="2.2] Cubic Equations
# ---
# * GSL::Poly::solve_cubic(same as solve_quadratic)
#
#   This method finds the real roots of the cubic equation,
#     x^3 + a x^2 + b x + c = 0
#
# ---
# * GSL::Poly::complex_solve_cubic(same as solve_cubic)
#
#   This method finds the complex roots of the cubic equation,
#     z^3 + a z^2 + b z + c = 0
#
# === {}[link:index.html"name="2.3] General Polynomial Equations
# ---
# * GSL::Poly::complex_solve(c0, c1, c2,,, )
# * GSL::Poly::solve(c0, c1, c2,,, )
#
#   Find the complex roots of the polynomial equation. Note that 
#   the coefficients are given by "ascending" order.
#
#   * Ex: x^2 - 3 x + 2 == 0 
#        >> GSL::Poly::complex_solve(2, -3, 1)    <--- different from Poly::quadratic_solve
#        [ [1.000e+00 0.000e+00] [2.000e+00 0.000e+00] ]
#        => #<GSL::Vector::Complex:0x75e614>
#
# == {}[link:index.html"name="3] GSL::Poly Class
# This class expresses polynomials of arbitrary orders.
#
# === {}[link:index.html"name="3.1] Constructors
# ---
# * GSL::Poly.alloc(c0, c1, c2, ....)
# * GSL::Poly[c0, c1, c2, ....]
#
#   This creates an instance of the <tt>GSL::Poly</tt> class, 
#   which represents a polynomial
#       c0 + c1 x + c2 x^2 + ....
#   This class is derived from <tt>GSL::Vector</tt>.
#
#   * Ex: x^2 - 3 x + 2
#       poly = GSL::Poly.alloc([2, -3, 1])
#
# === {}[link:index.html"name="3.2] Instance Methods
# ---
# * GSL::Poly#eval(x)
# * GSL::Poly#at(x)
#
#   Evaluates the polynomial 
#   c[0] + c[1] x + c[2] x^2 + ... + c[len-1] x^{len-1} 
#   using Horner's method for stability. The argument <tt>x</tt> is a
#   <tt>Numeric</tt>, <tt>GSL::Vector, Matrix</tt> or an <tt>Array</tt>.
#
# ---
# * GSL::Poly#solve_quadratic
#
#   Solve the quadratic equation.
#
#   * Ex:  z^2 - 3 z + 2 = 0:
#       >> a = GSL::Poly[2, -3, 1]
#       => GSL::Poly: 
#       [ 2.000e+00 -3.000e+00 1.000e+00 ]
#       >> a.solve_quadratic
#       => GSL::Vector: 
#       [ 1.000e+00 2.000e+00 ]
#
# ---
# * GSL::Poly#solve_cubic
#
#   Solve the cubic equation.
#
# ---
# * GSL::Poly#complex_solve
# * GSL::Poly#solve
# * GSL::Poly#roots
#
#   These methods find the complex roots of the quadratic equation,
#     c0 + c1 z + c2 z^2 + .... = 0
#
#   * Ex: z^2 - 3 z + 2 = 0:
#       >> a = GSL::Poly[2, -3, 1]
#       => GSL::Poly: 
#       [ 2.000e+00 -3.000e+00 1.000e+00 ]
#       >> a.solve
#       [ [1.000e+00 0.000e+00] [2.000e+00 0.000e+00] ]
#       => #<GSL::Vector::Complex:0x35db28>
#
# == {}[link:index.html"name="4] Polynomial fitting
# ---
# * GSL::Poly.fit(x, y, order)
# * GSL::Poly.wfit(x, w, y, order)
#
#   Finds the coefficient of a polynomial of order <tt>order</tt> 
#   that fits the vector data (<tt>x, y</tt>) in a least-square sense.
#   This provides a higher-level interface to the method
#   {GSL::Multifit#linear}[link:files/rdoc/fit_rdoc.html] in a case of polynomial fitting.
#
#   Example:
#     #!/usr/bin/env ruby
#     require("gsl")
#
#     x = GSL::Vector[1, 2, 3, 4, 5]
#     y = GSL::Vector[5.5, 43.1, 128, 290.7, 498.4]
#     # The results are stored in a polynomial "coef"
#     coef, cov, chisq, status = Poly.fit(x, y, 3) 
#
#     x2 = GSL::Vector.linspace(1, 5, 20)
#     graph([x, y], [x2, coef.eval(x2)], "-C -g 3 -S 4")
#
# == {}[link:index.html"name="5] Divided-difference representations
#
# ---
# * GSL::Poly::dd_init(xa, ya)
#
#   This method computes a divided-difference representation of the 
#   interpolating polynomial for the points <tt>(xa, ya)</tt>.
#
# ---
# * GSL::Poly::DividedDifference#eval(x)
#
#   This method evaluates the polynomial stored in divided-difference form 
#   <tt>self</tt> at the point <tt>x</tt>.
#
# ---
# * GSL::Poly::DividedDifference#taylor(xp)
#
#   This method converts the divided-difference representation of a polynomial 
#   to a Taylor expansion. On output the Taylor coefficients of the polynomial 
#   expanded about the point <tt>xp</tt> are returned.
#
# == {}[link:index.html"name="6] Extensions
# === {}[link:index.html"name="6.1] Special Polynomials
# ---
# * GSL::Poly.hermite(n)
#
#   This returns coefficients of the <tt>n</tt>-th order Hermite polynomial, <tt>H(x; n)</tt>.
#   For order of <tt>n</tt> >= 3, this method uses the recurrence relation
#        H(x; n+1) = 2 x H(x; n) - 2 n H(x; n-1)
#   * Ex:
#       >> GSL::Poly.hermite(2)
#       => GSL::Poly::Int: 
#       [ -2 0 4 ]                            <----- 4x^2 - 2
#       >> GSL::Poly.hermite(5)
#       => GSL::Poly::Int: 
#       [ 0 120 0 -160 0 32 ]                 <----- 32x^5 - 160x^3 + 120x
#       >> GSL::Poly.hermite(7)
#       => GSL::Poly::Int: 
#       [ 0 -1680 0 3360 0 -1344 0 128 ]
#
# ---
# * GSL::Poly.cheb(n)
# * GSL::Poly.chebyshev(n)
#
#   Return the coefficients of the <tt>n</tt>-th order Chebyshev polynomial, <tt>T(x; n</tt>.
#   For order of <tt>n</tt> >= 3, this method uses the recurrence relation
#        T(x; n+1) = 2 x T(x; n) - T(x; n-1)
#
# ---
# * GSL::Poly.cheb_II(n)
# * GSL::Poly.chebyshev_II(n)
#
#   Return the coefficients of the <tt>n</tt>-th order Chebyshev polynomial of type II,
#   <tt>U(x; n</tt>.
#        U(x; n+1) = 2 x U(x; n) - U(x; n-1)
#
# ---
# * GSL::Poly.bell(n)
#
#   Bell polynomial
#
# ---
# * GSL::Poly.bessel(n)
#
#   Bessel polynomial
#
# ---
# * GSL::Poly.laguerre(n)
#
#   Retunrs the coefficients of the <tt>n</tt>-th order Laguerre polynomial
#   multiplied by n!.
#
#   Ex:
#        rb(main):001:0> require("gsl")
#        => true
#        >> GSL::Poly.laguerre(0)
#        => GSL::Poly::Int: 
#        [ 1 ]                              <--- 1
#        >> GSL::Poly.laguerre(1)
#        => GSL::Poly::Int: 
#        [ 1 -1 ]                           <--- -x + 1
#        >> GSL::Poly.laguerre(2)
#        => GSL::Poly::Int: 
#        [ 2 -4 1 ]                         <--- (x^2 - 4x + 2)/2!
#        >> GSL::Poly.laguerre(3)
#        => GSL::Poly::Int:                   
#        [ 6 -18 9 -1 ]                     <--- (-x^3 + 9x^2 - 18x + 6)/3!
#        >> GSL::Poly.laguerre(4)
#        => GSL::Poly::Int:  
#        [ 24 -96 72 -16 1 ]                <--- (x^4 - 16x^3 + 72x^2 - 96x + 24)/4!
#  
# === {}[link:index.html"name="6.2] Polynomial Operations
# ---
# * GSL::Poly#conv
# * GSL::Poly#deconv
# * GSL::Poly#reduce
# * GSL::Poly#deriv
# * GSL::Poly#integ
# * GSL::Poly#compan
#
#
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