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 GSL::Vector object. The roots are returned as a GSL::Vector.
Ex: z^2 - 3z + 2 = 0
irb(main):006:0> 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 GSL::Vector. The roots are returned as a GSL::Vector::Complex of two elements.
Ex: z^2 - 3z + 2 = 0
irb(main):001:0> require("gsl")
=> true
irb(main):002:0> Poly::complex_solve_quadratic(1, -3, 2)
[ [1.000e+00 0.000e+00] [2.000e+00 0.000e+00] ]
=> #<GSL::Vector::Complex:0x764014>
irb(main):003:0> Poly::complex_solve_quadratic(1, -3, 2).real <--- Real part
=> GSL::Vector::View:
[ 1.000e+00 2.000e+00 ]
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
GSL::Poly::complex_solve(c0, c1, c2,,, )GSL::Poly::solve(c0, c1, c2,,, )Ex: x^2 - 3 x + 2 == 0
irb(main):004:0> 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>
This class expresses polynomials of arbitrary orders.
GSL::Poly.new(c0, c1, c2, ....)GSL::Poly[c0, c1, c2, ....]This creates an instance of the GSL::Poly class, which represents a polynomial
c0 + c1 x + c2 x^2 + ....
This class is derived from GSL::Vector.
Ex: x^2 - 3 x + 2
poly = GSL::Poly.new([2, -3, 1])
GSL::Poly#eval(x)GSL::Poly#at(x)Numeric, GSL::Vector, Matrix or an Array.GSL::Poly#solve_quadraticEx: z^2 - 3 z + 2 = 0:
irb(main):004:0> a = Poly[2, -3, 1] => GSL::Poly: [ 2.000e+00 -3.000e+00 1.000e+00 ] irb(main):005:0> a.solve_quadratic => GSL::Vector: [ 1.000e+00 2.000e+00 ]
GSL::Poly#solve_cubicGSL::Poly#complex_solveGSL::Poly#solveGSL::Poly#rootsThese methods find the complex roots of the quadratic equation,
c0 + c1 z + c2 z^2 + .... = 0
Ex: z^2 - 3 z + 2 = 0:
irb(main):009:0> a = Poly[2, -3, 1] => GSL::Poly: [ 2.000e+00 -3.000e+00 1.000e+00 ] irb(main):010:0> a.solve [ [1.000e+00 0.000e+00] [2.000e+00 0.000e+00] ] => #<GSL::Vector::Complex:0x35db28>
GSL::Poly.fit(x, y, order)GSL::Poly.wfit(x, w, y, order)Finds the coefficient of a polynomial of order order that fits the vector data (x, y) in a least-square sense. This provides a higher-level interface to the method GSL::Multifit#linear in a case of polynomial fitting.
Example:
#!/usr/bin/env ruby
require("gsl")
x = Vector[1, 2, 3, 4, 5]
y = 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 = Vector.linspace(1, 5, 20)
graph([x, y], [x2, coef.eval(x2)], "-C -g 3 -S 4")
GSL::Poly::dd_init(xa, ya)GSL::Poly::DividedDifference#eval(x)GSL::Poly::DividedDifference#taylor(xp)GSL::Poly.hermite(n)This returns coefficients of the n-th order Hermite polynomial, H(x; n). For order of n >= 3, this method uses the recurrence relation
H(x; n+1) = 2 x H(x; n) - 2 n H(x; n-1)
Ex:
irb(main):013:0> Poly.hermite(2) => GSL::Poly::Int: [ -2 0 4 ] <----- 4x^2 - 2 irb(main):014:0> Poly.hermite(5) => GSL::Poly::Int: [ 0 120 0 -160 0 32 ] <----- 32x^5 - 160x^3 + 120x irb(main):015:0> 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 n-th order Chebyshev polynomial, T(x; n. For order of n >= 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 n-th order Chebyshev polynomial of type II, U(x; n.
U(x; n+1) = 2 x U(x; n) - U(x; n-1)
GSL::Poly.bell(n)GSL::Poly.bessel(n)GSL::Poly.laguerre(n)Retunrs the coefficients of the n-th order Laguerre polynomial multiplied by n!.
Ex:
rb(main):001:0> require("gsl")
=> true
irb(main):002:0> Poly.laguerre(0)
=> GSL::Poly::Int:
[ 1 ] <--- 1
irb(main):003:0> Poly.laguerre(1)
=> GSL::Poly::Int:
[ 1 -1 ] <--- -x + 1
irb(main):004:0> Poly.laguerre(2)
=> GSL::Poly::Int:
[ 2 -4 1 ] <--- (x^2 - 4x + 2)/2!
irb(main):005:0> Poly.laguerre(3)
=> GSL::Poly::Int:
[ 6 -18 9 -1 ] <--- (-x^3 + 9x^2 - 18x + 6)/3!
irb(main):006:0> Poly.laguerre(4)
=> GSL::Poly::Int:
[ 24 -96 72 -16 1 ] <--- (x^4 - 16x^3 + 72x^2 - 96x + 24)/4!
GSL::Poly#convGSL::Poly#deconvGSL::Poly#reduceGSL::Poly#derivGSL::Poly#integGSL::Poly#compan