Contents:
GSL::Poly.eval(c, x)
Evaluates the polynomial c[0] + c[1]x + c[2]x^2 + .... The polynomial coefficients c can be an Array, a GSL::Vector, or an NArray. The evaluation point x is a Numeric, Array, GSL::Vector or NArray. From GSL 1.11, x can be a complex number, and c can be a complex polynomial given by a GSL::Vector::Complex or an Array.
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 c can be an Array, GSL::Poly or an NArray. If lenres is not given, lenres = LENGTH(c) + 1 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 lenres is not given, lenres = LENGTH(self) + 1 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 ]
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
>> 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 GSL::Vector. The roots are returned as a GSL::Vector::Complex 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 ]
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,,, )
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>
This class expresses polynomials of arbitrary orders.
GSL::Poly.alloc(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.alloc([2, -3, 1])
GSL::Poly#eval(x)
GSL::Poly#at(x)
Evaluates the polynomial c + c x + c x^2 + ... + c x^{len-1} using Horner's method for stability. The argument x is a Numeric, GSL::Vector, Matrix or an Array.
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>
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 = 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")
GSL::Poly::dd_init(xa, ya)
This method computes a divided-difference representation of the interpolating polynomial for the points (xa, ya).
GSL::Poly::DividedDifference#eval(x)
This method evaluates the polynomial stored in divided-difference form self at the point x.
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 xp are returned.
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:
>> 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 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)
Bell polynomial
GSL::Poly.bessel(n)
Bessel polynomial
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 >> 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!
GSL::Poly#conv
GSL::Poly#deconv
GSL::Poly#reduce
GSL::Poly#deriv
GSL::Poly#integ
GSL::Poly#compan
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