Polynomials¶ ↑

Contents:

Polynomial Evaluation¶ ↑

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 ]

Solving polynomial equations¶ ↑

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 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 ]

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
```

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>
```

Poly class¶ ↑

This class expresses polynomials of arbitrary orders.

Constructors¶ ↑

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])
```

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 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>

Polynomial fitting¶ ↑

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")

Divided-difference representations¶ ↑

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.

Extensions¶ ↑

Special Polynomials¶ ↑

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!

Polynomial Operations¶ ↑

GSL::Poly#conv
GSL::Poly#deconv
GSL::Poly#reduce
GSL::Poly#deriv
GSL::Poly#integ
GSL::Poly#compan

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