Polynomials Type

Namespace: Altaxo.Calc
Assembly: AltaxoCore.Redist.dll
Base Type: obj

Polynomials.

Constructors

Constructor Description

Polynomials()

Full Usage: Polynomials()

Static members

Static member Description

Polynomials.ChebyshevT(n, x, y, dy, d2y)

Full Usage: Polynomials.ChebyshevT(n, x, y, dy, d2y)

Parameters:
    n : int - Degree of the polynomial >= 0.
    x : float - Point in which the computation is performed, -1 <= x <= 1.
    y : byref<float> - Output: value of the polynomial in x.
    dy : byref<float> - Output: value of the first derivative in x.
    d2y : byref<float> - Output: value of the second derivative in x.

Computes the value of the Chebyshev polynomial of degree n and its first and second derivatives at a given point. 

            Note:
              This C++ implementation is based on the Fortran function
                 VACHPO
              from
                  "Fortran routines for spectral methods"
              by  Daniele Funaro
                  Department of Mathematics
                  University of Pavia
                  Via Abbiategrasso 209, 27100 Pavia, Italy
                  e-mails: fun18@ipvian.ian.pv.cnr.it
                           funaro@dragon.ian.pv.cnr.it

n : int

Degree of the polynomial >= 0.

x : float

Point in which the computation is performed, -1 <= x <= 1.

y : byref<float>

Output: value of the polynomial in x.

dy : byref<float>

Output: value of the first derivative in x.

d2y : byref<float>

Output: value of the second derivative in x.

Polynomials.HermiteH(n, x, y, dy, d2y)

Full Usage: Polynomials.HermiteH(n, x, y, dy, d2y)

Parameters:
    n : int - Degree of the polynomial >= 0.
    x : float - Point in which the computation is performed.
    y : byref<float> - Output: value of the polynomial in x.
    dy : byref<float> - Output: value of the first derivative in x.
    d2y : byref<float> - Output: value of the second derivative in x.

Computes the value of the Hermite polynomial of degree n and its first and second derivatives at a given point. 

            Note:
              This C++ implementation is based on the Fortran function
                 VAHEPO
              from
                  "Fortran routines for spectral methods"
              by  Daniele Funaro
                  Department of Mathematics
                  University of Pavia
                  Via Abbiategrasso 209, 27100 Pavia, Italy
                  e-mails: fun18@ipvian.ian.pv.cnr.it
                           funaro@dragon.ian.pv.cnr.it

n : int

Degree of the polynomial >= 0.

x : float

Point in which the computation is performed.

y : byref<float>

Output: value of the polynomial in x.

dy : byref<float>

Output: value of the first derivative in x.

d2y : byref<float>

Output: value of the second derivative in x.

Polynomials.JacobiP(n, a, b, x, y, dy, d2y)

Full Usage: Polynomials.JacobiP(n, a, b, x, y, dy, d2y)

Parameters:
    n : int - Degree of the polynomial >= 0.
    a : float - Parameter > -1.
    b : float - Parameter > -1.
    x : float - Point in which the computation is performed, -1 <= x <= 1.
    y : byref<float> - Output: value of the polynomial in x.
    dy : byref<float> - Output: value of the first derivative in x.
    d2y : byref<float> - Output: value of the second derivative in x.

Computes the value of the Jacobi polynomial of degree n and its first and second derivatives at a given point. 

            Note:
              This C++ implementation is based on the Fortran function
                  VAJAPO
              from
                  "Fortran routines for spectral methods"
              by  Daniele Funaro
                  Department of Mathematics
                  University of Pavia
                  Via Abbiategrasso 209, 27100 Pavia, Italy
                  e-mails: fun18@ipvian.ian.pv.cnr.it
                           funaro@dragon.ian.pv.cnr.it

n : int

Degree of the polynomial >= 0.

a : float

Parameter > -1.

b : float

Parameter > -1.

x : float

Point in which the computation is performed, -1 <= x <= 1.

y : byref<float>

Output: value of the polynomial in x.

dy : byref<float>

Output: value of the first derivative in x.

d2y : byref<float>

Output: value of the second derivative in x.

Polynomials.LaguerreL(n, a, x, y, dy, d2y)

Full Usage: Polynomials.LaguerreL(n, a, x, y, dy, d2y)

Parameters:
    n : int - Degree of the polynomial
    a : float - Parameter > -1
    x : float - Point in which the computation is performed, x >= 0
    y : byref<float> - Output: value of the polynomial in x
    dy : byref<float> - Output: value of the first derivative in x
    d2y : byref<float> - Output: value of the second derivative in x

Computes the value of the Laguerre polynomial of degree n and its first and second derivatives at a given point. 

            Note:
              This C++ implementation is based on the Fortran function
                  VALAPO
              from
                  "Fortran routines for spectral methods"
              by  Daniele Funaro
                  Department of Mathematics
                  University of Pavia
                  Via Abbiategrasso 209, 27100 Pavia, Italy
                  e-mails: fun18@ipvian.ian.pv.cnr.it
                           funaro@dragon.ian.pv.cnr.it

n : int

Degree of the polynomial

a : float

Parameter > -1

x : float

Point in which the computation is performed, x >= 0

y : byref<float>

Output: value of the polynomial in x

dy : byref<float>

Output: value of the first derivative in x

d2y : byref<float>

Output: value of the second derivative in x

Polynomials.LegendreP(n, x, y, dy, d2y)

Full Usage: Polynomials.LegendreP(n, x, y, dy, d2y)

Parameters:
    n : int - Degree of the polynomial >= 0.
    x : float - Point in which the computation is performed, -1 <= x <= 1.
    y : byref<float> - Output: value of the polynomial in x.
    dy : byref<float> - Output: value of the first derivative in x.
    d2y : byref<float> - Output: value of the second derivative in x.

Computes the value of the Legendre polynomial of degree n and its first and second derivatives at a given point. 

            Note:
              This C++ implementation is based on the Fortran function
                 VALEPO
              from
                  "Fortran routines for spectral methods"
              by  Daniele Funaro
                  Department of Mathematics
                  University of Pavia
                  Via Abbiategrasso 209, 27100 Pavia, Italy
                  e-mails: fun18@ipvian.ian.pv.cnr.it
                           funaro@dragon.ian.pv.cnr.it

n : int

Degree of the polynomial >= 0.

x : float

Point in which the computation is performed, -1 <= x <= 1.

y : byref<float>

Output: value of the polynomial in x.

dy : byref<float>

Output: value of the first derivative in x.

d2y : byref<float>

Output: value of the second derivative in x.

Polynomials.LegendreP(l, m, x)

Full Usage: Polynomials.LegendreP(l, m, x)

Parameters:
    l : int
    m : int
    x : float

Returns: float

l : int
m : int
x : float
Returns: float

Polynomials.SphericalHarmonicY(l, m, theta, phi)

Full Usage: Polynomials.SphericalHarmonicY(l, m, theta, phi)

Parameters:
    l : int - First integer.
    m : int - Second integer, must be in the range -l <= m <= l.
    theta : float - First angle.
    phi : float - Second angle.

Returns: Complex The spherical harmonics Y_lm(theta,phi).

Computes the spherical harmonics Y_lm(theta,phi) with l and m integers satisfying -l <= m <= l and arbitrary angles theta and phi.

l : int

First integer.

m : int

Second integer, must be in the range -l <= m <= l.

theta : float

First angle.

phi : float

Second angle.

Returns: Complex

The spherical harmonics Y_lm(theta,phi).