Spectral 1D

Implementation of Spectral Methods in 1 dimension.

Available polynomials:

• Jacobi Polynomials or Spectral1D.JACOBI
• Hermite Physicist or Spectral1D.HERMITEP
• Hermite Function or Spectral1D.HERMITEF
• Hermite Probabilistic or Spectral1D.HERMITEP_PROB
• Laguerre Polynomial or Spectral1D.LAGUERREP
• Laguerre Function or Spectral1D.LAGUERREF
• ORTHPOL package (generation of recursion coefficients using [Gau94]) or Spectral1D.ORTHPOL

Available quadrature rules (related to selected polynomials):

• Gauss or Spectral1D.GAUSS
• Gauss-Lobatto or Spectral1D.GAUSSLOBATTO
• Gauss-Radau or Spectral1D.GAUSSRADAU

Available quadrature rules (without polynomial selection):

• Kronrod-Patterson on the real line or Spectral1D.KPN (function Spectral1D.kpn(n))
• Kronrod-Patterson uniform or Spectral1D.KPU (function Spectral1D.kpu(n))
• Clenshaw-Curtis or Spectral1D.CC (function Spectral1D.cc(n))
• Fejer’s or Spectral1D.FEJ (function Spectral1D.fej(n))

Jacobi Polynomials

Jacobi polynomials are defined on the domain \(\Omega=[-1,1]\) by the recurrence relation

\[\begin{split}xP^{(\alpha,\beta)}_n(x) = & \frac{2(n+1)(n+\alpha+\beta+1)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)} P^{(\alpha,\beta)}_{n+1}(x) \\ & + \frac{\beta^2 - \alpha^2}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)} P^{(\alpha,\beta)}_{n}(x) \\ & + \frac{2(n+\alpha)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)} P^{(\alpha,\beta)}_{n-1}(x)\end{split}\]

with weight function

\[w(x;\alpha,\beta) = \frac{\Gamma(\alpha+\beta+2)}{2^{\alpha+\beta+1}\Gamma(\alpha+1)\Gamma(\beta+1)}(1-x)^\alpha (1+x)^\beta\]

Note

In probability theory, the Beta distribution is defined on \(\Psi=[0,1]\) and its the Probability Distribution Function is

\[\rho_B(x;\alpha,\beta) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)} x^{\alpha-1} (1-x)^{(\beta-1)}\]

The relation betwen \(w(x;\alpha,\beta)\) and \(\rho_B(x;\alpha,\beta)\) for \(x \in \Psi\) is

\[\rho_B(x;\alpha,\beta) = 2 * w(2*x-1;\beta-1,\alpha-1)\]

For example:

>>> from scipy import stats
>>> plot(xp,stats.beta(3,5).pdf(xp))
>>> plot(xp,2.*Bx(xx,4,2),'--')
>>> plot(xp,stats.beta(3,8).pdf(xp))
>>> plot(xp,2.*Bx(xx,7,2),'--')