Routines For Orthogonal Polynomial Calculus and Interpolation

Spencer Sherwin, Aeronautics, Imperial College London

Based on codes by Einar Ronquist and Ron Henderson

Abbreviations

z - Set of collocation/quadrature points
w - Set of quadrature weights
D - Derivative matrix
h - Lagrange Interpolant
I - Interpolation matrix
g - Gauss
gr - Gauss-Radau
gl - Gauss-Lobatto
j - Jacobi
m - point at minus 1 in Radau rules
p - point at plus 1 in Radau rules

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MAIN ROUTINES
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Points and Weights:

zwgj Compute Gauss-Jacobi points and weights
zwgrjm Compute Gauss-Radau-Jacobi points and weights (z=-1)
zwgrjp Compute Gauss-Radau-Jacobi points and weights (z= 1)
zwglj Compute Gauss-Lobatto-Jacobi points and weights

Derivative Matrices:

Dgj Compute Gauss-Jacobi derivative matrix
Dgrjm Compute Gauss-Radau-Jacobi derivative matrix (z=-1)
Dgrjp Compute Gauss-Radau-Jacobi derivative matrix (z= 1)
Dglj Compute Gauss-Lobatto-Jacobi derivative matrix

Lagrange Interpolants:

hgj Compute Gauss-Jacobi Lagrange interpolants
hgrjm Compute Gauss-Radau-Jacobi Lagrange interpolants (z=-1)
hgrjp Compute Gauss-Radau-Jacobi Lagrange interpolants (z= 1)
hglj Compute Gauss-Lobatto-Jacobi Lagrange interpolants

Interpolation Operators:

Imgj Compute interpolation operator gj->m
Imgrjm Compute interpolation operator grj->m (z=-1)
Imgrjp Compute interpolation operator grj->m (z= 1)
Imglj Compute interpolation operator glj->m

Polynomial Evaluation:

jacobfd Returns value and derivative of Jacobi poly. at point z
jacobd Returns derivative of Jacobi poly. at point z (valid at z=-1,1)

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LOCAL ROUTINES
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jacobz Returns Jacobi polynomial zeros
gammaf Gamma function for integer values and halves

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Useful references:

[1] Gabor Szego: Orthogonal Polynomials, American Mathematical Society, Providence, Rhode Island, 1939.
[2] Abramowitz & Stegun: Handbook of Mathematical Functions, Dover, New York, 1972.
[3] Canuto, Hussaini, Quarteroni & Zang: Spectral Methods in Fluid Dynamics, Springer-Verlag, 1988.
[4] Ghizzetti & Ossicini: Quadrature Formulae, Academic Press, 1970.
[5] Karniadakis & Sherwin: Spectral/hp element methods for CFD, 1999

NOTES

Legendre polynomial $\alpha = \beta = 0$
Chebychev polynomial $\alpha = \beta = -0.5$
All array subscripts start from zero, i.e. vector[0..N-1]