The Polylib library

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
• k - Kronrod
• 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
  • zwgk Compute Gauss-Kronrod-Jacobi points and weights
  • zwrk Compute Radau-Kronrod points and weights
  • zwlk Compute Lobatto-Kronrod points and weights
  • JacZeros Compute Gauss-Jacobi points and weights

Integration Matrices:

• Qg Compute Gauss integration matrix

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:

• polycoeff Returns value and derivative of Jacobi poly.
• 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
  • RecCoeff Calculates the recurrence coefficients for orthogonal poly
  • TriQL QL algorithm for symmetrix tridiagonal matrix
  • JKMatrix Generates the Jacobi-Kronrod matrix

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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

1. Legendre polynomial \( \alpha = \beta = 0 \)
2. Chebychev polynomial \( \alpha = \beta = -0.5 \)
3. All routines are double precision.
4. All array subscripts start from zero, i.e. vector[0..N-1]