4.10 Polylib

This directory contains polylib.h and polylib.cpp. These files contain foundational routines used for computing various operations related to Jacobi polynomials. The following abbreviations are used throughout the file:

• 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

Points and Weights: The following routines are used to compute 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

Derivative Matrices: The following routines are used to compute 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: The following routines are used to compute Lagrange interpolation matrices:

• 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: The following routines are used to compute various 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: The following routines are used to evaluate Jacobi polynomials.

• jacobfd – Returns value and derivative of Jacobi polynomial at point z
• jacobd – Returns derivative of Jacobi polynomial at point z (valid at z = -1,1)