archimedes.quadrature.gauss_radau¶
- archimedes.quadrature.gauss_radau(
- n: int,
- endpoint: str = 'left',
Gauss-Radau quadrature rule including exactly one endpoint.
Radau rules fix one endpoint of \([-1, 1]\) as a node and choose the remaining \(n - 1\) nodes to maximize the polynomial degree of exactness. Fixing the left endpoint, these are the roots of the Jacobi polynomial \(P_{n-1}^{(0,1)}(x)\); fixing the right endpoint, the roots of \(P_{n-1}^{(1,0)}(x)\). The rule is exact for polynomials up to degree \(2n - 2\).
- Parameters:
n (int) – Number of quadrature nodes, including the fixed endpoint.
endpoint ({"left", "right"}, optional) –
Which endpoint of \([-1, 1]\) to fix as a node:
"left"includes \(-1\) (LGR, the pseudospectral convention)."right"includes \(+1\) (Radau IIA, the IRK/DAE convention).
- Returns:
rule – Gauss-Radau rule with
nnodes on \([-1, 1]\), exact to degree \(2n - 2\).- Return type:
- Raises:
ValueError – If
n < 1, or ifendpointis not"left"or"right".