simplex_basis Namespace Reference

Classes
class	Basis
class	LineBasisTest
class	QuadratureRule

Functions
def	make_vandermonde (modes, nodes, sym)
def	lgl_points_1d_on_01 (N)
def	lgl_quad_rule_1d_01 (N)
def	gauss_legendre_quad_rule_1d (N)
def	coefs_of_polynomial (poly, degree)
	We're going to compute the coefficients of the polynomial, which is just an opaque lambda function to us, in the monomial basis, so that we can pass them to mp.polyroots.
def	mp_to_np (matrix)
def	np_to_mp (array)
def	lgl_simplex_nodes (dims, degree)

Variables
	exit
	dps
TriangleBasis	b = TriangleBasis(3)
TriangleBasis	ells = b.lagrange_interpolating_polynomials()

Function Documentation

coefs_of_polynomial()

def simplex_basis.coefs_of_polynomial	(		poly,
			degree
	)

We're going to compute the coefficients of the polynomial, which is just an opaque lambda function to us, in the monomial basis, so that we can pass them to mp.polyroots.

To do so, we use the relation

 p'.(x) = V * c,        (**)

where

• p'.(x) is the vector of evaluations of p' at some vector of points x,
• V is the Vandermonde matrix of monomials evaluated at x,
• c is the vector of monomial coefficients of p'.

To find c, we just have to solve the equation (**).

It doesn't matter what vector of point we choose for x as long as it results in a well-conditioned matrix V. We choose the Chebyshev nodes.

gauss_legendre_quad_rule_1d()

def simplex_basis.gauss_legendre_quad_rule_1d

(

N

)

lgl_points_1d_on_01()

def simplex_basis.lgl_points_1d_on_01

(

N

)

lgl_quad_rule_1d_01()

def simplex_basis.lgl_quad_rule_1d_01

(

N

)

lgl_simplex_nodes()

def simplex_basis.lgl_simplex_nodes	(		dims,
			degree
	)

make_vandermonde()

def simplex_basis.make_vandermonde	(		modes,
			nodes,
			sym
	)

Parameters

sym	the symbol indicating which family of polynomials to use: p – the "primary" polynomials of this set of modes dpdr – their derivatives with respect to r dpds – their derivatives with respect to s

mp_to_np()

def simplex_basis.mp_to_np

(

matrix

)

np_to_mp()

def simplex_basis.np_to_mp

(

array

)

Variable Documentation

b

TriangleBasis simplex_basis.b = TriangleBasis(3)

dps

simplex_basis.dps

ells

TriangleBasis simplex_basis.ells = b.lagrange_interpolating_polynomials()

exit

simplex_basis.exit