quad module

Created on Thu Aug 31 13:56:13 2023

@author: Pavan Pranjivan Mehta

class quad.quad_pts_weights

Bases: object

Computes Quadrature Points and weights

legendre()

Qudrature points and weights of Legendre Polynomials in [-1,1]

Parameters

degreeint

degree of the polynomial

Returns

Qudrature points and weights of Legendre polynomials within the interval [-1,1]

q: float

quadrature points

w: float

weights

jacobi(beta, degree)

Qudrature points and weights of Jacobi Polynomials, P_n^{alpha, beta} in [-1,1]; alpha, beta > -1.

Parameters

alphafloat

Paramter for Jacobi polynomial, alpha > -1

betafloat

Paramter for Jacobi polynomial, beta > -1

degreeint

degree of the polynomial

Returns

Qudrature points and weights of Jacobi polynomials within the interval [-1,1]

q: float

quadrature points

w: float

weights

shift_legendre()

Shifted Qudrature points and weights of Legendre Polynomials in [0,1]

Parameters

degreeint

degree of the polynomial

Returns

Qudrature points and weights of Legendre polynomials within the interval [0,1]

q: float

quadrature points

w: float

weights

shift_jacobi(beta, degree)

Shifted Qudrature points and weights of Jacobi Polynomials in [0,1]

Parameters

alphafloat

Paramter for Jacobi polynomial, lpha > -1

betafloat

Paramter for Jacobi polynomial, eta > -1

degreeint

degree of the polynomial

Returns

Qudrature points and weights of Jacobi polynomials within the interval [0,1]

q: float

quadrature points

w: float

weights

Q_poly(beta, degree)

Qudrature points and weights of Q Polynomials in [0,1]

Q := P_n (1 - 2x)

Parameters

alphafloat

Paramter for Jacobi polynomial, lpha > -1

betafloat

Paramter for Jacobi polynomial, eta > -1

degreeint

degree of the polynomial

Returns

Qudrature points and weights of Q polynomials within the interval [0,1]

q: float

quadrature points

w: float

weights

shift_arbitary_legendre(b, degree)

Shifted Qudrature points in arbitary domain and weights of Legendre Polynomials in [a,b]

Parameters

afloat

min of interval [a, b]

bfloat

max of interval [a, b]

degreeint

degree of the polynomial

Returns

Qudrature points and weights in arbitary domain of Legendre polynomials within the interval [a,b]

q: float

quadrature points

w: float

weights

shift_arbitary_jacobi(b, alpha, beta, degree)

Shifted Qudrature points in arbitary domain and weights of Jacobi Polynomials in [a,b]

Parameters

afloat

min of interval [a, b]

bfloat

max of interval [a, b]

alphafloat

Parameter for Jacobi polynomial, lpha > -1

betafloat

Parameter for Jacobi polynomial, eta > -1

degreeint

degree of the polynomial

Returns

Qudrature points and weights in arbitary domain of Jacobi polynomials within the interval [a,b]

q: float

quadrature points

w: float

weights

homogenous_legendre()

Qudrature points and weights of Homogenous basis using Legendre Polynomials in [-1,1]

phi_i = l_{k+2} - l_{k}, where l is legendre polynomial in [-1,1] of degree k

Parameters

degreeint

degree of the polynomial

Returns

Qudrature points and weights of Homogenous basis using Legendre polynomials within the interval [-1,1]

q2: float

quadrature points associted with degree+2 Legendre Polynomial

w2: float

weights associted with degree+2 Legendre Polynomial

q0: float

quadrature points associted with degree Legendre Polynomial

w0: float

weights associted with degree Legendre Polynomial

homogenous_shift_legendre()

Qudrature points and weights of Homogenous basis using Shited Legendre Polynomials in [0,1]

phi_i = l_{k+2} - l_{k}, where l is shifted legendre polynomial in [0,1] of degree k

Parameters

degreeint

degree of the polynomial

Returns

Qudrature points and weights of Homogenous basis using Legendre polynomials within the interval [0,1]

q2: float

quadrature points associted with degree+2 shifted Legendre Polynomial

q0: float

quadrature points associted with degree shifted Legendre Polynomial

w2: float

weights associted with degree+2 shifted Legendre Polynomial

w0: float

weights associted with degree shifted Legendre Polynomial

class quad.gauss_quad

Bases: object

Computes Gauss qundrature

legendre(degree)

Gauss Legendre Quadrature in [-1,1]

Parameters

f: float array

f(x) at quadrature points

degreeint

degree of the polynomial

Returns

Value of Gauss Legendre Quadrature within the interval [-1,1]

shift_legendre(degree)

Gauss Legendre Quadrature (shifted) in [0,1]

Parameters

f: float array

f(x) at quadrature points

degreeint

degree of the polynomial

Returns

Value of Shifted Gauss Legendre Quadrature within the interval [0,1]

shift_jacobi(alpha, beta, degree)

Gauss Jacobi Quadrature (shifted) in [0,1]

Parameters

f: float array

f(x) at quadrature points

alphafloat

Parameter for Jacobi polynomial, lpha > -1

betafloat

Parameter for Jacobi polynomial, eta > -1

degreeint

degree of the polynomial

Returns

Value of Shifted Gauss Jacobi Quadrature within the interval [0,1]

Q_poly(alpha, beta, degree)

Gauss Q-poly Quadrature in [0,1]

Parameters

f: float array

f(x) at quadrature points

alphafloat

Parameter for Q polynomial, lpha > -1

betafloat

Parameter for Q polynomial, eta > -1

degreeint

degree of the polynomial

Returns

Value of Gauss Q-poly Quadrature within the interval [0,1]

shift_arbitary_legendre(a, b, degree)

Gauss Legendre Quadrature (shifted arbitary) in [a,b]

Parameters

f: float array

f(x) at quadrature points

a : min of interval [a, b]

b : max of interval [a, b]

degreeint

degree of the polynomial

Returns

Value of Shifted arbitary Gauss Legendre Quadrature within the interval [a,b]

shift_arbitary_jacobi(a, b, alpha, beta, degree)

Gauss Jacobi Quadrature (shifted arbitary) in [a,b]

Parameters

f: float array

f(x) at quadrature points

a : min of interval [a, b]

b : max of interval [a, b]

alphafloat

Parameter for Jacobi polynomial, lpha > -1

betafloat

Parameter for Jacobi polynomial, eta > -1

degreeint

degree of the polynomial

Returns

Value of Shifted arbitary Gauss Jacobi Quadrature within the interval [a,b]

homogenous_legendre(f0, degree)

Gauss Legendre Quadrature for homogenous in [-1,1]

phi_i = l_{k+2} - l_{k}, where l is legendre polynomial in [-1,1] of degree k

Parameters

f2: float array

f(x) at quadrature points with associted degree “k+2” legendre polynomials

f0: float array

f(x) at quadrature points with associted degree “k” legendre polynomials

degreeint

degree of the polynomial

Returns

Value of Gauss Legendre Quadrature for homogenous within the interval [-1,1]

homogenous_shift_legendre(f0, degree)

Gauss Legendre Quadrature for homogenous in [0,1]

phi_i = l_{k+2} - l_{k}, where l is shifted legendre polynomial in [0,1] of degree k

Parameters

f2: float array

f(x) at quadrature points with associted degree “k+2” shifted legendre polynomials

f0: float array

f(x) at quadrature points with associted degree “k” shifted legendre polynomials

degreeinf2 = np.sin(q2)

f0 = np.sin(q0)t degree of the polynomial

Returns

Value of Gauss Legendre Quadrature for homogenous within the interval [0,1]

class quad.test

Bases: object

gauss_legendre()

shift_gauss_legendre()

shift_arbirat_gauss_legendre(b=1)

gauss_legendre_homogeous()

gauss_shift_legendre_homogeous()