ortho.OrthogonalFunctions

class ortho.OrthogonalFunctions(start_index=1, num_func=9, end_interval=1, verbose=False)

Generates a set of orthonormal functions.

Parameters:

start_indexint, default=1

The index of the starting function, \(i_0\).

num_funcint, default=9

Number of orthogonal functions to generate

end_intervalfloat, default=1

The right interval of orthogonality, \(L\).

veboseboolean, default=False

Prints the generated functions. An example of output is

phi_1(t) =  sqrt(x)
phi_2(t) =  sqrt(6)*(5*x**(1/3) - 6*sqrt(x))/3
phi_3(t) =  sqrt(2)*(21*x**(1/4) - 40*x**(1/3) + 20*sqrt(x))/2

Notes

The orthogonal functions, \(\phi_i^{\perp}\), are generated based on the set of non-orthonormal functions \(\phi_i\) defined by the inverse-monomials

\[\phi_i(t) = t^{\frac{1}{i+1}}, \qquad i = i_0,\dots,i_0+n.\]

The orthonormalized functions \(\phi_i^{\perp}\) are the linear combination of the functions \(\phi_i\) by

\[\phi_i^{\perp}(t) = \alpha_i \sum_{j = i_0}^{i_0+n} a_{ij} \phi_j(t), \qquad i = i_0,\dots,i_0+n.\]

The functions \(\phi_i^{\perp}\) are orthonormal in the interval \(t \in [0,L]\) with respect to the weight function \(w(t) = t^{-1}\). That is,

\[\langle \phi_i^{\perp},\phi_j^{\perp} \rangle_{L^2([0,L], \mathrm{d}t/t)} = \int_0^L \phi_i^{\perp}(t) \phi_j^{\perp}(t) \frac{\mathrm{d}t}{t} = \delta_{ij},\]

where \(\delta_{ij}\) is the Kronecker delta function. The orthogonal functions are generated by Gram-Schmidt orthogonalization process.

Attributes:

sym_phisympy obj

Orthogonal functions \(\phi_i^{\perp}\)

sym_alphasympy obj

Coefficients \(\alpha_i\)

sym_coeffssympy obj

Coefficients \(a_{i,j}\)

alphalist

Coefficients \(\alpha_i\)

coeffslist of lists

Coefficients \(a_{i,j}\)

Methods

check([verbose])	Check the mutual orthogonality of the functions.
print()	Print coeffs of Functions.
plot([filename])	Plot the generated functions.