Dear UQWorld community,

I use UQLab to perform polynomial chaos expansions (PCEs). As I aim at further working with the resulting PCE coefficients in Matlab, I try to normalize the Jacobi polynomials in Matlab (function see jacobiP) such that they are consistent with the Jacobi polynomial definition in UQLab. How can I extract the proper normalizing constant?

I found a statement on the normalization of the Jacobi polynomials in the user manual “UQLab user manual Polynomial Chaos Expansions”, page 3, Table 1. What is the definition of the function B(.) there? To my knowledge, the Beta function needs two arguments, so should it be B(a,b) instead of B(a)*B(b)?

I figured out that the UQLab function *uq_eval_jacobi* gets the parameters (a,b) of the beta distribution as input. Do I have to use the parameters (a-1,b-1) as input to the Jacobi polynomial as it is defined in the Matlab function *jacobiP* then?

Exemplary, for order n=2, I compared the Matlab output (normalized to unit norm) and the UQLab output of the Jacobi polynomials for x ranging from -1 to 1.

j2_mat(x) = jacobiP(2,a-1,b-1,x)/normConst(2),

where normConst(2)= sqrt((2^(a+b-1)/(2\cdot2+a+b-1)*(Gamma(2+a)*Gamma(2+b)/(Gamma(2+a+b-1)*Gamma(2+1))). This is the definition of J_{a,b,2} in “UQLab user manual Polynomial Chaos Expansions”, page 3, Table 1, for the values a-1 and b-1.

j2_UQLab(x) = uq_eval_jacobi(2, x, [a,b], true).

Why do j2_mat and j2_UQLab lead to different results?

The same scaling procedure worked perfectly fine for the Legendre polynomials by setting normConst(2) = sqrt(1/(2*2+1)) as also specified in “UQLab user manual Polynomial Chaos Expansions”, page 3, Table 1.

What is the normalization constant of the Jacobi polynomials inherent in UQLab?

I would appreciate any help on that issue.

Thanks for taking time to reply.