Parameter: chmax
Definition
The maximum allowable expansion order $n$ for the second kind Chebyshev orthogonal polynomials. It must be greater than 2.
The recursive definition for the second kind Chebyshev polynomials is as follows:
\[U_{0} = 1\]
\[U_{1} = 2x\]
\[U_{n+1}(x) = 2xU_{n}(x) - U_{n-1}(x)\]
The second kind Chebyshev polynomials are orthogonal with respect to the weight
\[\sqrt{1-x^2}\]
on the interval [-1,1], i.e., we have:
\[\begin{equation} \int_{-1}^1 U_n(x)U_m(x)\sqrt{1-x^2}\,dx = \begin{cases} 0 &: n\ne m, \\ \pi/2 &: n=m. \end{cases} \end{equation}\]
Type
Integer
Default value
32
Component
Only for the GARDENIA, NARCISSUS, LAVENDER, CAMELLIA, and MANJUSHAKA components.
Behavior
The parameter is used as a cutoff to limit the maximum expansion order for the second kind Chebyshev orthogonal polynomials.
Comment
Only when isort = 3 or isort = 6 this parameter is useful. See isort parameter for more details. How to choose a suitable chmax parameters is a tricky job. If chmax is too small, the calculated results won't be accurate. If chmax is too large, the so-called Gibbs oscillation will occur significantly. According to our experiences, 32 or 48 may be a reasonable choice. Though there is no upper limit for isort, the larger the value of chmax is, the heavier the computational burden will be.
See also chgrd for more details.
As for the applications of orthogonal polynomials in CT-QMC impurity solver, please refer to Lewin's[1] and Hartmann's[2] papers.
Reference