Parameter: lemax
Definition
The maximum allowable expansion order $n$ for the Legendre orthogonal polynomials. It must be greater than 2.
The recursive definition for the Legendre polynomials is as follows:
\[P_{0}(x) =1\]
\[P_{1}(x) = x\]
\[(n+1) P_{n+1}(x) = (2n+1)xP_{n}(x) - n P_{n-1}(x)\]
An important property of the Legendre polynomials is that they are orthogonal with respect to the $L^2$ inner product on the interval $−1 \leq x\leq 1$:
\[\int^{1}_{-1} P_{m}(x) P_{n}(x) dx = \frac{2}{2n+1}\delta_{mn}\]
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 Legendre orthogonal polynomials.
Comment
Only when isort = 2 or isort = 5 this parameter is useful. See isort parameter for more details. How to choose a suitable lemax parameters is a tricky job. If lemax is too small, the calculated results won't be accurate. If lemax is too large, the so-called Gibbs oscillation will occur dramatically. According to our experiences, 32 or 48 may be a reasonable choice. It is worthy to emphasis that due to the limitation of implementation, lemax must be less than 50.
See also legrd 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