Lower symmetric subrange Freud polynomials $\pi_n$.
These are orthogonal on $[-c, c]$ with respect to the weight function $w(t)=|t|^n e^{-|t|^*}$.
(a) Write a symbolic MATLAB routine smom_1s f .m that generates the first $2 N$ moments of $w$.
(b) Write a routine dig_1sf.m analogous to the routine dig_11sf.m of Ex. 2.18(b) and use it with nof dig $=16$ for $N=15$ and $N=30$, and for selected values of $c, \mu, v$ in the ranges Table 2.3. The intervals $\left(0, v^*\right)$ of monotonic decrease of all zeros of half-range Freud polynomials $\pi_n$ in dependence on $n$ and $\mu$
(FIGURE CAN'T COPY)
$1 \leq c \leq 6,-1 / 2 \leq \mu \leq 1 / 2$, and $1 \leq v \leq 10$, to determine how many digits are required at most.
(c) Write a routine sr_1sf.m generating in dig-digit arithmetic the $N \times 2$ array ab of recurrence coefficients to nof dig digits. Use it in a script zeros_1sf.m that is analogous to the previous three scripts zeros_ $\cdots$, but applicable to symmetric weight functions, and hence computes only the positive zeros.
(d) Run the script zeros_1sf.m of (c) with $N=15$ and $N=30$, using $c=1, \mu=0$, and nofdig $=16$. Provide plots of the positive zeros.
(e) Same as (d), but with $c=2$ and $N=2,7,15,30$. Comment on the results.
(f) Same as (e), but with $c=6, N=15$, and $N=30$.