Carlson's Quadrature Set B

Beside the quadrature Set A that is described and computed in the previous blog, Carlson (1963) described an alternative set of quadratures, Set B. This set corresponds to odd approximations. It "relates to Set A in the way Double-P relates to Single-P." (ibid). The principles on which this set is constructed are same as in the case of Set A. However, the equations slightly differ. First we determine the $W_l$ coefficients from the system:
$$ \sum_{l=1}^{n/2-1} W_l = \frac{n-3}{3}, $$ $$ W_l^2 = W_1^2 + (l-1)\Delta^\prime, $$ where $\Delta^\prime = 2/n$. This system is again solved by Newton-Raphson formula. The $W_l$ values are then used to get $w_l$ (as with Set A).



In the case of Set B, $\mu_1$ is free parameter, so we cannot make a qualified assumption on its value as we did it before. Instead we solve the following system; $$ \sum_{l=1}^{n/2} \;\;\;\;w_l |\mu_l| = 1/2, $$ $$ \mu_l^2 = \mu_1^2 + (l-1)\Delta, $$ with $$ \Delta = \frac{2}{n}(1-3\mu_1^2). $$ Note that the summation in the equation above runs this time from 1 to $n/2$. This system is solved numerically by Newton-Raphson too. It is a rather robust equations and it converges quickly for initial value of $\mu$ between 0 and 1.

The weights $w_m$ of rays are find in exactly the same way as in Set A.

My IDL code that evaluates the Set B of Carlson has four routines: the main procedure carlson_quadrature_b.pro, routines that compute $\omega_l$ and $\mu_l$ w1nr_b.pro and mu1nr_b.pro; and it shares variations3.pro with Set A (click on the procedure names to download them).

Here is an example for $n=8$,

carlson_quadrature_b, 8, wl=wl, mu=mu, wm=wm

gives the following values (Set B, n=8):

IDL> print, wl
      0.30807338      0.27921633      0.18401386      0.22869642
IDL> print, mu
      0.11104440      0.50307327      0.70273364      0.85708018
IDL> print, wm
     0.025259964     0.079376947      0.11434821
IDL> print, variations
     1     1     1     2     2     2     2     3     3     3     4     4
     2     3     4     1     2     3     4     1     2     3     1     2
     4     3     2     4     3     2     1     3     2     1     2     1

Let's finish with another quote of Carlson: "One expects then Set B, for fixed n, to provide better accuracy in applications than set A." Happy computing!