Monday, 5 November 2012

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).