BVT Algorithm Tests

Double precision implementations, using equation (21) with several Gauss-Legendre rules, and with the adaptive algorithm described in the previous section (absolute error tolerance set at $10^{-14}$), were tested. The test used $b_1 = -5, -4.75, \ldots, 5$, $b_2 = b_1, b_1+.25, \ldots, 5$ , $\rho = -n/(n+1), (-n+4)/(n+1), \ldots, n/(n+1)$ with $n = 64$, and $\nu = 1, 2, \ldots , 25$, with results given in Table 2, which include average computation time in seconds for the different integration methods. Additional tests with $n = 128, 256, 512$, to check for sensitivity of the algorithms to nearly singular problems, did not produce significantly different results.

Table 2: Generalized Plackett Formula BVT Method Results

	$\epsilon$	Maximum Error	Average Time
6-Point Rule	0	$1\cdot10^{-3}$	$2\cdot10^{-6}$ s
12-Point Rule	0	$1\cdot10^{-4}$	$6\cdot10^{-6}$ s
24-Point Rule	0	$1\cdot10^{-6}$	$1\cdot10^{-5}$ s
24-Point Rule	.01	$2\cdot10^{-4}$	$1\cdot10^{-5}$ s
48-Point Rule	0	$6\cdot10^{-11}$	$2\cdot10^{-5}$ s
48-Point Rule	.01	$1\cdot10^{-5}$	$2\cdot10^{-5}$ s
Adaptive	0	$3\cdot10^{-16}$	$3\cdot10^{-5}$ s
Adaptive	.01	$6\cdot10^{-16}$	$3\cdot10^{-5}$ s

A quadruple precision implementation of an algorithm based on the Dunnett and Sobel (1954) paper was used for an accurate comparison. A double precision implementation of the Dunnett and Sobel (1954) paper algorithm reliably achieves double precision results and typically takes about $3\cdot10^{-6}$ seconds, approximately one tenth of the time taken by the adaptive algorithm using equation (21). This time difference and results in Table 2 do not support the use of methods that use equation (21) with numerical integration for the efficient computation of BVT probabilities.