For k = 3

The coordinates of these added points are

x_i = (x_i1 , x_i2 , x_i3), i = 1, …, 6,

where x_i1 =sinθ_i2cosθ_i1, xi2 =sinθ_i2sinθ_i1, and x_i3 =sinθ_i2.

After 200 iterations, the optimal added points obtained by BA sampler are

The maximum of P_eff value is 0.6680.

For k = 3, the determinants of X^’X for 200 × 10 × 14 = 28000 steps.