Dot product test

In Appendix A transpose operators to hyperbola and parabola stacking in continuous domain are derived. For hyperbola stacking $\bold H$ it is weighted parabola stacking in $(\tau,m)$-space, which is symbolically denoted as $t/\tau\, \bold P$, and for parabola stacking $\bold P$ in $(\tau,m)$-space it is weighted stacking $\tau/t\, \bold H$in (t,x)-space. A dot product test was used to test these results (Table 1). The following form of the dot product test was used:

$$(\bold L^{\bold T}\bold d,\bold L^{\bold T}\bold d) = ({\bf LL^Td},\bold d),$$ (13)

where $\bold L$ is ${\bf H^T}$, ${\bf P}$ or whatever operator was used.

 

2|c|Operator in 2|c|Dot product

$$({\bf L^Td},{\bf L^Td}) ({\bf LL^Td},\bold d)$$ time domain sloth domain

$$\bold H {\bf H^T}$$ 110.5207 110.5218

$${\bf P^T} {\bf P}$$ 106.6519 106.6530

$$\sqrt{\tau/t}\,\bold H \sqrt{t/\tau}\,\bold P$$ 107.6680 108.4489

$$\bold H t/\tau\, \bold P$$ 110.5207 111.3971

$$\tau/t\, \bold H \bold P$$ 105.1228 105.8237

$$\bold H \bold P$$ 110.5207 108.4480

From the table we can see that adjoint operators derived in the continuous domain are much less precise (relative error about 0.7-$0.8\%$) than adjoint operators derived in the discrete domain (relative error $0.001\%$). Operators $\bold H$ and $\bold P$ in the last line of the table are not adjoint to each other, but they were tested to support the theoretical results from Appendix A (relative error about $2\%$).