This function compares the input parameters U and V with the restriction that both of them must be one of the three recognized structures.

If they are not of the same structure, then this routine reports the structure each has.

If they are both spin coefficient tables or both curvature component table, then the routine compares each pair of corresponding elements of U and V, and returns the sequence of indices of unmatching elements.

If U and V are both tensor_types, then this routine first compares their ranks, dimensions, then characters and returns an appropriate flag if detects any distinction.  If U and V share the same rank, dimension, and character then compare() goes on to compare each "independently defined" corresponding pair of components of the two tensor_types as well as the indexing functions of the component arrays. compare() then returns the sequence of the indices of the unmatching components, and the name 'indexfunc' if the two component arrays have different indexing functions.

The name 'Identical' is returned if U and V are found identical in each aspect checked.

Simplification :

compare() has one simplifier, `tensor/compare/simp`, used to simplify the difference of each pair of corresponding components/elements of the two objects being compared.  The simplified difference is then checked if vanishing or not.

`tensor/compare/simp` is initialized to `tensor/simp`, but it is recommended that it be customized to suit the needs of a particular comparison.

Caution :  When compare() reports that two objects differ in a set of their components/elements, what it really means is that the employed simplifier fails to simplify the pairwise differences of those reported components.  The user is hence advised to double check with another simplifier (more powerful presumably) or by inspecting each pair of the reported components.

This function is part of the tensor package, and can be used in the form compare(..) only after performing the command with(tensor), or with(tensor, compare).  The function can always be accessed in the long form tensor[compare].

$\mathrm{with}\left(\mathrm{tensor}\right)\:$

$\mathrm{I3}≔\mathrm{array}\left(\mathrm{identity}\,1..3\,1..3\right)\:$

$\mathrm{I4}≔\mathrm{array}\left(\mathrm{identity}\,1..4\,1..4\right)\:$

$U≔\mathrm{create}\left(\left[1\,1\right]\,\mathrm{op}\left(\mathrm{I3}\right)\right)$

$U≔table\left(\left[\mathrm{index\_char}=\left[1\,1\right]\,\mathrm{compts}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\right]\right)$

$\mathrm{compare}\left(U\,U\right)$

$\mathrm{Identical}$

$V≔\mathrm{create}\left(\left[1\,1\right]\,\mathrm{op}\left(\mathrm{I4}\right)\right)$

$V≔table\left(\left[\mathrm{index\_char}=\left[1\,1\right]\,\mathrm{compts}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\right]\right)$

$\mathrm{compare}\left(U\,V\right)$

$\mathrm{Dimensions\; unmatched}$

$Z≔\mathrm{create}\left(\left[1\,1\right]\,\mathrm{array}\left(\mathrm{sparse}\,1..4\,1..4\right)\right)$

$Z≔table\left(\left[\mathrm{index\_char}=\left[1\,1\right]\,\mathrm{compts}=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\right]\right)$

$\mathrm{compare}\left(V\,Z\right)$

$\mathrm{indexfunc}$

$\mathrm{T\_compts}≔\mathrm{array}\left(\mathrm{symmetric}\,1..3\,1..3\,\left[\left(1\,1\right)=x\,\left(2\,2\right)=y\,\left(3\,3\right)=z\,\left(1\,2\right)=0\,\left(1\,3\right)=0\,\left(2\,3\right)=0\right]\right)\:$

$T≔\mathrm{create}\left(\left[1\,1\right]\,\mathrm{op}\left(\mathrm{T\_compts}\right)\right)$

$T≔table\left(\left[\mathrm{index\_char}=\left[1\,1\right]\,\mathrm{compts}=\left[\begin{array}{ccc}x& 0& 0\\ 0& y& 0\\ 0& 0& z\end{array}\right]\right]\right)$

$\mathrm{compare}\left(T\,U\right)$

$\mathrm{indexfunc},\left[1\,1\right],\left[2\,2\right],\left[3\,3\right]$

$\mathrm{S\_tble}≔\mathrm{table}\left(\left['\mathrm{\lambda }'=0\,'\mathrm{\kappa }'=0\,'\mathrm{\sigma }'=0\,'\mathrm{\rho }'=0\,'\mathrm{\tau }'=0\,'\mathrm{\epsilon }'=0\,'\mathrm{\alpha }'=0\,'\mathrm{\beta }'=0\,'\mathrm{\gamma }'=0\,'\mathrm{\nu }'=0\,'\mathrm{\mu }'=0\,'\mathrm{pi}'=0\right]\right)$

$\mathrm{S\_tble}≔table\left(\left[\mathrm{\gamma }=0\,\mathrm{\beta }=0\,\mathrm{\nu }=0\,\mathrm{\alpha }=0\,\mathrm{\π}=0\,\mathrm{\kappa }=0\,\mathrm{\lambda }=0\,\mathrm{\epsilon }=0\,\mathrm{\mu }=0\,\mathrm{\tau }=0\,\mathrm{\rho }=0\,\mathrm{\sigma }=0\right]\right)$

$\mathrm{compare}\left(\mathrm{S\_tble}\,U\right)$

$\mathrm{S\_tble\; is\; a\; spin\; coefficient\; table;\; whereas\; U\; is\; a\; tensor\_type.}$

$\mathrm{compare}\left(\mathrm{S\_tble}\,\mathrm{S\_tble}\right)$

$\mathrm{Identical}$

$\mathrm{S2\_tble}≔\mathrm{table}\left(\left['\mathrm{\lambda }'=1\,'\mathrm{\kappa }'=1\,'\mathrm{\sigma }'=1\,'\mathrm{\rho }'=1\,'\mathrm{\tau }'=0\,'\mathrm{\epsilon }'=0\,'\mathrm{\alpha }'=0\,'\mathrm{\beta }'=0\,'\mathrm{\gamma }'=0\,'\mathrm{\nu }'=0\,'\mathrm{\mu }'=0\,'\mathrm{pi}'=0\right]\right)$

$\mathrm{S2\_tble}≔table\left(\left[\mathrm{\gamma }=0\,\mathrm{\beta }=0\,\mathrm{\nu }=0\,\mathrm{\alpha }=0\,\mathrm{\π}=0\,\mathrm{\kappa }=1\,\mathrm{\lambda }=1\,\mathrm{\epsilon }=0\,\mathrm{\mu }=0\,\mathrm{\tau }=0\,\mathrm{\rho }=1\,\mathrm{\sigma }=1\right]\right)$

$\mathrm{compare}\left(\mathrm{S\_tble}\,\mathrm{S2\_tble}\right)$

$\mathrm{\kappa },\mathrm{\lambda },\mathrm{\rho },\mathrm{\sigma }$

$\mathrm{PHI}≔\mathrm{array}\left(0..2\,0..2\right)\:$

$$\begin{gathered} \mathbf{for}i\mathbf{from}0\mathbf{to}2\mathbf{do} \\ \mathbf{for}j\mathbf{from}0\mathbf{to}2\mathbf{do} \\ \mathrm{PHI}\left[i,j\right]≔1 \\ \mathbf{end}\mathbf{do} \\ \mathbf{end}\mathbf{do}\: \end{gathered}$$$\mathrm{PSI}≔\mathrm{array}\left(0..4\,\left[0\,1\,2\,3\,4\right]\right)\:$$\mathrm{C\_tble}≔\mathrm{table}\left(\left['\mathrm{\Phi }'=\mathrm{op}\left(\mathrm{PHI}\right)\,'\mathrm{\Psi }'=\mathrm{op}\left(\mathrm{PSI}\right)\,'R'=r\right]\right)$

$\mathrm{C\_tble}≔table\left(\left[\mathrm{\Psi }=array\left(0..4\,\left[\left(0\right)=0\,\left(1\right)=1\,\left(2\right)=2\,\left(3\right)=3\,\left(4\right)=4\right]\right)\,\mathrm{\Phi }=array\left(0..2\,0..2\,\left[\left(0\,0\right)=1\,\left(0\,1\right)=1\,\left(0\,2\right)=1\,\left(1\,0\right)=1\,\left(1\,1\right)=1\,\left(1\,2\right)=1\,\left(2\,0\right)=1\,\left(2\,1\right)=1\,\left(2\,2\right)=1\right]\right)\,R=r\right]\right)$

$\mathrm{C2\_tble}≔\mathrm{table}\left(\left['\mathrm{\Phi }'=\mathrm{op}\left(\mathrm{PHI}\right)\,'\mathrm{\Psi }'=\mathrm{op}\left(\mathrm{PSI}\right)\,'R'=2r\right]\right)$

$\mathrm{C2\_tble}≔table\left(\left[\mathrm{\Psi }=array\left(0..4\,\left[\left(0\right)=0\,\left(1\right)=1\,\left(2\right)=2\,\left(3\right)=3\,\left(4\right)=4\right]\right)\,\mathrm{\Phi }=array\left(0..2\,0..2\,\left[\left(0\,0\right)=1\,\left(0\,1\right)=1\,\left(0\,2\right)=1\,\left(1\,0\right)=1\,\left(1\,1\right)=1\,\left(1\,2\right)=1\,\left(2\,0\right)=1\,\left(2\,1\right)=1\,\left(2\,2\right)=1\right]\right)\,R=2r\right]\right)$

$\mathrm{compare}\left(\mathrm{C\_tble}\,\mathrm{C2\_tble}\right)$

$R$

$\mathrm{compare}\left(\mathrm{C\_tble}\,\mathrm{C\_tble}\right)$

$\mathrm{Identical}$

$\mathrm{compare}\left(\mathrm{C\_tble}\,\mathrm{S\_tble}\right)$

$\mathrm{C\_tble\; is\; a\; curvature\; component\; table;\; whereas\; S\_tble\; is\; a\; spin\; coefficient\; table.}$