The function prod(A, B, [a1,b1], [a2,b2], ...) computes the inner product of the A and B with contraction taking place over the pairs of indices a1 (from A) and b1 (from B), a2 (from A) and b2 (from B), and so on.

The function prod(A, B) computes the outer product of A and B.

The indices in each pair must be of opposite index character.

There must not be any duplicates in the given indices from each tensor (it is impossible to contract over a single index more than once).  Thus, the call prod(A, B, [1,2], [1,3]) is illegal.  However, the call prod(A, B, [1,1]) is not illegal (provided the indices are of opposite index character) since there is no repetition of indices from A and no repetition of indices from B.

The return value is the resultant tensor_type object of rank equal to rank(A) + rank(B) - 2 * (# of pairs in the call).

Simplification:  This routine uses the `tensor/prod/simp` routine for simplification purposes.  The simplification routine is applied to each component of the result after it is computed.  By default, `tensor/prod/simp` is initialized to the `tensor/simp` routine. It is recommended that the `tensor/prod/simp` routine be customized to suit the needs of the particular problem.

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

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

$T≔\mathrm{create}\left(\left[1\,-1\right]\,\mathrm{array}\left(\left[\left[w\,x\,0\right]\,\left[y\,z\,0\right]\,\left[0\,y^2\,xyw\right]\right]\right)\right)$

$T≔table\left(\left[\mathrm{compts}=\left[\begin{array}{ccc}w& x& 0\\ y& z& 0\\ 0& y^2& xyw\end{array}\right]\,\mathrm{index\_char}=\left[1\,\mathrm{-1}\right]\right]\right)$

$U≔\mathrm{create}\left(\left[1\right]\,\mathrm{array}\left(\left[l\,m\,n\right]\right)\right)$

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

$\mathrm{prod}\left(T\,U\,\left[2\,1\right]\right)$

$table\left(\left[\mathrm{compts}=\left[\begin{array}{ccc}wl+mx& yl+mz& nwxy+m{y}^2\end{array}\right]\,\mathrm{index\_char}=\left[1\right]\right]\right)$

$V≔\mathrm{create}\left(\left[-1\right]\,\mathrm{array}\left(\left[a\,b\,c\right]\right)\right)$

$V≔table\left(\left[\mathrm{compts}=\left[\begin{array}{ccc}a& b& c\end{array}\right]\,\mathrm{index\_char}=\left[\mathrm{-1}\right]\right]\right)$

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

$table\left(\left[\mathrm{compts}=\left[\begin{array}{ccc}al& am& an\\ bl& bm& bn\\ cl& cm& cn\end{array}\right]\,\mathrm{index\_char}=\left[\mathrm{-1}\,1\right]\right]\right)$

$\mathrm{prod}\left(V\,U\,\left[1\,1\right]\right)$

$table\left(\left[\mathrm{compts}=al+bm+cn\,\mathrm{index\_char}=\left[\right]\right]\right)$

$A≔\mathrm{create}\left(\left[\right]\,a\right)$

$A≔table\left(\left[\mathrm{compts}=a\,\mathrm{index\_char}=\left[\right]\right]\right)$

$B≔\mathrm{create}\left(\left[\right]\,b\right)$

$B≔table\left(\left[\mathrm{compts}=b\,\mathrm{index\_char}=\left[\right]\right]\right)$

$\mathrm{prod}\left(A\,B\right)$

$table\left(\left[\mathrm{compts}=ab\,\mathrm{index\_char}=\left[\right]\right]\right)$