Generalized Products
--------------------

The `V O' (`calc-outer-product') [`outer'] command applies a given
binary operator to all possible pairs of elements from two vectors, to
produce a matrix.  For example, `V O *' with `[a, b]' and `[x, y, z]'
on the stack produces a multiplication table: `[[a x, a y, a z], [b x,
b y, b z]]'.  Element R,C of the result matrix is obtained by applying
the operator to element R of the lefthand vector and element C of the
righthand vector.

The `V I' (`calc-inner-product') [`inner'] command computes the
generalized inner product of two vectors or matrices, given a
"multiplicative" operator and an "additive" operator.  These can each
actually be any binary operators; if they are `*' and `+',
respectively, the result is a standard matrix multiplication.  Element
R,C of the result matrix is obtained by mapping the multiplicative
operator across row R of the lefthand matrix and column C of the
righthand matrix, and then reducing with the additive operator.  Just
as for the standard `*' command, this can also do a vector-matrix or
matrix-vector inner product, or a vector-vector generalized dot
product.

Since `V I' requires two operators, it prompts twice.  In each case,
you can use any of the usual methods for entering the operator.  If
you use `$' twice to take both operator formulas from the stack, the
first (multiplicative) operator is taken from the top of the stack and
the second (additive) operator is taken from second-to-top.