The V R (
reduce] command applies a given
binary operator across all the elements of a vector. A binary operator is
a function such as
max which takes two arguments. For
+ over a vector computes the sum of the elements
of the vector. Reducing
- computes the first element minus each of
the remaining elements. Reducing
max computes the maximum element
and so on. In general, reducing
f over the vector ‘[a, b, c, d]’
produces ‘f(f(f(a, b), c), d)’.
The I V R [
rreduce] command is similar to V R except
that works from right to left through the vector. For example, plain
V R - on the vector ‘[a, b, c, d]’ produces ‘a - b - c - d’
but I V R - on the same vector produces ‘a - (b - (c - d))’,
or ‘a - b + c - d’. This “alternating sum” occurs frequently
in power series expansions.
The V U (
accum] command does an
accumulation operation. Here Calc does the corresponding reduction
operation, but instead of producing only the final result, it produces
a vector of all the intermediate results. Accumulating
the vector ‘[a, b, c, d]’ produces the vector
‘[a, a + b, a + b + c, a + b + c + d]’.
The I V U [
raccum] command does a right-to-left accumulation.
For example, I V U - on the vector ‘[a, b, c, d]’ produces the
vector ‘[a - b + c - d, b - c + d, c - d, d]’.
As for V M, V R normally reduces a matrix elementwise. For
example, given the matrix ‘[[a, b, c], [d, e, f]]’, V R + will
compute ‘a + b + c + d + e + f’. You can type V R _ or
V R : to modify this behavior. The V R _ [
command reduces “across” the matrix; it reduces each row of the matrix
as a vector, then collects the results. Thus V R _ + of this
matrix would produce ‘[a + b + c, d + e + f]’. Similarly, V R :
reduced] reduces down; V R : + would produce ‘[a + d,
b + e, c + f]’.
There is a third “by rows” mode for reduction that is occasionally
useful; V R = [
reducer] simply reduces the operator over
the rows of the matrix themselves. Thus V R = + on the above
matrix would get the same result as V R : +, since adding two
row vectors is equivalent to adding their elements. But V R = *
would multiply the two rows (to get a single number, their dot product),
while V R : * would produce a vector of the products of the columns.
These three matrix reduction modes work with V R and I V R, but they are not currently supported with V U or I V U.
The obsolete reduce-by-columns function,
reducec, is still
supported but there is no way to get it through the V R command.
The commands C-x * : and C-x * _ are equivalent to typing C-x * r to grab a rectangle of data into Calc, and then typing V R : + or V R _ +, respectively, to sum the columns or rows of the matrix. See Grabbing From Buffers.