calc: Reducing
10.8.3 Reducing
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The ‘V R’ (‘calc-reduce’) [‘reduce’] command applies a given binary
operator across all the elements of a vector. A binary operator is a
function such as ‘+’ or ‘max’ which takes two arguments. For example,
reducing ‘+’ 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’ (‘calc-accumulate’) [‘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 ‘+’ over 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 _’ [‘reducea’] 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.
Grabbing From Buffers.
Info Catalog
calc: Mapping
calc: Reducing and Mapping
calc: Nesting and Fixed Points