The V M (calc-map) [map] command applies a given
operator elementwise to one or more vectors. For example, mapping
A [abs] produces a vector of the absolute values of the
elements in the input vector. Mapping + pops two vectors from
the stack, which must be of equal length, and produces a vector of the
pairwise sums of the elements. If either argument is a non-vector, it
is duplicated for each element of the other vector. For example,
[1,2,3] 2 V M ^ squares the elements of the specified vector.
With the 2 listed first, it would have computed a vector of powers of
two. Mapping a user-defined function pops as many arguments from the
stack as the function requires. If you give an undefined name, you will
be prompted for the number of arguments to use.
If any argument to V M is a matrix, the operator is normally mapped across all elements of the matrix. For example, given the matrix [[1, -2, 3], [-4, 5, -6]], V M A takes six absolute values to produce another @c{$3\times2$} 3x2 matrix, [[1, 2, 3], [4, 5, 6]].
The command V M _ [mapr] (i.e., type an underscore at the
operator prompt) maps by rows instead. For example, V M _ A views
the above matrix as a vector of two 3-element row vectors. It produces
a new vector which contains the absolute values of those row vectors,
namely [3.74, 8.77]. (Recall, the absolute value of a vector is
defined as the square root of the sum of the squares of the elements.)
Some operators accept vectors and return new vectors; for example,
v v reverses a vector, so V M _ v v would reverse each row
of the matrix to get a new matrix, [[3, -2, 1], [-6, 5, -4]].
Sometimes a vector of vectors (representing, say, strings, sets, or lists) happens to look like a matrix. If so, remember to use V M _ if you want to map a function across the whole strings or sets rather than across their individual elements.
The command V M : [mapc] maps by columns. Basically, it
transposes the input matrix, maps by rows, and then, if the result is a
matrix, transposes again. For example, V M : A takes the absolute
values of the three columns of the matrix, treating each as a 2-vector,
and V M : v v reverses the columns to get the matrix
[[-4, 5, -6], [1, -2, 3]].
(The symbols _ and : were chosen because they had row-like and column-like appearances, and were not already taken by useful operators. Also, they appear shifted on most keyboards so they are easy to type after V M.)
The _ and : modifiers have no effect on arguments that are not matrices (so if none of the arguments are matrices, they have no effect at all). If some of the arguments are matrices and others are plain numbers, the plain numbers are held constant for all rows of the matrix (so that 2 V M _ ^ squares every row of a matrix; squaring a vector takes a dot product of the vector with itself).
If some of the arguments are vectors with the same lengths as the rows (for V M _) or columns (for V M :) of the matrix arguments, those vectors are also held constant for every row or column.
Sometimes it is useful to specify another mapping command as the operator to use with V M. For example, V M _ V A + applies V A + to each row of the input matrix, which in turn adds the two values on that row. If you give another vector-operator command as the operator for V M, it automatically uses map-by-rows mode if you don't specify otherwise; thus V M V A + is equivalent to V M _ V A +. (If you really want to map-by-elements another mapping command, you can use a triple-nested mapping command: V M V M V A + means to map V M V A + over the rows of the matrix; in turn, V A + is mapped over the elements of each row.)
Previous versions of Calc had "map across" and "map down" modes
that are now considered obsolete; the old "map across" is now simply
V M V A, and "map down" is now V M : V A. The algebraic
functions mapa and mapd are still supported, though.
Note also that, while the old mapping modes were persistent (once you
set the mode, it would apply to later mapping commands until you reset
it), the new : and _ modifiers apply only to the current
mapping command. The default V M always means map-by-elements.
See section Algebraic Manipulation, for the a M command, which is like V M but for equations and inequalities instead of vectors. See section Storing Variables, for the s m command which modifies a variable's stored value using a V M-like operator.