The v l (
vlen] command computes the
length of a vector. The length of a non-vector is considered to be zero.
Note that matrices are just vectors of vectors for the purposes of this
With the Hyperbolic flag, H v l [
mdims] computes a vector
of the dimensions of a vector, matrix, or higher-order object. For
example, ‘mdims([[a,b,c],[d,e,f]])’ returns ‘[2, 3]’ since
its argument is a
The v f (
find] command searches
along a vector for the first element equal to a given target. The target
is on the top of the stack; the vector is in the second-to-top position.
If a match is found, the result is the index of the matching element.
Otherwise, the result is zero. The numeric prefix argument, if given,
allows you to select any starting index for the search.
The v a (
rearranges a vector to have a certain number of columns and rows. The
numeric prefix argument specifies the number of columns; if you do not
provide an argument, you will be prompted for the number of columns.
The vector or matrix on the top of the stack is flattened into a
plain vector. If the number of columns is nonzero, this vector is
then formed into a matrix by taking successive groups of n elements.
If the number of columns does not evenly divide the number of elements
in the vector, the last row will be short and the result will not be
suitable for use as a matrix. For example, with the matrix
‘[[1, 2], [3, 4]]’ on the stack, v a 4 produces
‘[[1, 2, 3, 4]]’ (a
matrix), v a 1 produces ‘[, , , ]’ (a
matrix), v a 2 produces ‘[[1, 2], [3, 4]]’ (the original
matrix), v a 3 produces ‘[[1, 2, 3], ]’ (not a
matrix), and v a 0 produces the flattened list
‘[1, 2, 3, 4]’.
The V S (
sort] command sorts the elements of
a vector into increasing order. Real numbers, real infinities, and
constant interval forms come first in this ordering; next come other
kinds of numbers, then variables (in alphabetical order), then finally
come formulas and other kinds of objects; these are sorted according
to a kind of lexicographic ordering with the useful property that
one vector is less or greater than another if the first corresponding
unequal elements are less or greater, respectively. Since quoted strings
are stored by Calc internally as vectors of ASCII character codes
(see Strings), this means vectors of strings are also sorted into
alphabetical order by this command.
The I V S [
rsort] command sorts a vector into decreasing order.
The V G (
produces an index table or permutation vector which, if applied to the
input vector (as the index of C-u v r, say), would sort the vector.
A permutation vector is just a vector of integers from 1 to n, where
each integer occurs exactly once. One application of this is to sort a
matrix of data rows using one column as the sort key; extract that column,
grade it with V G, then use the result to reorder the original matrix
with C-u v r. Another interesting property of the
V G command
is that, if the input is itself a permutation vector, the result will
be the inverse of the permutation. The inverse of an index table is
a rank table, whose kth element says where the kth original
vector element will rest when the vector is sorted. To get a rank
table, just use V G V G.
With the Inverse flag, I V G produces an index table that would sort the input into decreasing order. Note that V S and V G use a “stable” sorting algorithm, i.e., any two elements which are equal will not be moved out of their original order. Generally there is no way to tell with V S, since two elements which are equal look the same, but with V G this can be an important issue. In the matrix-of-rows example, suppose you have names and telephone numbers as two columns and you wish to sort by phone number primarily, and by name when the numbers are equal. You can sort the data matrix by names first, and then again by phone numbers. Because the sort is stable, any two rows with equal phone numbers will remain sorted by name even after the second sort.
The V H (
histogram] command builds a
histogram of a vector of numbers. Vector elements are assumed to be
integers or real numbers in the range [0..n) for some “number of
bins” n, which is the numeric prefix argument given to the
command. The result is a vector of n counts of how many times
each value appeared in the original vector. Non-integers in the input
are rounded down to integers. Any vector elements outside the specified
range are ignored. (You can tell if elements have been ignored by noting
that the counts in the result vector don’t add up to the length of the
If no prefix is given, then you will be prompted for a vector which will be used to determine the bins. (If a positive integer is given at this prompt, it will be still treated as if it were given as a prefix.) Each bin will consist of the interval of numbers closest to the corresponding number of this new vector; if the vector ‘[a, b, c, ...]’ is entered at the prompt, the bins will be ‘(-inf, (a+b)/2]’, ‘((a+b)/2, (b+c)/2]’, etc. The result of this command will be a vector counting how many elements of the original vector are in each bin.
The result will then be a vector with the same length as this new vector; each element of the new vector will be replaced by the number of elements of the original vector which are closest to it.
With the Hyperbolic flag, H V H pulls two vectors from the stack. The second-to-top vector is the list of numbers as before. The top vector is an equal-sized list of “weights” to attach to the elements of the data vector. For example, if the first data element is 4.2 and the first weight is 10, then 10 will be added to bin 4 of the result vector. Without the hyperbolic flag, every element has a weight of one.
The v t (
trn] command computes
the transpose of the matrix at the top of the stack. If the argument
is a plain vector, it is treated as a row vector and transposed into
a one-column matrix.
The v v (
rev] command reverses
a vector end-for-end. Given a matrix, it reverses the order of the rows.
(To reverse the columns instead, just use v t v v v t. The same
principle can be used to apply other vector commands to the columns of
The v m (
vmask] command uses
one vector as a mask to extract elements of another vector. The mask
is in the second-to-top position; the target vector is on the top of
the stack. These vectors must have the same length. The result is
the same as the target vector, but with all elements which correspond
to zeros in the mask vector deleted. Thus, for example,
‘vmask([1, 0, 1, 0, 1], [a, b, c, d, e])’ produces ‘[a, c, e]’.
See Logical Operations.
The v e (
expands a vector according to another mask vector. The result is a
vector the same length as the mask, but with nonzero elements replaced
by successive elements from the target vector. The length of the target
vector is normally the number of nonzero elements in the mask. If the
target vector is longer, its last few elements are lost. If the target
vector is shorter, the last few nonzero mask elements are left
unreplaced in the result. Thus ‘vexp([2, 0, 3, 0, 7], [a, b])’
produces ‘[a, 0, b, 0, 7]’.
With the Hyperbolic flag, H v e takes a filler value from the top of the stack; the mask and target vectors come from the third and second elements of the stack. This filler is used where the mask is zero: ‘vexp([2, 0, 3, 0, 7], [a, b], z)’ produces ‘[a, z, c, z, 7]’. If the filler value is itself a vector, then successive values are taken from it, so that the effect is to interleave two vectors according to the mask: ‘vexp([2, 0, 3, 7, 0, 0], [a, b], [x, y])’ produces ‘[a, x, b, 7, y, 0]’.
Another variation on the masking idea is to combine ‘[a, b, c, d, e]’
with the mask ‘[1, 0, 1, 0, 1]’ to produce ‘[a, 0, c, 0, e]’.
You can accomplish this with V M a &, mapping the logical “and”
operation across the two vectors. See Logical Operations. Note that
? : operation also discussed there allows other types of
masking using vectors.