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Many of the commands described here begin with the `v` prefix.
(For convenience, the shift-`V` prefix is equivalent to `v`.)
The commands usually apply to both plain vectors and matrices; some
apply only to matrices or only to square matrices. If the argument
has the wrong dimensions the operation is left in symbolic form.

Vectors are entered and displayed using ``[a,b,c]`' notation.
Matrices are vectors of which all elements are vectors of equal length.
(Though none of the standard Calc commands use this concept, a
three-dimensional matrix or rank-3 tensor could be defined as a
vector of matrices, and so on.)

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Calc's "pack" and "unpack" commands collect stack entries to build composite objects such as vectors and complex numbers. They are described in this chapter because they are most often used to build vectors.

The `v p` (`calc-pack`

) [`pack`

] command collects several
elements from the stack into a matrix, complex number, HMS form, error
form, etc. It uses a numeric prefix argument to specify the kind of
object to be built; this argument is referred to as the "packing mode."
If the packing mode is a nonnegative integer, a vector of that
length is created. For example, `C-u 5 v p` will pop the top
five stack elements and push back a single vector of those five
elements. (`C-u 0 v p` simply creates an empty vector.)

The same effect can be had by pressing `[` to push an incomplete
vector on the stack, using `TAB` (`calc-roll-down`

) to sneak
the incomplete object up past a certain number of elements, and
then pressing `]` to complete the vector.

Negative packing modes create other kinds of composite objects:

- -1
- Two values are collected to build a complex number. For example,
`5`creates the complex number (5, 7). The result is always a rectangular complex number. The two input values must both be real numbers, i.e., integers, fractions, or floats. If they are not, Calc will instead build a formula like ``RET`7 C-u -1 v p`a + (0, 1) b`'. (The other packing modes also create a symbolic answer if the components are not suitable.) - -2
- Two values are collected to build a polar complex number.
The first is the magnitude; the second is the phase expressed
in either degrees or radians according to the current angular
mode.
- -3
- Three values are collected into an HMS form. The first
two values (hours and minutes) must be integers or
integer-valued floats. The third value may be any real
number.
- -4
- Two values are collected into an error form. The inputs
may be real numbers or formulas.
- -5
- Two values are collected into a modulo form. The inputs
must be real numbers.
- -6
- Two values are collected into the interval `
`[a .. b]`'. The inputs may be real numbers, HMS or date forms, or formulas. - -7
- Two values are collected into the interval `
`[a .. b)`'. - -8
- Two values are collected into the interval `
`(a .. b]`'. - -9
- Two values are collected into the interval `
`(a .. b)`'. - -10
- Two integer values are collected into a fraction.
- -11
- Two values are collected into a floating-point number.
The first is the mantissa; the second, which must be an
integer, is the exponent. The result is the mantissa
times ten to the power of the exponent.
- -12
- This is treated the same as
*-11*by the`v p`command. When unpacking,*-12*specifies that a floating-point mantissa is desired. - -13
- A real number is converted into a date form.
- -14
- Three numbers (year, month, day) are packed into a pure date form.
- -15
- Six numbers are packed into a date/time form.

With any of the two-input negative packing modes, either or both
of the inputs may be vectors. If both are vectors of the same
length, the result is another vector made by packing corresponding
elements of the input vectors. If one input is a vector and the
other is a plain number, the number is packed along with each vector
element to produce a new vector. For example, `C-u -4 v p`
could be used to convert a vector of numbers and a vector of errors
into a single vector of error forms; `C-u -5 v p` could convert
a vector of numbers and a single number `M` into a vector of
numbers modulo `M`.

If you don't give a prefix argument to `v p`, it takes
the packing mode from the top of the stack. The elements to
be packed then begin at stack level 2. Thus
`1 RET 2 RET 4 n v p` is another way to
enter the error form `

If the packing mode taken from the stack is a vector, the result is a
matrix with the dimensions specified by the elements of the vector,
which must each be integers. For example, if the packing mode is
``[2, 3]`', then six numbers will be taken from the stack and
returned in the form ``[[a, b, c], [d, e, f]]`'.

If any elements of the vector are negative, other kinds of
packing are done at that level as described above. For
example, ``[2, 3, -4]`' takes 12 objects and creates a
2x3 matrix of error forms: ``[[a +/- b, c +/- d ... ]]`'.
Also, ``[-4, -10]`' will convert four integers into an
error form consisting of two fractions: ``a:b +/- c:d`'.

There is an equivalent algebraic function,
``pack( mode, items)`' where

The `v u` (`calc-unpack`

) command takes the vector, complex
number, HMS form, or other composite object on the top of the stack and
"unpacks" it, pushing each of its elements onto the stack as separate
objects. Thus, it is the "inverse" of `v p`. If the value
at the top of the stack is a formula, `v u` unpacks it by pushing
each of the arguments of the top-level operator onto the stack.

You can optionally give a numeric prefix argument to `v u`
to specify an explicit (un)packing mode. If the packing mode is
negative and the input is actually a vector or matrix, the result
will be two or more similar vectors or matrices of the elements.
For example, given the vector ``[a +/- b, c^2, d +/- 7]`',
the result of `C-u -4 v u` will be the two vectors
``[a, c^2, d]`' and ``[b, 0, 7]`'.

Note that the prefix argument can have an effect even when the input is
not a vector. For example, if the input is the number *-5*, then
`c-u -1 v u` yields *-5* and 0 (the components of *-5*
when viewed as a rectangular complex number); `C-u -2 v u` yields 5
and 180 (assuming degrees mode); and `C-u -10 v u` yields *-5*
and 1 (the numerator and denominator of *-5*, viewed as a rational
number). Plain `v u` with this input would complain that the input
is not a composite object.

Unpacking mode *-11* converts a float into an integer mantissa and
an integer exponent, where the mantissa is not divisible by 10
(except that 0.0 is represented by a mantissa and exponent of 0).
Unpacking mode *-12* converts a float into a floating-point mantissa
and integer exponent, where the mantissa (for non-zero numbers)
is guaranteed to lie in the range [1 .. 10). In both cases,
the mantissa is shifted left or right (and the exponent adjusted
to compensate) in order to satisfy these constraints.

Positive unpacking modes are treated differently than for `v p`.
A mode of 1 is much like plain `v u` with no prefix argument,
except that in addition to the components of the input object,
a suitable packing mode to re-pack the object is also pushed.
Thus, `C-u 1 v u` followed by `v p` will re-build the
original object.

A mode of 2 unpacks two levels of the object; the resulting re-packing mode will be a vector of length 2. This might be used to unpack a matrix, say, or a vector of error forms. Higher unpacking modes unpack the input even more deeply.

There are two algebraic functions analogous to `v u`.
The ``unpack( mode, item)`' function unpacks the

The `unpackt`

function is like `unpack`

but instead
of returning a simple vector of items, it returns a vector of
two things: The mode, and the vector of items. For example,
``unpackt(1, 2:3 +/- 1:4)`' returns ``[-4, [2:3, 1:4]]`',
and ``unpackt(2, 2:3 +/- 1:4)`' returns ``[[-4, -10], [2, 3, 1, 4]]`'.
The identity for re-building the original object is
``apply(pack, unpackt( n, x)) = x`'. (The

`apply`

function builds a function call given the function
name and a vector of arguments.)
Subscript notation is a useful way to extract a particular part
of an object. For example, to get the numerator of a rational
number, you can use ``unpack(-10, x)_1`'.

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Vectors and matrices can be added, subtracted, multiplied, and divided; see section 9.1 Basic Arithmetic.

The `|` (`calc-concat`

) command "concatenates" two vectors
into one. For example, after `[ 1 , 2 ] [ 3 , 4 ] |`, the stack
will contain the single vector ``[1, 2, 3, 4]`'. If the arguments
are matrices, the rows of the first matrix are concatenated with the
rows of the second. (In other words, two matrices are just two vectors
of row-vectors as far as `|` is concerned.)

If either argument to `|` is a scalar (a non-vector), it is treated
like a one-element vector for purposes of concatenation: `1 [ 2 , 3 ] |`
produces the vector ``[1, 2, 3]`'. Likewise, if one argument is a
matrix and the other is a plain vector, the vector is treated as a
one-row matrix.

The `H |` (`calc-append`

) [`append`

] command concatenates
two vectors without any special cases. Both inputs must be vectors.
Whether or not they are matrices is not taken into account. If either
argument is a scalar, the `append`

function is left in symbolic form.
See also `cons`

and `rcons`

below.

The `I |` and `H I |` commands are similar, but they use their
two stack arguments in the opposite order. Thus `I |` is equivalent
to `TAB |`, but possibly more convenient and also a bit faster.

The `v d` (`calc-diag`

) [`diag`

] function builds a diagonal
square matrix. The optional numeric prefix gives the number of rows
and columns in the matrix. If the value at the top of the stack is a
vector, the elements of the vector are used as the diagonal elements; the
prefix, if specified, must match the size of the vector. If the value on
the stack is a scalar, it is used for each element on the diagonal, and
the prefix argument is required.

To build a constant square matrix, e.g., a
3x3 matrix filled with ones,
use `0 M-3 v d 1 +`, i.e., build a zero matrix first and then add a
constant value to that matrix. (Another alternative would be to use
`v b` and `v a`; see below.)

The `v i` (`calc-ident`

) [`idn`

] function builds an identity
matrix of the specified size. It is a convenient form of `v d`
where the diagonal element is always one. If no prefix argument is given,
this command prompts for one.

In algebraic notation, ``idn(a,n)`' acts much like ``diag(a,n)`',
except that a is required to be a scalar (non-vector) quantity.
If n is omitted, ``idn(a)`' represents a times an
identity matrix of unknown size. Calc can operate algebraically on
such generic identity matrices, and if one is combined with a matrix
whose size is known, it is converted automatically to an identity
matrix of a suitable matching size. The `v i` command with an
argument of zero creates a generic identity matrix, ``idn(1)`'.
Note that in dimensioned matrix mode (see section 8.4.6 Matrix and Scalar Modes), generic
identity matrices are immediately expanded to the current default
dimensions.

The `v x` (`calc-index`

) [`index`

] function builds a vector
of consecutive integers from 1 to `n`, where `n` is the numeric
prefix argument. If you do not provide a prefix argument, you will be
prompted to enter a suitable number. If `n` is negative, the result
is a vector of negative integers from `n` to *-1*.

With a prefix argument of just `C-u`, the `v x` command takes
three values from the stack: `n`, `start`, and `incr` (with
`incr` at top-of-stack). Counting starts at `start` and increases
by `incr` for successive vector elements. If `start` or `n`
is in floating-point format, the resulting vector elements will also be
floats. Note that `start` and `incr` may in fact be any kind
of numbers or formulas.

When `start` and `incr` are specified, a negative `n` has a
different interpretation: It causes a geometric instead of arithmetic
sequence to be generated. For example, ``index(-3, a, b)`' produces
``[a, a b, a b^2]`'. If you omit `incr` in the algebraic form,
``index( n, start)`', the default value for

The `v b` (`calc-build-vector`

) [`cvec`

] function builds a
vector of `n` copies of the value on the top of the stack, where `n`
is the numeric prefix argument. In algebraic formulas, ``cvec(x,n,m)`'
can also be used to build an `n`-by-`m` matrix of copies of `x`.
(Interactively, just use `v b` twice: once to build a row, then again
to build a matrix of copies of that row.)

The `v h` (`calc-head`

) [`head`

] function returns the first
element of a vector. The `I v h` (`calc-tail`

) [`tail`

]
function returns the vector with its first element removed. In both
cases, the argument must be a non-empty vector.

The `v k` (`calc-cons`

) [`cons`

] function takes a value `h`
and a vector `t` from the stack, and produces the vector whose head is
`h` and whose tail is `t`. This is similar to `|`, except
if `h` is itself a vector, `|` will concatenate the two vectors
whereas `cons`

will insert `h` at the front of the vector `t`.

Each of these three functions also accepts the Hyperbolic flag [`rhead`

,
`rtail`

, `rcons`

] in which case `t` instead represents
the *last* single element of the vector, with `h`
representing the remainder of the vector. Thus the vector
``[a, b, c, d] = cons(a, [b, c, d]) = rcons([a, b, c], d)`'.
Also, ``head([a, b, c, d]) = a`', ``tail([a, b, c, d]) = [b, c, d]`',
``rhead([a, b, c, d]) = [a, b, c]`', and ``rtail([a, b, c, d]) = d`'.

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The `v r` (`calc-mrow`

) [`mrow`

] command extracts one row of
the matrix on the top of the stack, or one element of the plain vector on
the top of the stack. The row or element is specified by the numeric
prefix argument; the default is to prompt for the row or element number.
The matrix or vector is replaced by the specified row or element in the
form of a vector or scalar, respectively.

With a prefix argument of `C-u` only, `v r` takes the index of
the element or row from the top of the stack, and the vector or matrix
from the second-to-top position. If the index is itself a vector of
integers, the result is a vector of the corresponding elements of the
input vector, or a matrix of the corresponding rows of the input matrix.
This command can be used to obtain any permutation of a vector.

With `C-u`, if the index is an interval form with integer components,
it is interpreted as a range of indices and the corresponding subvector or
submatrix is returned.

Subscript notation in algebraic formulas (``a_b`') stands for the
Calc function `subscr`

, which is synonymous with `mrow`

.
Thus, ``[x, y, z]_k`' produces x, y, or z if
k is one, two, or three, respectively. A double subscript
(``M_i_j`', equivalent to ``subscr(subscr(M, i), j)`') will
access the element at row i, column j of a matrix.
The `a _` (`calc-subscript`

) command creates a subscript
formula ``a_b`' out of two stack entries. (It is on the `a`
"algebra" prefix because subscripted variables are often used
purely as an algebraic notation.)

Given a negative prefix argument, `v r` instead deletes one row or
element from the matrix or vector on the top of the stack. Thus
`C-u 2 v r` replaces a matrix with its second row, but `C-u -2 v r`
replaces the matrix with the same matrix with its second row removed.
In algebraic form this function is called `mrrow`

.

Given a prefix argument of zero, `v r` extracts the diagonal elements
of a square matrix in the form of a vector. In algebraic form this
function is called `getdiag`

.

The `v c` (`calc-mcol`

) [`mcol`

or `mrcol`

] command is
the analogous operation on columns of a matrix. Given a plain vector
it extracts (or removes) one element, just like `v r`. If the
index in `C-u v c` is an interval or vector and the argument is a
matrix, the result is a submatrix with only the specified columns
retained (and possibly permuted in the case of a vector index).

To extract a matrix element at a given row and column, use `v r` to
extract the row as a vector, then `v c` to extract the column element
from that vector. In algebraic formulas, it is often more convenient to
use subscript notation: ``m_i_j`' gives row i, column j
of matrix m.

The `v s` (`calc-subvector`

) [`subvec`

] command extracts
a subvector of a vector. The arguments are the vector, the starting
index, and the ending index, with the ending index in the top-of-stack
position. The starting index indicates the first element of the vector
to take. The ending index indicates the first element *past* the
range to be taken. Thus, ``subvec([a, b, c, d, e], 2, 4)`' produces
the subvector ``[b, c]`'. You could get the same result using
``mrow([a, b, c, d, e], [2 .. 4))`'.

If either the start or the end index is zero or negative, it is
interpreted as relative to the end of the vector. Thus
``subvec([a, b, c, d, e], 2, -2)`' also produces ``[b, c]`'. In
the algebraic form, the end index can be omitted in which case it
is taken as zero, i.e., elements from the starting element to the
end of the vector are used. The infinity symbol, `inf`

, also
has this effect when used as the ending index.

With the Inverse flag, `I v s` [`rsubvec`

] removes a subvector
from a vector. The arguments are interpreted the same as for the
normal `v s` command. Thus, ``rsubvec([a, b, c, d, e], 2, 4)`'
produces ``[a, d, e]`'. It is always true that `subvec`

and
`rsubvec`

return complementary parts of the input vector.

See section 12.1 Selecting Sub-Formulas, for an alternative way to operate on vectors one element at a time.

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The `v l` (`calc-vlength`

) [`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
command.

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
2x3 matrix.

The `v f` (`calc-vector-find`

) [`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` (`calc-arrange-vector`

) [`arrange`

] command
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
1x4 matrix), `v a 1` produces
``[[1], [2], [3], [4]]`' (a
4x1 matrix), `v a 2` produces
``[[1, 2], [3, 4]]`' (the original
2x2 matrix), `v a 3` produces
``[[1, 2, 3], [4]]`' (not a matrix), and `v a 0` produces
the flattened list ``[1, 2, 3, 4]`'.

The `V S` (`calc-sort`

) [`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 section 6.7 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` (`calc-grade`

) [`grade`

, `rgrade`

] command
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 `k`th element says where the `k`th 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` (`calc-histogram`

) [`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
input vector.)

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` (`calc-transpose`

) [`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` (`calc-reverse-vector`

) [`vec`

] 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
a matrix.)

The `v m` (`calc-mask-vector`

) [`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 section 12.10 Logical Operations.

The `v e` (`calc-expand-vector`

) [`vexp`

] command
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 section 12.10 Logical Operations. Note that
the `? :`

operation also discussed there allows other types of
masking using vectors.

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Basic arithmetic operations like addition and multiplication are defined for vectors and matrices as well as for numbers. Division of matrices, in the sense of multiplying by the inverse, is supported. (Division by a matrix actually uses LU-decomposition for greater accuracy and speed.) See section 9.1 Basic Arithmetic.

The following functions are applied element-wise if their arguments are
vectors or matrices: `change-sign`

, `conj`

, `arg`

,
`re`

, `im`

, `polar`

, `rect`

, `clean`

,
`float`

, `frac`

. See section Index of Algebraic Functions.

The `V J` (`calc-conj-transpose`

) [`ctrn`

] command computes
the conjugate transpose of its argument, i.e., ``conj(trn(x))`'.

The `A` (`calc-abs`

) [`abs`

] command computes the
Frobenius norm of a vector or matrix argument. This is the square
root of the sum of the squares of the absolute values of the
elements of the vector or matrix. If the vector is interpreted as
a point in two- or three-dimensional space, this is the distance
from that point to the origin.

The `v n` (`calc-rnorm`

) [`rnorm`

] command computes
the row norm, or infinity-norm, of a vector or matrix. For a plain
vector, this is the maximum of the absolute values of the elements.
For a matrix, this is the maximum of the row-absolute-value-sums,
i.e., of the sums of the absolute values of the elements along the
various rows.

The `V N` (`calc-cnorm`

) [`cnorm`

] command computes
the column norm, or one-norm, of a vector or matrix. For a plain
vector, this is the sum of the absolute values of the elements.
For a matrix, this is the maximum of the column-absolute-value-sums.
General k-norms for k other than one or infinity are
not provided.

The `V C` (`calc-cross`

) [`cross`

] command computes the
right-handed cross product of two vectors, each of which must have
exactly three elements.

The `&` (`calc-inv`

) [`inv`

] command computes the
inverse of a square matrix. If the matrix is singular, the inverse
operation is left in symbolic form. Matrix inverses are recorded so
that once an inverse (or determinant) of a particular matrix has been
computed, the inverse and determinant of the matrix can be recomputed
quickly in the future.

If the argument to `&` is a plain number x, this
command simply computes 1/x. This is okay, because the
``/`' operator also does a matrix inversion when dividing one
by a matrix.

The `V D` (`calc-mdet`

) [`det`

] command computes the
determinant of a square matrix.

The `V L` (`calc-mlud`

) [`lud`

] command computes the
LU decomposition of a matrix. The result is a list of three matrices
which, when multiplied together left-to-right, form the original matrix.
The first is a permutation matrix that arises from pivoting in the
algorithm, the second is lower-triangular with ones on the diagonal,
and the third is upper-triangular.

The `V T` (`calc-mtrace`

) [`tr`

] command computes the
trace of a square matrix. This is defined as the sum of the diagonal
elements of the matrix.

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Calc includes several commands which interpret vectors as *sets* of
objects. A set is a collection of objects; any given object can appear
only once in the set. Calc stores sets as vectors of objects in
sorted order. Objects in a Calc set can be any of the usual things,
such as numbers, variables, or formulas. Two set elements are considered
equal if they are identical, except that numerically equal numbers like
the integer 4 and the float 4.0 are considered equal even though they
are not "identical." Variables are treated like plain symbols without
attached values by the set operations; subtracting the set ``[b]`'
from ``[a, b]`' always yields the set ``[a]`' even though if
the variables ``a`' and ``b`' both equalled 17, you might
expect the answer ``[]`'.

If a set contains interval forms, then it is assumed to be a set of real numbers. In this case, all set operations require the elements of the set to be only things that are allowed in intervals: Real numbers, plus and minus infinity, HMS forms, and date forms. If there are variables or other non-real objects present in a real set, all set operations on it will be left in unevaluated form.

If the input to a set operation is a plain number or interval form
`a`, it is treated like the one-element vector ``[ a]`'.
The result is always a vector, except that if the set consists of a
single interval, the interval itself is returned instead.

See section 12.10 Logical Operations, for the `in`

function which tests if
a certain value is a member of a given set. To test if the set A
is a subset of the set B, use ``vdiff(A, B) = []`'.

The `V +` (`calc-remove-duplicates`

) [`rdup`

] command
converts an arbitrary vector into set notation. It works by sorting
the vector as if by `V S`, then removing duplicates. (For example,
`[a, 5, 4, a, 4.0]` is sorted to ``[4, 4.0, 5, a, a]`' and then
reduced to ``[4, 5, a]`'). Overlapping intervals are merged as
necessary. You rarely need to use `V +` explicitly, since all the
other set-based commands apply `V +` to their inputs before using
them.

The `V V` (`calc-set-union`

) [`vunion`

] command computes
the union of two sets. An object is in the union of two sets if and
only if it is in either (or both) of the input sets. (You could
accomplish the same thing by concatenating the sets with `|`,
then using `V +`.)

The `V ^` (`calc-set-intersect`

) [`vint`

] command computes
the intersection of two sets. An object is in the intersection if
and only if it is in both of the input sets. Thus if the input
sets are disjoint, i.e., if they share no common elements, the result
will be the empty vector ``[]`'. Note that the characters `V`
and `^` were chosen to be close to the conventional mathematical
notation for set union
and intersection
.

The `V -` (`calc-set-difference`

) [`vdiff`

] command computes
the difference between two sets. An object is in the difference
A - B if and only if it is in A but not in B.
Thus subtracting ``[y,z]`' from a set will remove the elements
``y`' and ``z`' if they are present. You can also think of this
as a general *set complement* operator; if A is the set of
all possible values, then A - B is the "complement" of B.
Obviously this is only practical if the set of all possible values in
your problem is small enough to list in a Calc vector (or simple
enough to express in a few intervals).

The `V X` (`calc-set-xor`

) [`vxor`

] command computes
the "exclusive-or," or "symmetric difference" of two sets.
An object is in the symmetric difference of two sets if and only
if it is in one, but *not* both, of the sets. Objects that
occur in both sets "cancel out."

The `V ~` (`calc-set-complement`

) [`vcompl`

] command
computes the complement of a set with respect to the real numbers.
Thus ``vcompl(x)`' is equivalent to ``vdiff([-inf .. inf], x)`'.
For example, ``vcompl([2, (3 .. 4]])`' evaluates to
``[[-inf .. 2), (2 .. 3], (4 .. inf]]`'.

The `V F` (`calc-set-floor`

) [`vfloor`

] command
reinterprets a set as a set of integers. Any non-integer values,
and intervals that do not enclose any integers, are removed. Open
intervals are converted to equivalent closed intervals. Successive
integers are converted into intervals of integers. For example, the
complement of the set ``[2, 6, 7, 8]`' is messy, but if you wanted
the complement with respect to the set of integers you could type
`V ~ V F` to get ``[[-inf .. 1], [3 .. 5], [9 .. inf]]`'.

The `V E` (`calc-set-enumerate`

) [`venum`

] command
converts a set of integers into an explicit vector. Intervals in
the set are expanded out to lists of all integers encompassed by
the intervals. This only works for finite sets (i.e., sets which
do not involve ``-inf`' or ``inf`').

The `V :` (`calc-set-span`

) [`vspan`

] command converts any
set of reals into an interval form that encompasses all its elements.
The lower limit will be the smallest element in the set; the upper
limit will be the largest element. For an empty set, ``vspan([])`'
returns the empty interval ``[0 .. 0)`'.

The `V #` (`calc-set-cardinality`

) [`vcard`

] command counts
the number of integers in a set. The result is the length of the vector
that would be produced by `V E`, although the computation is much
more efficient than actually producing that vector.

Another representation for sets that may be more appropriate in some
cases is binary numbers. If you are dealing with sets of integers
in the range 0 to 49, you can use a 50-bit binary number where a
particular bit is 1 if the corresponding element is in the set.
See section 9.7 Binary Number Functions, for a list of commands that operate on
binary numbers. Note that many of the above set operations have
direct equivalents in binary arithmetic: `b o` (`calc-or`

),
`b a` (`calc-and`

), `b d` (`calc-diff`

),
`b x` (`calc-xor`

), and `b n` (`calc-not`

),
respectively. You can use whatever representation for sets is most
convenient to you.

The `b u` (`calc-unpack-bits`

) [`vunpack`

] command
converts an integer that represents a set in binary into a set
in vector/interval notation. For example, ``vunpack(67)`'
returns ``[[0 .. 1], 6]`'. If the input is negative, the set
it represents is semi-infinite: ``vunpack(-4) = [2 .. inf)`'.
Use `V E` afterwards to expand intervals to individual
values if you wish. Note that this command uses the `b`
(binary) prefix key.

The `b p` (`calc-pack-bits`

) [`vpack`

] command
converts the other way, from a vector or interval representing
a set of nonnegative integers into a binary integer describing
the same set. The set may include positive infinity, but must
not include any negative numbers. The input is interpreted as a
set of integers in the sense of `V F` (`vfloor`

). Beware
that a simple input like ``[100]`' can result in a huge integer
representation (
2^100, a 31-digit integer, in this case).

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The commands in this section take vectors as arguments and compute
various statistical measures on the data stored in the vectors. The
references used in the definitions of these functions are Bevington's
*Data Reduction and Error Analysis for the Physical Sciences*,
and *Numerical Recipes* by Press, Flannery, Teukolsky and
Vetterling.

The statistical commands use the `u` prefix key followed by
a shifted letter or other character.

See section 11.4 Manipulating Vectors, for a description of `V H`
(`calc-histogram`

).

See section 12.8 Curve Fitting, for the `a F` command for doing
least-squares fits to statistical data.

See section 10.7 Probability Distribution Functions, for several common probability distribution functions.

11.7.1 Single-Variable Statistics | ||

11.7.2 Paired-Sample Statistics |

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These functions do various statistical computations on single
vectors. Given a numeric prefix argument, they actually pop
`n` objects from the stack and combine them into a data
vector. Each object may be either a number or a vector; if a
vector, any sub-vectors inside it are "flattened" as if by
`v a 0`; see section 11.4 Manipulating Vectors. By default one object
is popped, which (in order to be useful) is usually a vector.

If an argument is a variable name, and the value stored in that variable is a vector, then the stored vector is used. This method has the advantage that if your data vector is large, you can avoid the slow process of manipulating it directly on the stack.

These functions are left in symbolic form if any of their arguments are not numbers or vectors, e.g., if an argument is a formula, or a non-vector variable. However, formulas embedded within vector arguments are accepted; the result is a symbolic representation of the computation, based on the assumption that the formula does not itself represent a vector. All varieties of numbers such as error forms and interval forms are acceptable.

Some of the functions in this section also accept a single error form
or interval as an argument. They then describe a property of the
normal or uniform (respectively) statistical distribution described
by the argument. The arguments are interpreted in the same way as
the `M` argument of the random number function `k r`. In
particular, an interval with integer limits is considered an integer
distribution, so that ``[2 .. 6)`' is the same as ``[2 .. 5]`'.
An interval with at least one floating-point limit is a continuous
distribution: ``[2.0 .. 6.0)`' is *not* the same as
``[2.0 .. 5.0]`'!

The `u #` (`calc-vector-count`

) [`vcount`

] command
computes the number of data values represented by the inputs.
For example, ``vcount(1, [2, 3], [[4, 5], [], x, y])`' returns 7.
If the argument is a single vector with no sub-vectors, this
simply computes the length of the vector.

The `u +` (`calc-vector-sum`

) [`vsum`

] command
computes the sum of the data values. The `u *`
(`calc-vector-prod`

) [`vprod`

] command computes the
product of the data values. If the input is a single flat vector,
these are the same as `V R +` and `V R *`
(see section 11.8 Reducing and Mapping Vectors).

The `u X` (`calc-vector-max`

) [`vmax`

] command
computes the maximum of the data values, and the `u N`
(`calc-vector-min`

) [`vmin`

] command computes the minimum.
If the argument is an interval, this finds the minimum or maximum
value in the interval. (Note that ``vmax([2..6)) = 5`' as
described above.) If the argument is an error form, this returns
plus or minus infinity.

The `u M` (`calc-vector-mean`

) [`vmean`

] command
computes the average (arithmetic mean) of the data values.
If the inputs are error forms
``x +/- s`', this is the weighted
mean of the x values with weights
1 / s^2.
If the inputs are not error forms, this is simply the sum of the
values divided by the count of the values.

Note that a plain number can be considered an error form with
error
s = 0. If the input to `u M` is a mixture of
plain numbers and error forms, the result is the mean of the
plain numbers, ignoring all values with non-zero errors. (By the
above definitions it's clear that a plain number effectively
has an infinite weight, next to which an error form with a finite
weight is completely negligible.)

This function also works for distributions (error forms or
intervals). The mean of an error form `*a* `+/-` *b*' is simply
a. The mean of an interval is the mean of the minimum
and maximum values of the interval.

The `I u M` (`calc-vector-mean-error`

) [`vmeane`

]
command computes the mean of the data points expressed as an
error form. This includes the estimated error associated with
the mean. If the inputs are error forms, the error is the square
root of the reciprocal of the sum of the reciprocals of the squares
of the input errors. (I.e., the variance is the reciprocal of the
sum of the reciprocals of the variances.)
If the inputs are plain
numbers, the error is equal to the standard deviation of the values
divided by the square root of the number of values. (This works
out to be equivalent to calculating the standard deviation and
then assuming each value's error is equal to this standard
deviation.)

The `H u M` (`calc-vector-median`

) [`vmedian`

]
command computes the median of the data values. The values are
first sorted into numerical order; the median is the middle
value after sorting. (If the number of data values is even,
the median is taken to be the average of the two middle values.)
The median function is different from the other functions in
this section in that the arguments must all be real numbers;
variables are not accepted even when nested inside vectors.
(Otherwise it is not possible to sort the data values.) If
any of the input values are error forms, their error parts are
ignored.

The median function also accepts distributions. For both normal (error form) and uniform (interval) distributions, the median is the same as the mean.

The `H I u M` (`calc-vector-harmonic-mean`

) [`vhmean`

]
command computes the harmonic mean of the data values. This is
defined as the reciprocal of the arithmetic mean of the reciprocals
of the values.

The `u G` (`calc-vector-geometric-mean`

) [`vgmean`

]
command computes the geometric mean of the data values. This
is the *N*th root of the product of the values. This is also
equal to the `exp`

of the arithmetic mean of the logarithms
of the data values.

The `H u G` [`agmean`

] command computes the "arithmetic-geometric
mean" of two numbers taken from the stack. This is computed by
replacing the two numbers with their arithmetic mean and geometric
mean, then repeating until the two values converge.

Another commonly used mean, the RMS (root-mean-square), can be computed
for a vector of numbers simply by using the `A` command.

The `u S` (`calc-vector-sdev`

) [`vsdev`

] command
computes the standard deviation
of the data values. If the
values are error forms, the errors are used as weights just
as for `u M`. This is the *sample* standard deviation,
whose value is the square root of the sum of the squares of the
differences between the values and the mean of the N values,
divided by N-1.

This function also applies to distributions. The standard deviation of a single error form is simply the error part. The standard deviation of a continuous interval happens to equal the difference between the limits, divided by sqrt(12). The standard deviation of an integer interval is the same as the standard deviation of a vector of those integers.

The `I u S` (`calc-vector-pop-sdev`

) [`vpsdev`

]
command computes the *population* standard deviation.
It is defined by the same formula as above but dividing
by N instead of by N-1. The population standard
deviation is used when the input represents the entire set of
data values in the distribution; the sample standard deviation
is used when the input represents a sample of the set of all
data values, so that the mean computed from the input is itself
only an estimate of the true mean.

For error forms and continuous intervals, `vpsdev`

works
exactly like `vsdev`

. For integer intervals, it computes the
population standard deviation of the equivalent vector of integers.

The `H u S` (`calc-vector-variance`

) [`vvar`

] and
`H I u S` (`calc-vector-pop-variance`

) [`vpvar`

]
commands compute the variance of the data values. The variance
is the square
of the standard deviation, i.e., the sum of the
squares of the deviations of the data values from the mean.
(This definition also applies when the argument is a distribution.)

The `vflat`

algebraic function returns a vector of its
arguments, interpreted in the same way as the other functions
in this section. For example, ``vflat(1, [2, [3, 4]], 5)`'
returns ``[1, 2, 3, 4, 5]`'.

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The functions in this section take two arguments, which must be vectors of equal size. The vectors are each flattened in the same way as by the single-variable statistical functions. Given a numeric prefix argument of 1, these functions instead take one object from the stack, which must be an Nx2 matrix of data values. Once again, variable names can be used in place of actual vectors and matrices.

The `u C` (`calc-vector-covariance`

) [`vcov`

] command
computes the sample covariance of two vectors. The covariance
of vectors `x` and `y` is the sum of the products of the
differences between the elements of `x` and the mean of `x`
times the differences between the corresponding elements of `y`
and the mean of `y`, all divided by N-1. Note that
the variance of a vector is just the covariance of the vector
with itself. Once again, if the inputs are error forms the
errors are used as weight factors. If both `x` and `y`
are composed of error forms, the error for a given data point
is taken as the square root of the sum of the squares of the two
input errors.

The `I u C` (`calc-vector-pop-covariance`

) [`vpcov`

]
command computes the population covariance, which is the same as the
sample covariance computed by `u C` except dividing by N
instead of N-1.

The `H u C` (`calc-vector-correlation`

) [`vcorr`

]
command computes the linear correlation coefficient of two vectors.
This is defined by the covariance of the vectors divided by the
product of their standard deviations. (There is no difference
between sample or population statistics here.)

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The commands in this section allow for more general operations on the elements of vectors.

The simplest of these operations is `V A` (`calc-apply`

)
[`apply`

], which applies a given operator to the elements of a vector.
For example, applying the hypothetical function `f`

to the vector
``[1, 2, 3]`' would produce the function call ``f(1, 2, 3)`'.
Applying the `+`

function to the vector ``[a, b]`' gives
``a + b`'. Applying `+`

to the vector ``[a, b, c]`' is an
error, since the `+`

function expects exactly two arguments.

While `V A` is useful in some cases, you will usually find that either
`V R` or `V M`, described below, is closer to what you want.

11.8.1 Specifying Operators | ||

11.8.2 Mapping | ||

11.8.3 Reducing | ||

11.8.4 Nesting and Fixed Points | ||

11.8.5 Generalized Products |

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Commands in this section (like `V A`) prompt you to press the key
corresponding to the desired operator. Press `?` for a partial
list of the available operators. Generally, an operator is any key or
sequence of keys that would normally take one or more arguments from
the stack and replace them with a result. For example, `V A H C`
uses the hyperbolic cosine operator, `cosh`

. (Since `cosh`

expects one argument, `V A H C` requires a vector with a single
element as its argument.)

You can press `x` at the operator prompt to select any algebraic
function by name to use as the operator. This includes functions you
have defined yourself using the `Z F` command. (See section 19.4 Programming with Formulas.) If you give a name for which no function has been
defined, the result is left in symbolic form, as in ``f(1, 2, 3)`'.
Calc will prompt for the number of arguments the function takes if it
can't figure it out on its own (say, because you named a function that
is currently undefined). It is also possible to type a digit key before
the function name to specify the number of arguments, e.g.,
`V M 3 x f RET` calls `f`

with three arguments even if it
looks like it ought to have only two. This technique may be necessary
if the function allows a variable number of arguments. For example,
the `v e` [`vexp`

] function accepts two or three arguments;
if you want to map with the three-argument version, you will have to
type `V M 3 v e`.

It is also possible to apply any formula to a vector by treating that
formula as a function. When prompted for the operator to use, press
`'` (the apostrophe) and type your formula as an algebraic entry.
You will then be prompted for the argument list, which defaults to a
list of all variables that appear in the formula, sorted into alphabetic
order. For example, suppose you enter the formula ``x + 2y^x`'.
The default argument list would be ``(x y)`', which means that if
this function is applied to the arguments ``[3, 10]`' the result will
be ``3 + 2*10^3`'. (If you plan to use a certain formula in this
way often, you might consider defining it as a function with `Z F`.)

Another way to specify the arguments to the formula you enter is with
`$`, `$$`, and so on. For example, `V A ' $$ + 2$^$$`
has the same effect as the previous example. The argument list is
automatically taken to be ``($$ $)`'. (The order of the arguments
may seem backwards, but it is analogous to the way normal algebraic
entry interacts with the stack.)

If you press `$` at the operator prompt, the effect is similar to
the apostrophe except that the relevant formula is taken from top-of-stack
instead. The actual vector arguments of the `V A $` or related command
then start at the second-to-top stack position. You will still be
prompted for an argument list.

A function can be written without a name using the notation ``<#1 - #2>`',
which means "a function of two arguments that computes the first
argument minus the second argument." The symbols ``#1`' and ``#2`'
are placeholders for the arguments. You can use any names for these
placeholders if you wish, by including an argument list followed by a
colon: ``<x, y : x - y>`'. When you type `V A ' $$ + 2$^$$ RET`,
Calc builds the nameless function ``<#1 + 2 #2^#1>`' as the function
to map across the vectors. When you type `V A ' x + 2y^x RET RET`,
Calc builds the nameless function ``<x, y : x + 2 y^x>`'. In both
cases, Calc also writes the nameless function to the Trail so that you
can get it back later if you wish.

If there is only one argument, you can write ``#`' in place of ``#1`'.
(Note that ``< >`' notation is also used for date forms. Calc tells
that ``< stuff>`' is a nameless function by the presence of
`

You can type a nameless function directly to `V A '`, or put one on
the stack and use it with `V A $`. Calc will not prompt for an
argument list in this case, since the nameless function specifies the
argument list as well as the function itself. In `V A '`, you can
omit the ``< >`' marks if you use ``#`' notation for the arguments,
so that `V A ' #1+#2 RET` is the same as `V A ' <#1+#2> RET`,
which in turn is the same as `V A ' $$+$ RET`.

The internal format for ``<x, y : x + y>`' is ``lambda(x, y, x + y)`'.
(The word `lambda`

derives from Lisp notation and the theory of
functions.) The internal format for ``<#1 + #2>`' is ``lambda(ArgA,
ArgB, ArgA + ArgB)`'. Note that there is no actual Calc function called
`lambda`

; the whole point is that the `lambda`

expression is
used in its symbolic form, not evaluated for an answer until it is applied
to specific arguments by a command like `V A` or `V M`.

(Actually, `lambda`

does have one special property: Its arguments
are never evaluated; for example, putting ``<(2/3) #>`' on the stack
will not simplify the ``2/3`' until the nameless function is actually
called.)

As usual, commands like `V A` have algebraic function name equivalents.
For example, `V A k g` with an argument of ``v`' is equivalent to
``apply(gcd, v)`'. The first argument specifies the operator name,
and is either a variable whose name is the same as the function name,
or a nameless function like ``<#^3+1>`'. Operators that are normally
written as algebraic symbols have the names `add`

, `sub`

,
`mul`

, `div`

, `pow`

, `neg`

, `mod`

, and
`vconcat`

.

The `call`

function builds a function call out of several arguments:
``call(gcd, x, y)`' is the same as ``apply(gcd, [x, y])`', which
in turn is the same as ``gcd(x, y)`'. The first argument of `call`

,
like the other functions described here, may be either a variable naming a
function, or a nameless function (``call(<#1+2#2>, x, y)`' is the same
as ``x + 2y`').

(Experts will notice that it's not quite proper to use a variable to name
a function, since the name `gcd`

corresponds to the Lisp variable
`var-gcd`

but to the Lisp function `calcFunc-gcd`

. Calc
automatically makes this translation, so you don't have to worry
about it.)

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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
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 12.2 Algebraic Manipulation, for the `a M` command, which is like
`V M` but for equations and inequalities instead of vectors.
See section 14.1 Storing Variables, for the `s m` command which modifies a
variable's stored value using a `V M`-like operator.

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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 `M-# :` and `M-# _` are equivalent to typing
`M-# 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 section 16.3 Grabbing from Other Buffers.

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The `H V R` [`nest`

] command applies a function to a given
argument repeatedly. It takes two values, ``a`' and ``n`', from
the stack, where ``n`' must be an integer. It then applies the
function nested ``n`' times; if the function is ``f`' and ``n`'
is 3, the result is ``f(f(f(a)))`'. The number ``n`' may be
negative if Calc knows an inverse for the function ``f`'; for
example, ``nest(sin, a, -2)`' returns ``arcsin(arcsin(a))`'.

The `H V U` [`anest`

] command is an accumulating version of
`nest`

: It returns a vector of ``n+1`' values, e.g.,
``[a, f(a), f(f(a)), f(f(f(a)))]`'. If ``n`' is negative and
``F`' is the inverse of ``f`', then the result is of the
form ``[a, F(a), F(F(a)), F(F(F(a)))]`'.

The `H I V R` [`fixp`

] command is like `H V R`, except
that it takes only an ``a`' value from the stack; the function is
applied until it reaches a "fixed point," i.e., until the result
no longer changes.

The `H I V U` [`afixp`

] command is an accumulating `fixp`

.
The first element of the return vector will be the initial value ``a`';
the last element will be the final result that would have been returned
by `fixp`

.

For example, 0.739085 is a fixed point of the cosine function (in radians):
``cos(0.739085) = 0.739085`'. You can find this value by putting, say,
1.0 on the stack and typing `H I V U C`. (We use the accumulating
version so we can see the intermediate results: ``[1, 0.540302, 0.857553,
0.65329, ...]`'. With a precision of six, this command will take 36 steps
to converge to 0.739085.)

Newton's method for finding roots is a classic example of iteration
to a fixed point. To find the square root of five starting with an
initial guess, Newton's method would look for a fixed point of the
function ``(x + 5/x) / 2`'. Putting a guess of 1 on the stack
and typing `H I V R ' ($ + 5/$)/2 RET` quickly yields the result
2.23607. This is equivalent to using the `a R` (`calc-find-root`

)
command to find a root of the equation ``x^2 = 5`'.

These examples used numbers for ``a`' values. Calc keeps applying
the function until two successive results are equal to within the
current precision. For complex numbers, both the real parts and the
imaginary parts must be equal to within the current precision. If
``a`' is a formula (say, a variable name), then the function is
applied until two successive results are exactly the same formula.
It is up to you to ensure that the function will eventually converge;
if it doesn't, you may have to press `C-g` to stop the Calculator.

The algebraic `fixp`

function takes two optional arguments, ``n`'
and ``tol`'. The first is the maximum number of steps to be allowed,
and must be either an integer or the symbol ``inf`' (infinity, the
default). The second is a convergence tolerance. If a tolerance is
specified, all results during the calculation must be numbers, not
formulas, and the iteration stops when the magnitude of the difference
between two successive results is less than or equal to the tolerance.
(This implies that a tolerance of zero iterates until the results are
exactly equal.)

Putting it all together, ``fixp(<(# + A/#)/2>, B, 20, 1e-10)`'
computes the square root of ``A`' given the initial guess ``B`',
stopping when the result is correct within the specified tolerance, or
when 20 steps have been taken, whichever is sooner.

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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.

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Commands for controlling vector and matrix display use the `v` prefix
instead of the usual `d` prefix. But they are display modes; in
particular, they are influenced by the `I` and `H` prefix keys
in the same way (see section 8.7 Display Modes). Matrix display is also
influenced by the `d O` (`calc-flat-language`

) mode;
see section 8.8.1 Normal Language Modes.

The commands `v <` (`calc-matrix-left-justify`

), `v >`
(`calc-matrix-right-justify`

), and `v =`
(`calc-matrix-center-justify`

) control whether matrix elements
are justified to the left, right, or center of their columns.

The `v [` (`calc-vector-brackets`

) command turns the square
brackets that surround vectors and matrices displayed in the stack on
and off. The `v {` (`calc-vector-braces`

) and `v (`
(`calc-vector-parens`

) commands use curly braces or parentheses,
respectively, instead of square brackets. For example, `v {` might
be used in preparation for yanking a matrix into a buffer running
Mathematica. (In fact, the Mathematica language mode uses this mode;
see section 8.8.5 Mathematica Language Mode.) Note that, regardless of the
display mode, either brackets or braces may be used to enter vectors,
and parentheses may never be used for this purpose.

The `v ]` (`calc-matrix-brackets`

) command controls the
"big" style display of matrices. It prompts for a string of code
letters; currently implemented letters are `R`

, which enables
brackets on each row of the matrix; `O`

, which enables outer
brackets in opposite corners of the matrix; and `C`

, which
enables commas or semicolons at the ends of all rows but the last.
The default format is ``RO`'. (Before Calc 2.00, the format
was fixed at ``ROC`'.) Here are some example matrices:

[ [ 123, 0, 0 ] [ [ 123, 0, 0 ], [ 0, 123, 0 ] [ 0, 123, 0 ], [ 0, 0, 123 ] ] [ 0, 0, 123 ] ] RO ROC |

[ 123, 0, 0 [ 123, 0, 0 ; 0, 123, 0 0, 123, 0 ; 0, 0, 123 ] 0, 0, 123 ] O OC |

[ 123, 0, 0 ] 123, 0, 0 [ 0, 123, 0 ] 0, 123, 0 [ 0, 0, 123 ] 0, 0, 123 R blank |

Note that of the formats shown here, ``RO`', ``ROC`', and
``OC`' are all recognized as matrices during reading, while
the others are useful for display only.

The `v ,` (`calc-vector-commas`

) command turns commas on and
off in vector and matrix display.

In vectors of length one, and in all vectors when commas have been
turned off, Calc adds extra parentheses around formulas that might
otherwise be ambiguous. For example, ``[a b]`' could be a vector
of the one formula ``a b`', or it could be a vector of two
variables with commas turned off. Calc will display the former
case as ``[(a b)]`'. You can disable these extra parentheses
(to make the output less cluttered at the expense of allowing some
ambiguity) by adding the letter `P`

to the control string you
give to `v ]` (as described above).

The `v .` (`calc-full-vectors`

) command turns abbreviated
display of long vectors on and off. In this mode, vectors of six
or more elements, or matrices of six or more rows or columns, will
be displayed in an abbreviated form that displays only the first
three elements and the last element: ``[a, b, c, ..., z]`'.
When very large vectors are involved this will substantially
improve Calc's display speed.

The `t .` (`calc-full-trail-vectors`

) command controls a
similar mode for recording vectors in the Trail. If you turn on
this mode, vectors of six or more elements and matrices of six or
more rows or columns will be abbreviated when they are put in the
Trail. The `t y` (`calc-trail-yank`

) command will be
unable to recover those vectors. If you are working with very
large vectors, this mode will improve the speed of all operations
that involve the trail.

The `v /` (`calc-break-vectors`

) command turns multi-line
vector display on and off. Normally, matrices are displayed with one
row per line but all other types of vectors are displayed in a single
line. This mode causes all vectors, whether matrices or not, to be
displayed with a single element per line. Sub-vectors within the
vectors will still use the normal linear form.

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