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# 11. Vector/Matrix Functions

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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## 11.1 Packing and Unpacking

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 RET 7 C-u -1 v p 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 `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 `1 +/- 2'.

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 mode is a packing mode (an integer or a vector of integers) and items is a vector of objects to be packed (re-packed, really) according to that mode. For example, `pack([3, -4], [a,b,c,d,e,f])' yields `[a +/- b, c +/- d, e +/- f]'. The function is left in symbolic form if the packing mode is illegal, or if the number of data items does not match the number of items required by the mode.

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 item using the given mode, returning the result as a vector of components. Here the mode must be an integer, not a vector. For example, `unpack(-4, a +/- b)' returns `[a, b]', as does `unpack(1, a +/- b)'.

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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## 11.2 Building Vectors

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 incr is one for positive n or two for negative n.

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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## 11.3 Extracting Vector Elements

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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## 11.4 Manipulating Vectors

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 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 (`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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## 11.5 Vector and Matrix Arithmetic

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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## 11.6 Set Operations using Vectors

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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## 11.7 Statistical Operations on Vectors

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.

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### 11.7.1 Single-Variable Statistics

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 Nth 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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### 11.7.2 Paired-Sample Statistics

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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## 11.8 Reducing and Mapping Vectors

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.

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### 11.8.1 Specifying Operators

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 `#' signs inside stuff, or by the fact that stuff begins with a list of variables followed by a colon.)

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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### 11.8.2 Mapping

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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### 11.8.3 Reducing

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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### 11.8.4 Nesting and Fixed Points

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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### 11.8.5 Generalized Products

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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## 11.9 Vector and Matrix Display Formats

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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