wiki:ArraysCowan

Specification

Terminology

An array is an object of a disjoint type, a container with elements arranged according to a rectilinear coordinate system. An array can have any number of dimensions or axes, including zero; this number is called the rank of the array. Arrays of rank zero contain exactly one element. Note that "rank" is a Fortran, Common Lisp, and APL term that has nothing to do with matrix ranks in the sense of linear algebra.

Each axis has an extent represented by two exact integers, the first representing the smallest possible coordinate for that axis, and the second representing the largest possible coordinate plus one. Extent is also used, by a mild abuse of language, for the difference between the two values. The smallest coordinates are collected into a Scheme vector known as the lower bound of the array; the largest coordinates are collected into another Scheme vector known as the upper bound of the array. An index of the array is any Scheme vector of exact integers which has the same number of elements as the array's rank, and whose values lie between the lower bound (inclusive) and the upper bound (exclusive) of the corresponding axis.

An array can be a general array, meaning each element can be any Scheme object, or it can be a specialized array, meaning that each element can only belong to a given restricted type. This is accomplished by separating array objects from the underlying storage objects, which can be Scheme vectors or numeric vectors or other objects. Any object which can map a non-negative exact integer into an appropriate value can serve as a storage object by writing a storage class for it. See StorageClassesCowan for the storage class API. Note that if the extent of any axis is non-positive, the array has no elements.

In order to map an array's index (a Scheme vector of exact integers) into a storage index (a single exact integer), each array maintains another associated vector of exact integers, the stride, as well as a single additional exact integer, the offset. Multiplying each element of the stride by the corresponding element of the array index, summing the results, and adding the offset produces the corresponding storage index. Therefore, the offset is the storage index of the element whose array index consists of all zeros.

Procedures that accept an array object and return a new array sharing the same storage object but with different upper bound, lower bound, stride, and/or offset are known as array transformations, and this SRFI provides a number of them. The SRFI also provides other procedures for operating on arrays, all of which have the property that they are meaningful no matter what the elements of the array are. So array-map can be used to sum two matrices, since that is done element-wise over the + operation; but there is no operation provided for ordinary matrix multiplication, because it depends on the array elements being solely numeric.

In the same way that the names start and end are applied to optional numerical indexes that default to the smallest element of a sequence (list, vector, string, or whatever) and the largest element plus one, in this SRFI they default to the lower bound and the upper bound of an array.

In certain procedures, the elements of an array are processed in lexicographic order, also known as row-major order. This means the order in which the highest-numbered axis changes most rapidly, and the other axes change only when the following axis returns to its lowest value.

General requirements

Where a procedure that is passed to any procedure defined in this SRFI receives an index as an argument, it is an error for that procedure to mutate the index.

Note that array objects are immutable, but their storage objects are usually mutable. It is possible to create arrays that are prohibited from mutating their storage objects.

The procedures of this SRFI that are not transformations may return arrays whose stride is implementation-dependent (so that the order of elements in the storage object may be either row-major or column-major), but the offset is always 0.

Predicates

(array? obj)

Returns #t if obj is an array, and #f otherwise.

(array-mutable? array)

Returns #t if array can mutate its storage object, and #f otherwise.

Constructors

(make-array storage-class lower-bound upper-bound [ fill ])

Returns a newly allocated mutable array (with a newly allocated storage object of the specified storage-class) that has the specified bounds. The values of the elements are unspecified if the fill argument is omitted.

(array-tabulate proc storage-class lower-bound upper-bound mutable?)

Returns a newly allocated array (with a newly allocated storage object of the specified storage-class) that has the specified bounds. The values of the elements are computed by calling proc on every possible index of the array in lexicographic order. If mutable? is true, the array can mutate its storage object.

(array-broadcast array obj)

Returns a newly allocated array whose bounds and storage class are the same as array and all of whose elements are obj.

Metadata accessors

(array-storage-class array)

Returns the storage class with which array was created.

(array-storage-objectarray)

Returns the storage object underlying this array. Note that this can be #f in the case of storage classes without actual storage.

(array-rank array)

Return the rank (number of dimensions) of array.

(array-lower-bound array)

Returns the index specifying the lower bound of array. It is an error to mutate this index.

(array-upper-bound array)

Returns the index specifying the upper bound of array. It is an error to mutate this index.

(array-stride array)

Return the stride of array. It is an error to mutate this vector.

(array-offset array)

Returns the offset of array.

(array-index->storage-index array index)

Converts the index of array to the equivalent storage index.

Accessors

(array-ref array index)

Returns the value of the element of array specified by index. Note that this is different from the array-ref of most Lisp systems, which specifies the index as a sequence of arguments.

(array-recursive-ref array index ...)

Applies array-ref to the array using the first index. If there are more arguments, apply array-ref to the result using the next index. Repeat until there are no more indexes, returning the last result. It is an error if any intermediate result is not an array. (APL enlist.)

(array-for-each proc array [ start [ end ] ])

Iterates over the elements of array in lexicographic order, starting at the index start and ending just before the index end, and calling proc on each element. Each invocation of proc receives the value of the element at that index. The value returned by proc is discarded.

(array-for-each-index proc array [ start [ end ] ])

Iterates over the indexes of array in lexicographic order, starting at the index start and ending at the index end, and calling proc on each index. The value returned by proc is discarded.

Mutators

These procedures return an unspecified value.

(array-set! array index value)

Sets the value of the element of array specified by index to value. Note that this is different from the array-set! of most Lisp systems, which specifies the index as a sequence of arguments.

(array-tabulate! proc array [ start [ end ] ])

Iterates over the indexes of array in lexicographical order starting at the index start (each dimension is inclusive) and ending at the index end (each dimension is exclusive), and calling proc on each index. Whatever proc returns becomes the value of the array at the index.

The whole array

(array-map proc array ...)

Returns a newly allocated array with the same bounds as the arrays; it is an error if they do not all have the same bounds. For each valid index value, proc is invoked, passing each corresponding element of the arrays. Whatever proc returns becomes the value of the storage element corresponding to that index in the result array. The order of invocations of proc is not specified.

(array-map! proc array ...)

It is an error if the arrays do not all have the same bounds. For each valid index value, proc is invoked, passing each corresponding element of the arrays. Whatever proc returns becomes the value of the storage element corresponding to that index in the first specified array. The order of invocations of proc is not specified. Returns an undefined value.

(array-fold proc seed array ...)

Returns a newly allocated array with the same bounds as the arrays; it is an error if they do not all have the same bounds. For each valid index value, proc is invoked in lexicographic order, passing each corresponding element of the arrays and a seed, whose initial value is seed. Proc returns two values, the value of the storage element corresponding to that index in the result array, and the new value of the seed.

(array-reduce proc array axis [ n ])

Returns a newly allocated array whose rank is one less than the rank of array, by combining all the groups of elements of length n (where the default is the extent of axis) along axis using proc, a two-argument procedure. The order and number of invocations of proc is unspecified. If there is only one such element, it is unchanged. (APL reduce.)

(array-cumulate proc array axis)

Returns a newly allocated array whose bounds are the same as the bounds of array. Each element along axis is constructed by reducing (as if by array-reduce) successive prefixes of the elements of array along that axis. (APL scan.)

(array-count pred array)

Returns an exact integer containing the number of elements in array that satisfy pred.

(array-index pred array)

Returns the index of the first element of array in lexicographic order that satisfies pred.

(array-compress array booleans axis)

Returns an array with the same bounds as array except possibly along the axis dimension. The array is sliced along axis and the elements of booleans (a vector of boolean values) are used to decide which slices to incorporate into the result: if the corresponding boolean is #t, the slice is incorporated, otherwise not. (APL compress.)

(array-expand array booleans nil axis)

Returns an array with the same bounds as array except possibly along the axis dimension. Array is sliced along axis and the elements of booleans (a vector of boolean values) are used to decide where, if anywhere, nil is to be interpolated: if the corresponding boolean is #t, nil is interpolated, otherwise the next slice is incorporated. It is an error if the size of booleans is not to the extent of the axis dimension in the result. It is also an error if nil does not have the same bounds as a slice. (APL expand.)

(array-rearrange array vector axis)

Returns an array with the same bounds as array. Array is sliced along the axis dimension, and the slices are reassembled in the order given by vector. The slice whose number appears in the first element of vector appears first in the result, and so on. It is an error if vector is not a vector of exact integers. (Generalized version of APL rotate.)

Transformations

These procedures return arrays which share their storage object with the array argument.

(array-transform proc array)

The procedure proc specifies an affine function that returns an index of array when given an index of the returned array. The array does not retain a dependency on proc. (SRFI 25 share-array.)

(array-diagonal array)

Returns a one-dimensional array which contains the diagonal elements of array (that is, the elements whose indices are all the same integer).

(array-reshape lower-bound upper-bound array)

Returns an array with the specified bounds whose elements in lexicographic order are the same as the elements of array in lexicographic order. It is an error if there are too many or too few elements. (APL reshape.)

(array-restride stride offset array)

Returns an array with the specified stride and offset whose elements in lexicographic order are the same as the elements of array in lexicographic order. The strides and offset can be arbitrary, and no attempt is made to recalculate the bounds, so the result might be unusable or only partly usable.

(array-reverse array axis)

Returns an array with the same bounds as array, but whose elements on the specified axis are reversed. (APL reverse.)

(array-transpose array)

Returns an array whose axes appear in the reverse order of the axes of array. This implies that the upper and lower bound are the reverse of the bounds of array. (APL monadic transpose.)

(array-rearrange-axes array vector)

Returns an array whose axes are an arbitrary permutation of the axes of array. Vector specifies how to do the permutation: the axis whose number appears in the first element of vector appears as the first axis of the result, and so on. (APL dyadic transpose with integer-valued vector.)

(array-slice array start end)

Returns a smaller array with the same rank as array whose elements begin at the "lower left" corner specified by start and end at the "upper right" corner specified byend. Unlike array-copy, the result shares its storage object with array. (APL take and drop.)

(array-squeeze array vector)

Returns an array with the ranks specified by the elements of vector removed from array. It is an error if the extents of the specified ranks are not equal to 1. (NumPy squeeze)

(array-unsqueeze array rank)

Returns an array whose rank is one greater than the rank of array. This is accomplished by inserting a new axis numbered 'axis' whose extent is (0:1). (NumPy expand.)

Copying and conversion

These procedures return arrays which do not share their storage objects with any existing arrays.

When these procedures return arrays, both the array and the underlying storage object are newly allocated

(array-copy array mutable? [ start [ end ] ])

Returns an array containing the elements of array starting at start (inclusive) and ending at end (exclusive). The lower bound of the resulting array is all zeros, and the upper bound is determined by subtracting start from end element-wise. The stride and offset are implementation-defined. The resulting array is mutable if mutable? is true. The returned array has the same storage class as array.

(array-copy! to at from [ start [ end ] ])

Copies the elements of array from from index start (inclusive) to index end (inclusive) onto array to starting at index at. It is an error if there are not enough elements in to to make this possible. The storage objects of from and to need not belong to the same storage classes.

(array-append axis array ...)

Returns a newly allocated array consisting of the arrays concatenated along axis. It is an error unless the storage classes of the arrays are the same, and the result has the same storage class. It is also an error unless the bounds of all the arrays are the same, with the possible exception of axis. The axisth element of the lower bound of the result is 0; the corresponding element of the upper bound is the sum of the extents of the arrays.

(array-repeat array axis repeat)

Append repeat copies of array along axis axis, as if by array-append. (Variant of NumPy repeat.)

(array-reclassify array storage-class)

Returns an array with the same bounds and elements as array, but whose storage class is specified by storage-class.

(array->nested-vector array)

(array->nested-list array)

Returns a newly allocated vector/list whose elements are also newly constructed vectors/lists, and so on as far down as necessary to cover every axis of the array. Bottom-level Scheme vectors/lists contain the elements of array. Thus, if array has rank 1, the result is a vector/list; if the array has rank 2, the result is a vector/list whose elements are vectors/lists, and so on. As a special case, if array has rank 0, the sole element is returned.

(nested-vector->array nested-vector storage-class rank)

(nested-list->array nested-list storage-class rank)

Returns a newly allocated array of the storage-class with rank rank whose elements are initialized from the vector nested-vector or list nested-list. It is an error if this argument is not rectangular. As a special case, if rank is 0, the sole element is nested-vector or nested-list, which in that case need not be a Scheme vector/list.

Inner and outer products

These procedures return arrays which do not share their storage objects with any existing arrays.

(array-inner-product storage-class proc1 proc2 array1 array2)

Returns a newly allocated array using the storage-class whose bounds are equal to the bounds of array1 without their last elements, concatenated with the bounds of array2 without their first elements. It is an error if the omitted upper bounds are not numerically equal; it is also an error if the omitted lower bounds are not numerically equal. It is an error if both arrays have rank 0.

Each element of the result array results from applying proc2 to the corresponding elements of the last vectors of array1 and the first vectors of array2 and then reducing them with proc1 to a single value. The order and number of invocations of the procedures is unspecified.

In particular, if both arrays have rank 1, the last and first vectors are the whole of the arrays, and the result has rank 0; if both arrays have rank 2, the last vectors of array1 are the column-wise vectors, and the first vectors of array2 are the row-wise vectors, and the result has rank 2. (APL inner product.)

Example: (array-inner-product vector-storage-class + * array1 array2), where the arrays have rank 1, computes the usual dot product of two vectors.

(array-outer-product storage-class proc array1 array2)

Returns a newly allocated array using the storage-class whose bounds are the concatenation of the bounds of array1 and array2. Each element of the new array is the result of applying proc to every element of array1 and every element of array2. The order and number of invocations of proc is unspecified. (APL outer product.)

Input/output

The external representation of an array consists of a # character, followed by a sequence of digits indicating the rank of the array, followed by the letter a, followed by a coded representation of the storage class, all with no intervening whitespace. This prefix is followed, after optional whitespace, by the representation of a nested list produced as if by array->nested-list. The prefix is interpreted case-insensitively.

Standard numeric storage classes are encoded by using the first few characters of the name of the storage class. Thus, the representation of an array of rank 2 using u32-storage-class begins with #2au32. Other storage classes, including vector-storage-class, sparse-storage-class, and user-created storage classes, are encoded using the empty string.

(array-read [ input-port ])

Reads the external representation of an array from input-port (the current input port if input-port is not specified) and returns the corresponding array. If the coded representation of the storage class is not recognized, vector-storage-class is used; this permits the addition of new coded storage classes in a backward compatible way.

(array-write array [ stream ])

Writes the external representation of array from output-port (the current output port if output-port is not specified) and returns an unspecified value.

This SRFI recommends, but does not require, that the standard Scheme procedures read, write, and display be extended to deal with external representations of arrays. On R7RS systems, if read accepts the external representation of arrays, it must also be allowed in Scheme code, in which case array constants are self-quoting.

Last modified 8 months ago Last modified on 08/07/16 12:02:13