Introduction

This library is a collection of functions and macros for manipulating Common Lisp arrays and performing numerical calculations with them.

For example, arrays can be created:

;; uniform and normal random numbers
(rand '(2 2)) ; => #2A((0.62944734 0.2709539) (0.81158376 0.6700171))

;; linear ranges
(linspace 1 10 7) ; => #(1 5/2 4 11/2 7 17/2 10)

;; Using a function, optionally given index position
(generate #'identity '(2 3) :position) ; => #2A((0 1 2) (3 4 5))

Arrays can be manipulated:

(defparameter A #2A((1 2) (3 4)))
(defparameter B #2A((2 3) (4 5)))

;; split along any dimension
(split A 1)  ; => #(#(1 2) #(3 4))

;; stack along any dimension
(stack 1 A B) ; => #2A((1 2 2 3) (3 4 4 5))

;; element-wise function map
(each #'+ #(0 1 2) #(2 3 5)) ; => #(2 4 7)

;; element-wise expressions
(vectorize (A B) (* A (sqrt B))) ; => #2A((1.4142135 3.4641016) (6.0 8.944272))

;; index operations e.g. matrix-matrix multiply:
(each-index (i j)
  (sum-index k
    (* (aref A i k) (aref B k j)))) ; => #2A((10 13) (22 29))

Installation

A version of this library is on QuickLisp, based on the original library. The QuickLisp version has some but not all functionality of this fork: It includes the generate, each (element-wise map), displace, flatten, split, combine, sub, partition, stack, reshape, margin and recycle functions.

  (ql:quickload :array-operations)

This fork keeps these functions, and adds each-index, sum-index, reduce-index, vectorize, vectorize-reduce, nested-loop, zeros, ones, rand, randn, linspace, argmax and argmin. To install, clone into your Quicklisp local project directory:

  $ git clone https://github.com/bendudson/array-operations.git ~/quicklisp/local-projects/

Then load as above with (ql:quickload :array-operations).

To run the test suite (using clunit):

  (ql:quickload :array-operations-tests)
  (array-operations-tests:run)

A quick tour of the library

Shorthand for frequently used Common Lisp array functions

The library defines the following short function names that are synomyms for Common Lisp operations:

array-operations	Common Lisp
size	array-total-size
rank	array-rank
dim	array-dimension
dims	array-dimensions
nrow	number of rows in matrix
ncol	number of columns in matrix

The array-operations package has the nickname aops, so you can use, for example, (aops:size my-array) without use'ing the package.

Displaced arrays for fun and profit

displaced array n. an array which has no storage of its own, but which is instead indirected to the storage of another array, called its target, at a specified offset, in such a way that any attempt to access the displaced array implicitly references the target array. (CLHS Glossary)

Displaced arrays are one of the niftiest features of Common Lisp. When an array is displaced to another array, it shares structure with (part of) that array. The two arrays do not need to have the same dimensions, in fact, the dimensions do not be related at all as long as the displaced array fits inside the original one. The row-major index of the former in the latter is called the offset of the displacement.

Displaced arrays are usually constructed using make-array, but this library also provides displace for that purpose:

  (defparameter *a* #2A((1 2 3) (4 5 6)))
  (aops:displace *a* 2 1) ; => #(2 3)

flatten displaces to a row-major array:

  (aops:flatten *a*) ; => #(1 2 3 4 5 6)

The real fun starts with split, which splits off subarrays nested within a given axis:

  (aops:split *a* 1) ; => #(#(1 2 3) #(4 5 6))
  (defparameter *b* #3A(((0 1) (2 3))
                        ((4 5) (6 7))))
  (aops:split *b* 0) ; => #3A(((0 1) (2 3)) ((4 5) (6 7)))
  (aops:split *b* 1) ; => #(#2A((0 1) (2 3)) #2A((4 5) (6 7)))
  (aops:split *b* 2) ; => #2A((#(0 1) #(2 3)) (#(4 5) #(6 7)))
  (aops:split *b* 3) ; => #3A(((0 1) (2 3)) ((4 5) (6 7)))

Note how splitting at 0 and the rank of the array returns the array itself.

Now consider sub, which returns a specific array, composed of the elements that would start with given subscripts:

  (aops:sub *b* 0) ; => #2A((0 1) (2 3))
  (aops:sub *b* 0 1) ; => #(2 3)
  (aops:sub *b* 0 1 0) ; => 2

There is also a (setf sub) function.

partition returns a consecutive chunk of an array separated along its first subscript:

  (aops:partition #2A((0 1)
                    (2 3)
                    (4 5)
                    (6 7)
                    (8 9))
                1 3) ; => #2A((2 3) (4 5))

and also has a (setf partition) pair.

combine is the opposite of split:

  (aops:combine #(#(0 1) #(2 3))) ; => #2A((0 1) (2 3))

subvec returns a displaced subvector:

  (aops:subvec #(0 1 2 3 4) 2 4) ; => #(2 3)

There is also a (setf subvec) function, which is like (setf subseq) except for demanding matching lengths.

Finally, reshape can be used to displace arrays into a different shape:

  (aops:reshape *a* '(3 2)) ; => #2A((1 2) (3 4) (5 6))

You can use t for one of the dimensions, to be filled in automatically:

  (aops:reshape *b* '(1 t)) ; => #2A((0 1 2 3 4 5 6 7))

reshape-col and reshape-row reshape your array into a column or row matrix, respectively.

Dimension specifications

Functions in the library accept the following in place of dimensions:

• a list of dimensions (as for make-array),
• a positive integer, which is used as a single-element list,
• another array, the dimensions of which are used.

The last one allows you to specify dimensions with other arrays. For example, to reshape an array a1 to look like a2, you can use

  (aops:reshape a1 a2)

instead of the longer form

  (aops:reshape a1 (aops:dims a2))

Array creation and transformations

When the resulting element type cannot be inferred, functions that create and transform arrays are provided in pairs: one of these will allow you to specify the array-element-type of the result, while the other assumes it is t. The former ends with a *, and the element-type is always its first argument. I give examples for the versions without *, use the other when you are optimizing your code and you are sure you can constrain to a given element-type.

Element traversal order of these functions is unspecified. The reason for this is that the library may use parallel code in the future, so it is unsafe to rely on a particular element traversal order.

The following functions all make a new array, taking the dimensions as input. The version ending in * also takes the array type as first argument. There are also versions ending in ! which do not make a new array, but take an array as first argument, which is modified and returned.

Function	Description
zeros	Filled with zeros
ones	Filled with ones
rand	Filled with uniformly distrubuted random numbers between 0 and 1
randn	Normally distributed with mean 0 and standard deviation 1
linspace	Evenly spaced numbers in given range

For example:

  (aops:rand '(2 2))  ; => #2A((0.6686077 0.59425664) (0.7987722 0.6930506))

  (aops:rand* 'single-float '(2 2)) ; => #2A((0.39332366 0.5557821) (0.48831415 0.10924244))

  (let ((a (make-array '(2 2) :element-type 'double-float)))
    ;; Modify array A, filling with random numbers
    (aops:rand! a))  ; => #2A((0.6324615478515625d0 0.4636608362197876d0)
                              (0.4145939350128174d0 0.5124958753585815d0))

generate (and generate*) allow you to generate arrays using functions.

  (aops:generate (lambda () (random 10)) 3) ; => #(6 9 5)
  (aops:generate #'identity '(2 3) :position) ; => #2A((0 1 2) (3 4 5))
  (aops:generate #'identity '(2 2) :subscripts)
  ;; => #2A(((0 0) (0 1)) ((1 0) (1 1)))
  (aops:generate #'cons '(2 2) :position-and-subscripts)
  ;; => #2A(((0 0 0) (1 0 1)) ((2 1 0) (3 1 1)))

Depending on the last argument, the function will be called with the (row-major) position, the subscripts, both, or no argument.

permute can permutate subscripts (you can also invert, complement, and complete permutations, look at the docstring and the unit tests). Transposing is a special case of permute:

  (defparameter *a* #2A((1 2 3) (4 5 6)))
  (aops:permute '(0 1) *a*) ; => #2A((1 2 3) (4 5 6))
  (aops:permute '(1 0) *a*) ; => #2A((1 4) (2 5) (3 6))

each applies a function to its (array) arguments elementwise:

  (aops:each #'+ #(0 1 2) #(2 3 5)) ; => #(2 4 7)

vectorize is a macro which performs elementwise operations

  (defparameter a #(1 2 3 4))
  (aops:vectorize (a) (* 2 a)) ; => #(2 4 6 8)

  (defparameter b #(2 3 4 5))
  (aops:vectorize (a b) (* a (sin b))) ; => #(0.9092974 0.28224 -2.2704074 -3.8356972)

There is also a version vectorize* which takes a type argument for the resulting array, and a version vectorize! which sets elements in a given array.

The semantics of margin are more difficult to explain, so perhaps an example will be more useful. Suppose that you want to calculate column sums in a matrix. You could permute (transpose) the matrix, split its subarrays at rank one (so you get a vector for each row), and apply the function that calculates the sum. margin automates that for you:

  (aops:margin (lambda (column)
               (reduce #'+ column))
             #2A((0 1)
                 (2 3)
                 (5 7)) 0) ; => #(7 11)

But the function is much more general than this: the arguments inner and outer allow arbitrary permutations before splitting.

Finally, recycle allows you to recycle arrays along inner and outer dimensions:

  (aops:recycle #(2 3) :inner 2 :outer 4)
  ; => #3A(((2 2) (3 3)) ((2 2) (3 3)) ((2 2) (3 3)) ((2 2) (3 3)))

Indexing operations

nested-loop is a simple macro which iterates over a set of indices with a given range

  (defparameter A #2A((1 2) (3 4)))

  (aops:nested-loop (i j) (array-dimensions A)
    (setf (aref A i j) (* 2 (aref A i j))))
  A ; => #2A((2 4) (6 8))

  (aops:nested-loop (i j) '(2 3)
    (format t "(~a ~a) " i j)) ; => (0 0) (0 1) (0 2) (1 0) (1 1) (1 2) 

sum-index is a macro which uses a code walker to determine the dimension sizes, summing over the given index or indices

  (defparameter A #2A((1 2) (3 4)))

  ;; Trace
  (aops:sum-index i (aref A i i)) ; => 5

  ;; Sum array
  (aops:sum-index (i j) (aref A i j)) ; => 10

  ;; Sum array
  (aops:sum-index i (row-major-aref A i)) ; => 10

The main use for sum-index is in combination with each-index.

each-index is a macro which creates an array and iterates over the elements. Like sum-index it is given one or more index symbols, and uses a code walker to find array dimensions.

  (defparameter A #2A((1 2) (3 4)))
  (defparameter B #2A((5 6) (7 8)))

  ;; Transpose
  (aops:each-index (i j) (aref A j i)) ; => #2A((1 3) (2 4))

  ;; Sum columns
  (aops:each-index i
    (aops:sum-index j
      (aref A j i))) ; => #(4 6)

  ;; Matrix-matrix multiply
  (aops:each-index (i j)
     (aops:sum-index k
        (* (aref A i k) (aref B k j)))) ; => #2A((19 22) (43 50))

reduce-index is a more general version of sum-index, which applies a reduction operation over one or more indices.

  (defparameter A #2A((1 2) (3 4)))
  
  ;; Sum all values in an array
  (aops:reduce-index #'+ i (row-major-aref A i)) ; => 10
  
  ;; Maximum value in each row
  (aops:each-index i
    (aops:reduce-index #'max j
      (aref A i j)))  ; => #(2 4)

Reductions

Some reductions over array elements can be done using the CL reduce function, together with aops:flatten, which returns a displaced vector:

  (defparameter a #2A((1 2) (3 4)))
  (reduce #'max (aops:flatten a)) ; => 4

argmax and argmin find the row-major-aref index where an array is maximum or minimum. They both return two values: the first value is the index; the second is the array value at that index.

  (defparameter a #(1 2 5 4 2))
  (aops:argmax a) ; => 2 5
  (aops:argmin a) ; => 0 1

More complicated reductions can be done with vectorize-reduce, for example the maximum absolute difference between arrays:

  (defparameter a #2A((1 2) (3 4)))
  (defparameter b #2A((2 2) (1 3)))

  (aops:vectorize-reduce #'max (a b) (abs (- a b))) ; => 2

See also reduce-index above.

Scalars as 0-dimensional arrays

Library functions treat non-array objects as if they were equivalent to 0-dimensional arrays: for example, (aops:split array (rank array)) returns an array that effectively equivalent (eq) to array. Another example is recycle:

  (aops:recycle 4 :inner '(2 2)) ; => #2A((4 4) (4 4))

Stacking

You can also stack compatible arrays along any axis:

  (defparameter *a1* #(0 1 2))
  (defparameter *a2* #(3 5 7))
  (aops:stack 0 *a1* *a2*) ; => #(0 1 2 3 5 7)
  (aops:stack 1
            (aops:reshape-col *a1*)
            (aops:reshape-col *a2*)) ; => #2A((0 3) (1 5) (2 7))