6.3 Broadcasting

6.3 Broadcasting🔗ℹ

It is often useful to apply a pointwise operation to two or more arrays in a many-to-one manner. Library support for this, which math/array provides, is called broadcasting.

Creating a 6×6 identity matrix:

> (define diag (diagonal-array 2 6 1 0))
> (array-shape diag)
- : Indexes
'#(6 6)
> diag
- : #(struct:Array
      (Indexes Index (Boxof Boolean) (-> Void) (-> Indexes (U One Zero)))
      #<syntax:build/user/8.15.0.2/pkgs/math-lib/math/private/array/typed-array-struct.rkt:56:13 prop:equal+hash>
      #<syntax:build/user/8.15.0.2/pkgs/math-lib/math/private/array/typed-array-struct.rkt:55:13 prop:custom-write>
      #<syntax:build/user/8.15.0.2/pkgs/math-lib/math/private/array/typed-array-struct.rkt:54:13 prop:custom-print-quotable>)
(array
#[#[1 0 0 0 0 0]
   #[0 1 0 0 0 0]
   #[0 0 1 0 0 0]
   #[0 0 0 1 0 0]
   #[0 0 0 0 1 0]
   #[0 0 0 0 0 1]])

Multiplying each element by 10:

> (array-shape (array 10))
- : Indexes
'#()
> (array* diag (array 10))
- : #(struct:Array
      (Indexes Index (Boxof Boolean) (-> Void) (-> Indexes Index))
      #<syntax:build/user/8.15.0.2/pkgs/math-lib/math/private/array/typed-array-struct.rkt:56:13 prop:equal+hash>
      #<syntax:build/user/8.15.0.2/pkgs/math-lib/math/private/array/typed-array-struct.rkt:55:13 prop:custom-write>
      #<syntax:build/user/8.15.0.2/pkgs/math-lib/math/private/array/typed-array-struct.rkt:54:13 prop:custom-print-quotable>)
(array
#[#[10 0 0 0 0 0]
   #[0 10 0 0 0 0]
   #[0 0 10 0 0 0]
   #[0 0 0 10 0 0]
   #[0 0 0 0 10 0]
   #[0 0 0 0 0 10]])

Adding (array #[0 1 2 3 4 5]) pointwise to every row:

> (array+ (array* diag (array 10))
          (array #[0 1 2 3 4 5]))
- : #(struct:Array
      (Indexes Index (Boxof Boolean) (-> Void) (-> Indexes Nonnegative-Fixnum))
      #<syntax:build/user/8.15.0.2/pkgs/math-lib/math/private/array/typed-array-struct.rkt:56:13 prop:equal+hash>
      #<syntax:build/user/8.15.0.2/pkgs/math-lib/math/private/array/typed-array-struct.rkt:55:13 prop:custom-write>
      #<syntax:build/user/8.15.0.2/pkgs/math-lib/math/private/array/typed-array-struct.rkt:54:13 prop:custom-print-quotable>)
(array
#[#[10 1 2 3 4 5]
   #[0 11 2 3 4 5]
   #[0 1 12 3 4 5]
   #[0 1 2 13 4 5]
   #[0 1 2 3 14 5]
   #[0 1 2 3 4 15]])

6.3.1 Broadcasting Rules🔗ℹ

Suppose we have two array shapes ds = (vector d0 d1 ...) and es = (vector e0 e1 ...). Broadcasting proceeds as follows:

The shorter shape is padded on the left with 1 until it is the same length as the longer shape.
For each axis k, dk and ek are compared. If dk = ek, the result axis is dk; if one axis is length 1, the result axis is the length of the other; otherwise fail.
Both arrays’ axes are stretched by (conceptually) copying the rows of axes with length 1.

Example: Suppose we have an array drr with shape ds = #(4 1 3) and another array err with shape es = #(3 3). Following the rules:

es is padded to get #(1 3 3).
The result axis is derived from #(4 1 3) and #(1 3 3) to get #(4 3 3).
drr’s second axis is stretched to length 3, and err’s new first axis (which is length 1 by rule 1) is stretched to length 4.

The same example, but more concrete:

> (define drr
    (array #[#[#["00" "01" "02"]]
             #[#["10" "11" "12"]]
             #[#["20" "21" "22"]]
             #[#["30" "31" "32"]]]))
> (array-shape drr)
- : Indexes
'#(4 1 3)
> (define err
    (array #[#["aa" "ab" "ac"]
             #["ba" "bb" "bc"]
             #["ca" "cb" "cc"]]))
> (array-shape err)
- : Indexes
'#(3 3)
> (define drr+err (array-map string-append drr err))
> (array-shape drr+err)
- : Indexes
'#(4 3 3)
> drr+err
- : #(struct:Array
      (Indexes Index (Boxof Boolean) (-> Void) (-> Indexes String))
      #<syntax:build/user/8.15.0.2/pkgs/math-lib/math/private/array/typed-array-struct.rkt:56:13 prop:equal+hash>
      #<syntax:build/user/8.15.0.2/pkgs/math-lib/math/private/array/typed-array-struct.rkt:55:13 prop:custom-write>
      #<syntax:build/user/8.15.0.2/pkgs/math-lib/math/private/array/typed-array-struct.rkt:54:13 prop:custom-print-quotable>)
(array
#[#[#["00aa" "01ab" "02ac"]
     #["00ba" "01bb" "02bc"]
     #["00ca" "01cb" "02cc"]]
   #[#["10aa" "11ab" "12ac"]
     #["10ba" "11bb" "12bc"]
     #["10ca" "11cb" "12cc"]]
   #[#["20aa" "21ab" "22ac"]
     #["20ba" "21bb" "22bc"]
     #["20ca" "21cb" "22cc"]]
   #[#["30aa" "31ab" "32ac"]
     #["30ba" "31bb" "32bc"]
     #["30ca" "31cb" "32cc"]]])

Notice how the row #["00" "01" "02"] in drr is repeated in the result because drr’s second axis was stretched during broadcasting. Also, the column #[#["aa"] #["ba"] #["ca"]] in err is repeated because err’s first axis was stretched.

For the above example, array-map does this before operating on drr and err:

> (define ds (array-shape-broadcast (list (array-shape drr)
                                          (array-shape err))))
> ds
- : Indexes
'#(4 3 3)
> (array-broadcast drr ds)
- : #(struct:Array
      (Indexes Index (Boxof Boolean) (-> Void) (-> Indexes String))
      #<syntax:build/user/8.15.0.2/pkgs/math-lib/math/private/array/typed-array-struct.rkt:56:13 prop:equal+hash>
      #<syntax:build/user/8.15.0.2/pkgs/math-lib/math/private/array/typed-array-struct.rkt:55:13 prop:custom-write>
      #<syntax:build/user/8.15.0.2/pkgs/math-lib/math/private/array/typed-array-struct.rkt:54:13 prop:custom-print-quotable>)
(array
#[#[#["00" "01" "02"]
     #["00" "01" "02"]
     #["00" "01" "02"]]
   #[#["10" "11" "12"]
     #["10" "11" "12"]
     #["10" "11" "12"]]
   #[#["20" "21" "22"]
     #["20" "21" "22"]
     #["20" "21" "22"]]
   #[#["30" "31" "32"]
     #["30" "31" "32"]
     #["30" "31" "32"]]])
> (array-broadcast err ds)
- : #(struct:Array
      (Indexes Index (Boxof Boolean) (-> Void) (-> Indexes String))
      #<syntax:build/user/8.15.0.2/pkgs/math-lib/math/private/array/typed-array-struct.rkt:56:13 prop:equal+hash>
      #<syntax:build/user/8.15.0.2/pkgs/math-lib/math/private/array/typed-array-struct.rkt:55:13 prop:custom-write>
      #<syntax:build/user/8.15.0.2/pkgs/math-lib/math/private/array/typed-array-struct.rkt:54:13 prop:custom-print-quotable>)
(array
#[#[#["aa" "ab" "ac"]
     #["ba" "bb" "bc"]
     #["ca" "cb" "cc"]]
   #[#["aa" "ab" "ac"]
     #["ba" "bb" "bc"]
     #["ca" "cb" "cc"]]
   #[#["aa" "ab" "ac"]
     #["ba" "bb" "bc"]
     #["ca" "cb" "cc"]]
   #[#["aa" "ab" "ac"]
     #["ba" "bb" "bc"]
     #["ca" "cb" "cc"]]])

6.3.2 Broadcasting Control🔗ℹ

The parameter array-broadcasting controls how pointwise operations broadcast arrays. Its default value is #t, which means that broadcasting proceeds as described in Broadcasting Rules. Another possible value is #f, which allows pointwise operations to succeed only if array shapes match exactly:

> (parameterize ([array-broadcasting #f])
(array* (index-array #(3 3)) (array 10)))
array-shape-broadcast: incompatible array shapes
(array-broadcasting #f): '#(3 3), '#()

Another option is R-style permissive broadcasting, which allows pointwise operations to always succeed, by repeating shorter axes’ rows instead of repeating just singleton axes’ rows:

> (define arr10 (array-map number->string (index-array #(10))))
> (define arr3 (array-map number->string (index-array #(3))))
> arr10
- : #(struct:Array
      (Indexes Index (Boxof Boolean) (-> Void) (-> Indexes String))
      #<syntax:build/user/8.15.0.2/pkgs/math-lib/math/private/array/typed-array-struct.rkt:56:13 prop:equal+hash>
      #<syntax:build/user/8.15.0.2/pkgs/math-lib/math/private/array/typed-array-struct.rkt:55:13 prop:custom-write>
      #<syntax:build/user/8.15.0.2/pkgs/math-lib/math/private/array/typed-array-struct.rkt:54:13 prop:custom-print-quotable>)
(array #["0" "1" "2" "3" "4" "5" "6" "7" "8" "9"])
> arr3
- : #(struct:Array
      (Indexes Index (Boxof Boolean) (-> Void) (-> Indexes String))
      #<syntax:build/user/8.15.0.2/pkgs/math-lib/math/private/array/typed-array-struct.rkt:56:13 prop:equal+hash>
      #<syntax:build/user/8.15.0.2/pkgs/math-lib/math/private/array/typed-array-struct.rkt:55:13 prop:custom-write>
      #<syntax:build/user/8.15.0.2/pkgs/math-lib/math/private/array/typed-array-struct.rkt:54:13 prop:custom-print-quotable>)
(array #["0" "1" "2"])
> (array-map string-append arr10 (array #["+" "-"]) arr3)
array-shape-broadcast: incompatible array shapes
(array-broadcasting #t): '#(10), '#(2), '#(3)
> (parameterize ([array-broadcasting 'permissive])
    (array-map string-append arr10 (array #["+" "-"]) arr3))
- : #(struct:Array
      (Indexes Index (Boxof Boolean) (-> Void) (-> Indexes String))
      #<syntax:build/user/8.15.0.2/pkgs/math-lib/math/private/array/typed-array-struct.rkt:56:13 prop:equal+hash>
      #<syntax:build/user/8.15.0.2/pkgs/math-lib/math/private/array/typed-array-struct.rkt:55:13 prop:custom-write>
      #<syntax:build/user/8.15.0.2/pkgs/math-lib/math/private/array/typed-array-struct.rkt:54:13 prop:custom-print-quotable>)
(array #["0+0" "1-1" "2+2" "3-0" "4+1" "5-2" "6+0" "7-1" "8+2" "9-0"])

Notice that (array #["+" "-"]) was repeated five times, and that arr3 was repeated three full times and once partially.

1	Constants and Elementary Functions
2	Flonums
3	Special Functions
4	Number Theory
5	Arbitrary-Precision Floating-Point Numbers (Bigfloats)
6	Arrays
7	Matrices and Linear Algebra
8	Statistics Functions
9	Probability Distributions
10	Stuff That Doesn’t Belong Anywhere Else

6.1	Quick Start
6.2	Definitions
6.3	Broadcasting
6.4	Slicing
6.5	Nonstrict Arrays
6.6	Types, Predicates and Accessors
6.7	Construction
6.8	Conversion
6.9	Comprehensions and Sequences
6.10	Pointwise Operations
6.11	Indexing and Slicing
6.12	Transformations
6.13	Folds, Reductions and Expansions
6.14	Other Array Operations
6.15	Subtypes
6.16	Strictness