Split

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↓

Split (↓) is a monadic primitive function which reduces the rank of its argument by converting one of its axes to one level of nesting. The axis to move defaults to the last axis, but a different axis can be chosen using function axis. It shares its glyph ↓ with the dyadic function Drop. Split is the inverse of Mix in the sense that the latter undoes the enclosing that Split introduced.

Examples

The result of Split on a non-scalar array is always a nested array whose elements are vectors. The rank of ↓[K]Y is ¯1+≢⍴Y (original rank minus 1), its shape is (K≠⍳≢⍴Y)/⍴Y (original shape with K-th axis removed), and the shape of each element is (⍴Y)[K]. 

      ⎕←Y←2 3 4⍴⎕A  ⍝ 3D array
ABCD
EFGH
IJKL
    
MNOP
QRST
UVWX
      ↓Y  ⍝ Last axis split; 2×3 array of length-4 vectors
┌────┬────┬────┐
│ABCD│EFGH│IJKL│
├────┼────┼────┤
│MNOP│QRST│UVWX│
└────┴────┴────┘
      ↓[2]Y  ⍝ 2nd axis split; 2×4 array of length-3 vectors
┌───┬───┬───┬───┐
│AEI│BFJ│CGK│DHL│
├───┼───┼───┼───┤
│MQU│NRV│OSW│PTX│
└───┴───┴───┴───┘
      ↓↓Y  ⍝ Split twice
┌────────────────┬────────────────┐
│┌────┬────┬────┐│┌────┬────┬────┐│
││ABCD│EFGH│IJKL│││MNOP│QRST│UVWX││
│└────┴────┴────┘│└────┴────┴────┘│
└────────────────┴────────────────┘

      (≡Y)(≢⍴Y)  ⍝ Original array is depth 1, rank 3
1 3
      (≡↓Y)(≢⍴↓Y)  ⍝ Split array is depth 1+1, rank 3-1
2 2
Works in: Dyalog APL

Split is a no-op to a scalar. 

      2≡↓2
1

Alternatives

Most dialects do not have Split. Instead, they can use Enclose (⊂) with bracket axis or the Rank operator: 

      ↓Y
┌────┬────┬────┐
│ABCD│EFGH│IJKL│
├────┼────┼────┤
│MNOP│QRST│UVWX│
└────┴────┴────┘
      ⊂[3]Y
┌────┬────┬────┐
│ABCD│EFGH│IJKL│
├────┼────┼────┤
│MNOP│QRST│UVWX│
└────┴────┴────┘
      ⊂⍤1⊢Y
┌────┬────┬────┐
│ABCD│EFGH│IJKL│
├────┼────┼────┤
│MNOP│QRST│UVWX│
└────┴────┴────┘

It is common to split a higher-rank array into its constituent major cells. The behaviour of Split on matrices might mislead to the belief that this is what the primitive does. However, it isn't so for vectors or arrays of higher rank than 2. Instead, the solution is to use or ⊂[1↓⍳≢⍴Y]Y or ⊂⍤¯1⊢Y: 

      ⊂[1↓⍳≢⍴Y]Y
┌────┬────┐
│ABCD│MNOP│
│EFGH│QRST│
│IJKL│UVWX│
└────┴────┘
      ⊂⍤¯1⊢Y
┌────┬────┐
│ABCD│MNOP│
│EFGH│QRST│
│IJKL│UVWX│
└────┴────┘

External links

Lessons

Documentation


APL built-ins [edit]
Primitives (Timeline) Functions
Scalar
Monadic Conjugate ∙ Negate ∙ Signum ∙ Reciprocal ∙ Magnitude ∙ Exponential ∙ Natural Logarithm ∙ Floor ∙ Ceiling ∙ Factorial ∙ Not ∙ Pi Times ∙ Roll ∙ Type ∙ Imaginary ∙ Square Root ∙ Round
Dyadic Add ∙ Subtract ∙ Times ∙ Divide ∙ Residue ∙ Power ∙ Logarithm ∙ Minimum ∙ Maximum ∙ Binomial ∙ Comparison functions ∙ Boolean functions (And, Or, Nand, Nor) ∙ GCD ∙ LCM ∙ Circular ∙ Complex ∙ Root
Non-Scalar
Structural Shape ∙ Reshape ∙ Tally ∙ Depth ∙ Ravel ∙ Enlist ∙ Table ∙ Catenate ∙ Reverse ∙ Rotate ∙ Transpose ∙ Raze ∙ Mix ∙ Split ∙ Enclose ∙ Nest ∙ Cut (K) ∙ Pair ∙ Link ∙ Partitioned Enclose ∙ Partition
Selection First ∙ Pick ∙ Take ∙ Drop ∙ Unique ∙ Identity ∙ Stop ∙ Select ∙ Replicate ∙ Expand ∙ Set functions (Intersection ∙ Union ∙ Without) ∙ Bracket indexing ∙ Index ∙ Cartesian Product ∙ Sort
Selector Index generator ∙ Grade ∙ Index Of ∙ Interval Index ∙ Indices ∙ Deal ∙ Prefix and suffix vectors
Computational Match ∙ Not Match ∙ Membership ∙ Find ∙ Nub Sieve ∙ Encode ∙ Decode ∙ Matrix Inverse ∙ Matrix Divide ∙ Format ∙ Execute ∙ Materialise ∙ Range
Operators Monadic Each ∙ Commute ∙ Constant ∙ Replicate ∙ Expand ∙ Reduce ∙ Windowed Reduce ∙ Scan ∙ Outer Product ∙ Key ∙ I-Beam ∙ Spawn ∙ Function axis ∙ Identity (Null, Ident)
Dyadic Bind ∙ Compositions (Compose, Reverse Compose, Beside, Withe, Atop, Over) ∙ Inner Product ∙ Determinant ∙ Power ∙ At ∙ Under ∙ Rank ∙ Depth ∙ Variant ∙ Stencil ∙ Cut ∙ Direct definition (operator) ∙ Identity (Lev, Dex)
Quad names Index origin ∙ Comparison tolerance ∙ Migration level ∙ Atomic vector
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