Inner Product: Difference between revisions

From APL Wiki
Jump to navigation Jump to search
No edit summary
(Editing; contrast shape rules with conformability)
Line 1: Line 1:
{{Built-in|Inner Product|<nowiki>.</nowiki>}}, is a [[dyadic operator]], which will produce a [[dyadic function]] when applied with two [[dyadic function]]s. In APL, the inner product is a generalisation of the [https://en.wikipedia.org/wiki/Matrix_multiplication matrix product], which allows not only addition-multiplication, but any [[dyadic function]]s given.
{{Built-in|Inner Product|<nowiki>.</nowiki>}} is a [[dyadic operator]] that produces a [[dyadic function]] when applied with two dyadic functions. It's a generalisation of the [[wikipedia:Matrix multiplication|matrix product]], allowing not just addition-multiplication, but any [[dyadic function]]s given as [[operand]]s.


== Examples ==
== Examples ==
Line 25: Line 25:
</source>
</source>


Note that the [[shape]]s of the arguments must be compatible with each other: The last [[axis]] of the left argument must have the same length as the first axis of the right argument, or formally, for <source lang=apl inline>X f.g Y</source> it must be that <source lang=apl inline>(¯1↑⍴X)≡(1↑⍴Y)</source>. The shape of the result is <source lang=apl inline>(¯1↓⍴X),(1↓⍴Y)</source>.
The [[shape]]s of the arguments must be compatible with each other: The last [[axis]] of the left argument must have the same length as the first axis of the right argument, or formally, for <source lang=apl inline>X f.g Y</source> it must be that <source lang=apl inline>(¯1↑⍴X)≡(1↑⍴Y)</source>. Although this rule differs from [[conformability]], the arguments may also be subject to [[scalar extension|scalar]] or [[singleton extension]]. The shape of the result is <source lang=apl inline>(¯1↓⍴X),(1↓⍴Y)</source>.


For example, when applying inner product on two [[matrix|matrices]], the number of columns in the left array must match with number of rows in the right array, otherwise we will get an error.
For example, when applying inner product on two [[matrix|matrices]], the number of columns in the left array must match with number of rows in the right array, otherwise we will get an error.

Revision as of 19:36, 6 September 2021

.

Inner Product (.) is a dyadic operator that produces a dyadic function when applied with two dyadic functions. It's a generalisation of the matrix product, allowing not just addition-multiplication, but any dyadic functions given as operands.

Examples

      x ← 1 2 3
      y ← 4 5 6
      x ,.(⊂,) y ⍝ visualizing of pairing
┌─────────────┐
│┌───┬───┬───┐│
││1 4│2 5│3 6││
│└───┴───┴───┘│
└─────────────┘
      x {⊂⍺,'+',⍵}.{⊂⍺,'×',⍵} y ⍝ visualizing function application in matrix multiplication
┌───────────────────────────┐
│┌─────────────────────────┐│
││┌─────┬─┬───────────────┐││
│││1 × 4│+│┌─────┬─┬─────┐│││
│││     │ ││2 × 5│+│3 × 6││││
│││     │ │└─────┴─┴─────┘│││
││└─────┴─┴───────────────┘││
│└─────────────────────────┘│
└───────────────────────────┘
      x+.×y ⍝ matrix multiplication
32

The shapes of the arguments must be compatible with each other: The last axis of the left argument must have the same length as the first axis of the right argument, or formally, for X f.g Y it must be that (¯1↑⍴X)≡(1↑⍴Y). Although this rule differs from conformability, the arguments may also be subject to scalar or singleton extension. The shape of the result is (¯1↓⍴X),(1↓⍴Y).

For example, when applying inner product on two matrices, the number of columns in the left array must match with number of rows in the right array, otherwise we will get an error. 

      ⎕  ← x ← 2 3⍴⍳10
1 2 3
4 5 6
      ⎕ ← y ← 4 2⍴⍳10
1 2
3 4
5 6
7 8
      x+.×y 
LENGTH ERROR
      x+.×y
        ∧
      ⎕ ← y ← 3 2⍴⍳10 ⍝ reshape y to be compatible with x
      x+.×y
22 28
49 64

External links

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
Retrieved from ‘https://aplwiki.com/index.php?title=Inner_Product&oldid=7320’