Determinant

Determinant (.) is a primitive dyadic operator that takes two dyadic functions for operands and produces a monadic function. Much like Inner Product for matrix products, it generalizes the matrix determinant, so that -.× computes the determinant and +.× the permanent. Determinant has been implemented in SHARP APL, NARS2000, and J; however, the J definition differs by factoring out a reduction from the definition, so that -/ .* is the matrix determinant.

Unlike the inner product operator, which performs $O(n^{3})$ arithmetic operations on square matrices of side length $n$ given arithmetic operands, definitions of the determinant use a super-polynomial number of evaluations, for example $O(n\cdot n!)$ in SHARP APL's definition. Computing the permanent in particular is not believed to be possible in polynomial time, which would rule out a polynomial-time definition for any generalization of it. The specific determinant function -.× can be computed much faster (for example by LU decomposition in $O(n^{3})$ ), and is always handled specially for performance. Other combinations of operands may also have optimized implementations.

The determinant operator was described by Ken Iverson in "Determinant-Like Functions Produced by the Dot Operator"^[1] and also implemented in SHARP APL^[2] in 1982. It also appears in Iverson's later publications Rationalized APL and A Dictionary of APL.

Documentation

NARS2000
J NuVoc, Dictionary

References

↑ Ken Iverson. SATN-42: Determinant-Like Functions Produced by the Dot Operator. 1982-04-01.
↑ IPSA Newsletter July/August 1982 (pdf)

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)
Operators	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

[1] Ken Iverson. SATN-42: Determinant-Like Functions Produced by the Dot Operator. 1982-04-01.

[2] IPSA Newsletter July/August 1982 (pdf)

[1]

[2]

Determinant

Documentation

References

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