Self-adjunctions and matrices

Došen, Kosta; Petric, Z.

doi:10.1016/s0022-4049(03)00084-7

Cited by 23 publications

(96 citation statements)

References 37 publications

(57 reference statements)

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“…where (dropping subscripts) η = ǫ ○ [41]. From this we infer not only the cancellationlaw of devectorization, but also a closed formula for vectorization,…”

Section: Devectorizationmentioning

confidence: 76%

“…Our final calculation shows how iterated biproducts "explain" the traditional forloop implementation of MMM. Interestingly enough, such iterative implementation is shown to stem from generalized divide-and-conquer (35): (40) and (41) (41) and generalized (27) and (26) …”

Section: Calculating Triple Nested Loopsmentioning

confidence: 99%

“…As we did for other matrix combinators, we shall capture such intuition formally in the form of the universal property which wraps up the isomorphism above, this time finding inspiration in [41]:…”

Section: Column-major Vectorizationmentioning

confidence: 99%

“…Summing up, we are in presence of an adjunction between functor FX = id k ⊗ X and itself -a self-adjunction [41] -inducing a monoidal closed structure in the category. The root for this is again the biproduct, entailing the same functor ⊕ (62) for both coproduct and product.…”

Section: Self-adjunctionmentioning

confidence: 99%

“…10], follows from universal properties (67,77) by index-free calculation. This is an advance over the traditional, index-wise matrix representations and proofs [33,35,41] where notation is often quite loose, full of dot-dot-dots. We will also stress on the role of matrix types in the reasoning.…”

Section: Unfolding Vectorization Algebramentioning

confidence: 99%

See 4 more Smart Citations

Typing linear algebra: A biproduct-oriented approach

Macedo

Oliveira²

2013

Science of Computer Programming

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Interested in formalizing the generation of fast running code for linear algebra applications, the authors show how an index-free, calculational approach to matrix algebra can be developed by regarding matrices as morphisms of a category with biproducts. This shifts the traditional view of matrices as indexed structures to a type-level perspective analogous to that of the pointfree algebra of programming. The derivation of fusion, cancellation and abide laws from the biproduct equations makes it easy to calculate algorithms implementing matrix multiplication, the central operation of matrix algebra, ranging from its divide-and-conquer version to its vectorization implementation.From errant attempts to learn how particular products and coproducts emerge from biproducts, not only blocked matrix algebra is rediscovered but also a way of extending other operations (e.g. Gaussian elimination) blockwise, in a calculational style, is found.The prospect of building biproduct-based type checkers for computer algebra systems such as MATLAB™ is also considered.Keywords: Linear algebra, categories of matrices, algebra of programming

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“…where (dropping subscripts) η = ǫ ○ [41]. From this we infer not only the cancellationlaw of devectorization, but also a closed formula for vectorization,…”

Section: Devectorizationmentioning

confidence: 76%

Section: Calculating Triple Nested Loopsmentioning

confidence: 99%

Section: Column-major Vectorizationmentioning

confidence: 99%

Section: Self-adjunctionmentioning

confidence: 99%

Section: Unfolding Vectorization Algebramentioning

confidence: 99%

See 3 more Smart Citations

Typing linear algebra: A biproduct-oriented approach

Macedo

Oliveira²

2013

Science of Computer Programming

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Matrices as Arrows!

Macedo

Oliveira

2010

Lecture Notes in Computer Science

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Motivated by the need to formalize generation of fast running code for linear algebra applications, we show how an index-free, calculational approach to matrix algebra can be developed by regarding matrices as morphisms of a category with biproducts. This shifts the traditional view of matrices as indexed structures to a type-level perspective analogous to that of the pointfree algebra of programming. The derivation of fusion, cancellation and abide laws from the biproduct equations makes it easy to calculate algorithms implementing matrix multiplication, the kernel operation of matrix algebra, ranging from its divide-and-conquer version to the conventional, iterative one. From errant attempts to learn how particular products and coproducts emerge from biproducts, we not only rediscovered block-wise matrix combinators but also found a way of addressing other operations calculationally such as e.g. Gaussian elimination. A strategy for addressing vectorization along the same lines is also given.

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