A Relaxed Algorithm for Online Matrix Inversion

Storjohann, Arne; Yang, Shiyun

doi:10.1145/2755996.2756672

Cited by 12 publications

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“…., the vector (A P Q ) −1 · b P ∈ K s×1 . Recently we have discovered an online algorithm for relaxed matrix inversion [13] that can be used to compute all these vectors in time O(r ω ), provided that the final matrix A P ∈ K r×m has generic column rank profile. By incorporating the online matrix inversion algorithm together with a Toeplitz preconditioner [8] into algorithm ImprovedRankProfile, we can compute the row rank profile of a full column rank matrix A in time (r ω + |A|) 1+o (1) .…”

Section: Discussionmentioning

confidence: 99%

Linear independence oracles and applications to rectangular and low rank linear systems

Storjohann

Yang

2014

Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation

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Randomized algorithms are given for linear algebra problems on an input matrix A ∈ K n×m over a field K. We give an algorithm that simultaneously computes the row and column rank profiles of A in 2r 3 + (r 2 + n + m + |A|) 1+o(1) field operations from K, where r is the rank of A and |A| denotes the number of nonzero entries of A. Here, the +o(1) in cost estimates captures some missing log n and log m factors. The rank profiles algorithm is randomized of the Monte Carlo type: the correct answer will be returned with probability at least 1/2. Given a b ∈ K n×1 , we give an algorithm that either computes a particular solution vector x ∈ K m×1 to the system Ax = b, or produces an inconsistency certificate vector u ∈ K 1×n such that uA = 0 and ub = 0. The linear solver examines at most r + 1 rows and r columns of A and has running time 2r 3 + (r 2 + n + m + |R| + |C|)1+o (1) field operations from K, where |R| and |C| are the number of nonzero entries in the rows and columns, respectively, that are examined. The solver is randomized of the Las Vegas type: an incorrect result is never returned but the algorithm may report FAIL with probability at most 1/2. These cost estimates are achieved by making use of a novel randomized online data structure for the detection of linearly independent rows and columns.

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Section: Discussionmentioning

confidence: 99%

Linear independence oracles and applications to rectangular and low rank linear systems

Storjohann

Yang

2014

Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation

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“…This is a direct consequence of [Cheung et al, 2013, Theorem 2.11], along with the classic algorithm of [Bunch and Hopcroft, 1974] for fast matrix inversion. Alternatively, one could use the approach of [Storjohann and Yang, 2015] to compute the lexicographically minimal set of linearly independent rows in M , as well as a representation of the inverse, in the same running time.…”

Section: Multiplication With Error Correction Algorithmmentioning

confidence: 99%

Error Correction in Fast Matrix Multiplication and Inverse

Roche

2018

Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation

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We present new algorithms to detect and correct errors in the product of two matrices, or the inverse of a matrix, over an arbitrary field. Our algorithms do not require any additional information or encoding other than the original inputs and the erroneous output. Their running time is softly linear in the number of nonzero entries in these matrices when the number of errors is sufficiently small, and they also incorporate fast matrix multiplication so that the cost scales well when the number of errors is large. These algorithms build on the recent result of Gasieniec, Levcopoulos, Lingas, Pagh, and Tokuyama [2017] on correcting matrix products, as well as existing work on verification algorithms, sparse low-rank linear algebra, and sparse polynomial interpolation. 14 for i Ð 1, 2, . . . , r do 15 Set pJ i , eqth entry of E to c for each term cx e of f i 16 J Ð FindNonzeroRowspV Þ Ñ pC´EqV´ApBV q, ǫ 1 q 17 if #J ą r{2 then 18 k Ð 2k 19 if k ě 2n#J then return C´AB 20 foreach i P J do 21 Clear entries from row i of E added on this iteration 22 return E 17 for i Ð 1, 2, . . . , r do 18 Set pJ i , eqth entry of E to c for each term cx e of f i 19 J Ð FindNonzeroRowspV Þ Ñ V´pB`EqpAV q, ǫ 1 q 20 if #J ą r{2 then 21 k Ð 2k 22 if k ě 2n#J then return A´1´B 23 foreach i P J do 24 Clear entries from row i of E added on this iteration 25 return E

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“…An alternative way to compute row and column rank profiles has recently been proposed by Storjohann andYang (2014, 2015), reducing the time complexity from a deterministic O(mnr ω−2 ) to a Monte Carlo probabilistic 2r 3 + O(r 2 (log m + log n) + mn) in Storjohann and Yang (2014) first, and to (r ω + mn) 1+o(1) in Storjohann and Yang (2015), under the condition that the base field contains at least 2 min(m, n)(1+⌈log 2 m⌉+⌈log 2 n⌉) elements. We show how these results can be extended to the computation of the rank profile matrix.…”

Section: Storjohann and Yang's Algorithmmentioning

confidence: 99%

“…The r 3 term in this complexity is from the iterative construction of the inverses of the non-singular sub-matrices of order s for 1 ≤ s ≤ r, by rank one updates. To reduce this complexity to O(r ω ), Storjohann and Yang (2015) propose a relaxed computation of this online matrix inversion. In order to group arithmetic operations into matrix multiplications, their approach is to anticipate part of the updates on the columns that will be appended in the future.…”

Section: Storjohann and Yang's Algorithmmentioning

confidence: 99%

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Fast computation of the rank profile matrix and the generalized Bruhat decomposition

Dumas

Pernet

Sultan

2017

Journal of Symbolic Computation

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The row (resp. column) rank profile of a matrix describes the stair-case shape of its row (resp. column) echelon form. We describe a new matrix invariant, the rank profile matrix, summarizing all information on the row and column rank profiles of all the leading sub-matrices. We show that this normal form exists and is unique over a field but also over any principal ideal domain and finite chain ring. We then explore the conditions for a Gaussian elimination algorithm to compute all or part of this invariant, through the corresponding PLUQ decomposition. This enlarges the set of known elimination variants that compute row or column rank profiles. As a consequence a new Crout base case variant significantly improves the practical efficiency of previously known implementations over a finite field. With matrices of very small rank, we also generalize the techniques of Storjohann and Yang to the computation of the rank profile matrix, achieving an (r ω + mn) 1+o(1) time complexity for an m × n matrix of rank r, where ω is the exponent of matrix multiplication. Finally, we give connections to the Bruhat decomposition, and several of its variants and generalizations. Consequently, the algorithmic improvements made for the PLUQ factorization, and their implementation, directly apply to these decompositions. In particular, we show how a PLUQ decomposition revealing the rank profile matrix also reveals both a row and a column echelon form of the input matrix or of any of its leading sub-matrices, by a simple post-processing made of row and column permutations.

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A Relaxed Algorithm for Online Matrix Inversion

Cited by 12 publications

References 14 publications

Linear independence oracles and applications to rectangular and low rank linear systems

Linear independence oracles and applications to rectangular and low rank linear systems

Error Correction in Fast Matrix Multiplication and Inverse

Fast computation of the rank profile matrix and the generalized Bruhat decomposition

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