Efficient matrix preconditioners for black box linear algebra

Chen, Li; Eberly, Wayne; Kaltofen, Erich; Saunders, B. David; Turner, William J.; Villard, Gilles

doi:10.1016/s0024-3795(01)00472-4

Cited by 52 publications

(72 citation statements)

References 10 publications

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“…Also, we have not addressed preconditioners. The preconditioners such as diagonal, Toeplitz, butterfly (Chen et al, 2002), either apply only for large fields or have only large field analyses. One can generally use an extension field to get the requisite cardinality, but the computational cost is high.…”

Section: Resultsmentioning

confidence: 99%

Probabilistic analysis of Wiedemann's algorithm for minimal polynomial computation

Harrison

Johnson

Saunders

2014

ACM Commun. Comput. Algebra

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Blackbox algorithms for linear algebra problems start with projection of the sequence of powers of a matrix to a sequence of vectors (Lanczos), a sequence of scalars (Wiedemann) or a sequence of smaller matrices (block methods). Such algorithms usually depend on the minimal polynomial of the resulting sequence being that of the given matrix. Here exact formulas are given for the probability that this occurs. They are based on the generalized Jordan normal form (direct sum of companion matrices of the elementary divisors) of the matrix. Sharp bounds follow from this for matrices of unknown elementary divisors. The bounds are valid for all finite field sizes and show that a small blocking factor can give high probability of success for all cardinalities and matrix dimensions.

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Section: Resultsmentioning

confidence: 99%

Probabilistic analysis of Wiedemann's algorithm for minimal polynomial computation

Harrison

Johnson

Saunders

2014

ACM Commun. Comput. Algebra

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“…We have described a framework that permits black box matrix multiplication, which can be employed, for example, in the pre-conditioners needed in some of the algorithms [3]. We have also given a means to incorporate external memory managers, which allows the use of external garbage collected libraries in LinBox and which can implement a memory model where allocation is distributed over several computers.…”

Section: Discussionmentioning

confidence: 99%

Generic matrix multiplication and memory management in linBox

Kaltofen

Morozov

Yuhasz

2005

Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation

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We describe the design and implementation of two components in the LinBox library. The first is an implementation of black box matrix multiplication as a lazy matrixtimes-matrix product. The implementation uses template meta-programming to set the intermediate vector type used during application of the matrix product. We also describe an interface mechanism that allows incorporation of external components with native memory management such as garbage collection into LinBox. An implementation of the interface based on SACLIB's field arithmetic procedures is presented.

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“…The first condition is avoided with probability no less than 1 − (bn log h) 1+o(1) /2 w . Here the quantity bn log h is proportional to the number of integers tested, the log h factor representing the prime density (5). The integer w is the number of witnesses for compositness that are probed for each integer.…”

Section: Baby Steps/giant Steps With Early Terminationmentioning

confidence: 99%