Fast deterministic computation of determinants of dense matrices

Abbott, John; Bronstein, Manuel; Mulders, Thom

doi:10.1145/309831.309934

Cited by 40 publications

(93 citation statements)

References 7 publications

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“…One such method is Clarkson's algorithm (Brönnimann & Yvinec 2000;Clarkson 1992), where the number of mantissa bits in the intermediate floating point scalars that are necessary for obtaining a correct sign depends on the orthogonal defect of the matrix. If the matrix has a large first invariant factor, Chinese remaindering can be employed in connection with computing the solution of a random linear system via Hensel lifting (Abbott et al 1999;Pan 1988).…”

Section: Introductionmentioning

confidence: 99%

On the complexity of computing determinants

Kaltofen

Villard²

2005

comput. complex.

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Abstract. We present new baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems. Our algorithms compute the determinant, characteristic polynomial, Frobenius normal form and Smith normal form of a dense n × n matrix A with integer entries in (n 3.2 log A ) 1+o(1) and (n 2.697263 log A ) 1+o(1) bit operations; here A denotes the largest entry in absolute value and the exponent adjustment by "+o(1)" captures additional factors C 1 (log n) C 2 (loglog A ) C 3 for positive real constants C 1 , C 2 , C 3 . The bit complexity (n 3.2 log A ) 1+o(1) results from using the classical cubic matrix multiplication algorithm. Our algorithms are randomized, and we can certify that the output is the determinant of A in a Las Vegas fashion. The second category of problems deals with the setting where the matrix A has elements from an abstract commutative ring, that is, when no divisions in the domain of entries are possible. We present algorithms that deterministically compute the determinant, characteristic polynomial and adjoint of A with n 3.2+o(1) and O(n 2.697263 ) ring additions, subtractions and multiplications.

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Section: Introductionmentioning

confidence: 99%

On the complexity of computing determinants

Kaltofen

Villard²

2005

comput. complex.

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“…Suppose that the dimension of the matrix is n × n and that the maximum bit length of all entries is b. The algorithm by [10] requires (n 3.5 b 2.5 ) 1+o (1) fixed precision, that is, bit operations. Here and in the following we use the exponent "+o(1)" to capture missing polylogarithmic factors O((log n) C 1 (log b) C 2 ), where C1, C2 are constants ("soft-O").…”

Section: Introductionmentioning

confidence: 99%

“…1+o (1) [20, Section 2]. Both algorithms use randomization and the algorithm in [10] is Monte Carlo-always fast and probably correct-and the one in [20] is Las Vegas-always correct and probably fast.…”

Section: Introductionmentioning

confidence: 99%

“…Both algorithms can be speeded by asymptotically fast subcubic matrix multiplication algorithmsà la Strassen [8,7,14]. By blocking [6,16,29,30] the baby steps/giant steps algorithm can be further improved, which yields the currently fastest known algorithms [20,Section 3] of bit complexity (n 3+1/3 b) 1+o (1) , that without subcubic matrix multiplication and without the FFTbased polynomial "half" GCD proceduresà la Knuth [23; 2, Chapter 8], and of bit complexity n 2.698 b 1+o(1) with subcubic matrix multiplication and FFT-based polynomial GCD procedures.…”

Section: Introductionmentioning

confidence: 99%

“…In our survey [21] we list Gaussian elimination combined with Chinese remaindering, Hensel lifting [1] and the above cited result [10] as propitious. For example, if the the determinant ∆ is a small integer, Chinese remaindering can employ what is know as "early termination."…”

Section: Introductionmentioning

confidence: 99%

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