The talk is organized as follows: Section 2 provides some necessary background ... with one-sided decomposition (H-inver
Two recursive pivot-free algorithms for matrix inversion Gennadi Malaschonok and Mikhail Zuyev Tambov State University, Laboratory for Algebraic Computations, e-mail:
[email protected] We present two deterministic recursive pivot-free algorithms for matrix inversion. These algorithms improve on previous methods that required invertible on-diagonal blocks, or required row- or column-based pivoting. These algorithms have the same complexity as matrix multiplication, and they are recursive algorithms that do not require pivoting. Therefore they are suitable for the parallel computer systems. Fast matrix multiplication and fast block matrix inversion were discovered by Strassen [1]. We can write inverse matrix A−1 in the form −1 I −A−1 C I 0 I 0 A 0 , 0 I 0 (D − BA−1 C)−1 −B I 0 I A C if A = is an invertible matrix with invertible block A. B D The complexity of Strassen’s recursive algorithm for block matrix inversion is the same as the complexity of an algorithm for matrix multiplication. Other recursive methods for adjoint and inverse matrix computation have the complexity of matrix multiplications too (see for example [3-6]). This is another form of the inverse matrix A−1 −1 I −A−1 C I 0 I 0 A 0 0 I 0 (B −1 D − A−1 C)−1 −I I 0 B −1 for invertible matrix A with invertible blocks A and B. Here block inversion may be done simultaneously for blocks A and B. So this form gives a method with better parallelization of the computational process. In all these algorithms it is assumed that principal minors are invertible and the leading elements are nonzero as in most of the direct algorithms for matrix inversion. It is necessary to find a suitable nonzero element and to perform permutations of matrix columns or rows. Such permutation is not a very difficult operation for sequential computations but it is a difficult operation for parallel computations. 1
The problem of obtaining such pivot-free algorithm and usefullness of such algorithm was studied in [2],[3] by S.Watt. He presented the algorithm that is based on the following identity for a nonsingular matrix: A−1 = (AT A)−1 AT . Here AT is the transposed matrix to A and all principal minors of the matrix At A are nonzero. It is nice and simple decision but unfortunately two additional specialized matrix multiplications are used in this algorithm. We suggest two algorithms for matrix inversion. These algorithms have the same complexity as matrix multiplication and do not require pivoting. For singular matrices they allow to obtain a nonsingular block of the biggest size. These algorithms may be used in any field, including the reals and the complexes. But we do not require numerical stability as usually for block method. So the most interesting cases are finite fields, field of rationals and their extensions. The preliminary short versions of this talk is in [6,7]. The talk is organized as follows: Section 2 provides some necessary background and notations. Section 3 presents our method with two-sided decomposition (E-inversion) with illustration and example. Section 4 presents our method with one-sided decomposition (H-inversion) with illustration and example.
Список литературы [1] Strassen V.: Gaussian Elimination is not optimal. Numerische Mathematik. 13, (1969) 354–356. [2] Watt S.M. Pivot-Free Block Matrix Inversion. Maple Conference 2006, July 23-26, Waterloo, Canada. http://www.csd.uwo.ca/ watt/pub/reprints/2006-mc-bminv-poster.pdf [3] Watt S.M. Pivot-Free Block Matrix Inversion. http://www.csd.uwo.ca/ watt/pub/reprints/2006-synasc-bminv.pdf [4] Malaschonok G.I.: Effective Matrix Methods in Commutative Domains. In: Formal Power Series and Algebraic Combinatorics. Springer, Berlin (2000) 506–517. [5] Akritas A. and Malaschonok G.: Computation of Adjoint Matrix. In: Fourth International Workshop on Computer Algebra Systems and Applications (CASA 2006), LNCS 3992. Springer, Berlin (2006) 486-489. [6] Malaschonok G.I.: Parallel Algorithms of Computer Algebra. Materials of the conference dedicated for the 75 years of the Mathematical and Physical Dep. of Tambov State University. /November 22-24, 2005/. Tambov. TSU (2005) 44-56. [7] Malaschonok G.I. and Zuyev M.S.: Generalized algorithm for computing of inverse matrix. 11-th conference "Derzhavinskie Chtenia". /February 2-6, 2006/. Tambov. TSU (2006) 58-62. 2
