Gennadi I. Malaschonok

Applications of singular-value decomposition (SVD)

Let A be an m×n matrix with m≥n. Then one form of the singular-value decomposition of A is A=UTΣV,where U and V are orthogonal and Σ is square diagonal. That is, UUT=Irank(A), VVT=Irank(A), U is rank(A)×m, V is rank(A)×n and Σ=σ10⋯000σ2⋯00⋮⋮⋱⋮⋮00⋯σrank(A)−1000⋯0σrank(A)is a rank(A)×rank(A) diagonal matrix. In addition σ1≥σ2≥⋯≥σrank(A)>0. The σi’s are called the singular values of A and their number is equal to the rank of A. The ratio σ1/σrank(A) can be regarded as a condition number of the matrix A.

Effective Matrix Methods in Commutative Domains

Effective matrix methods for solving standard linear algebra problems in a commutative domains are discussed. Two of them are new. There are a methods for computing adjoined matrices and solving system of linear equations in a commutative domains.

Fast generalized bruhat decomposition

The deterministic recursive pivot-free algorithms for computing the generalized Bruhat decomposition of the matrix in the field and for the computation of the inverse matrix are presented. This method has the same complexity as algorithm of matrix multiplication, and it is suitable for the parallel computer systems.

Efficient algorithms for computing the characteristic polynomial in a domain

Abstract Two new sequential methods are given for computing the characteristic polynomial of an endomorphism of a free finite rank- n module over a domain, that require O( n 3 ) ring operations with exact divisions.

Generalized Bruhat Decomposition in Commutative Domains

Deterministic recursive algorithms for the computation of generalized Bruhat decomposition of the matrix in commutative domain are presented. This method has the same complexity as the algorithm of matrix multiplication.

Computations in modules over commutative domains

This paper is a review of results on computational methods of linear algebra over commutative domains. Methods for the following problems are examined: solution of systems of linear equations, computation of determinants, computation of adjoint and inverse matrices, computation of the characteristic polynomial of a matrix.

Computation of the adjoint matrix

The best method for computing the adjoint matrix of an order n matrix in an arbitrary commutative ring requires O(nβ+1/3 log n log log n) operations, provided that the complexity of the algorithm for multiplying two matrices is γnβ+o(nβ). For a commutative domain – and under the same assumptions – the complexity of the best method is 6γnβ/(2β–2)+o(nβ). In the present work a new method is presented for the computation of the adjoint matrix in a commutative domain. Despite the fact that the number of operations required is now 1.5 times more, than that of the best method, this new method permits a better parallelization of the computational process and may be successfully employed for computations in parallel computational systems.

Fast Matrix Computation of Subresultant Polynomial Remainder Sequences

We present an improved (faster) variant of the matrix-triangularization subresultant prs method for the computation of a greatest common divisor of two polynomials A and B (of degrees dA and dB, respectively) along with their polynomial remainder sequence [1]. The computing time of our fast method is 0(n2+slog ∥C∥2), for standard arithmetic and 0(((n1+s+n 3 log ∥C∥)(log n+ log ∥C∥)2) for the Chinese remainder method, where n = d A + d B, ∥C∥ is the maximal coefficient of the two polynomials and the best known s < 2.356. By comparison, the computing time of the old version is 0(n 5 log ∥C∥2 ).

Triangular Decomposition of Matrices in a Domain

Deterministic recursive algorithms for the computation of matrix triangular decompositions with permutations like LU and Bruhat decomposition are presented for the case of commutative domains. This decomposition can be considered as a generalization of LU and Bruhat decompositions because they both may easily be obtained from this triangular decomposition. Algorithms have the same complexity as the algorithm of matrix multiplication.

Solution of Systems of Linear Diophantine Equations

Two new methods to solve linear systems of Diophantine equations are proposed - modular (CRT) and p-adic (Hensel). Each of them allows to obtain solutions of a system with the size n x m with the complexity O(nsm). For quasi-square systems, the p-adic method allows to obtain solution with the complexity O(n 3), and the modular method with complexity O(n s+1). Both estimates have the accuracy up to the logarithmic multipliers, s being the power in the estimation of matrix multiplication time.