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Blockwise analysis for solving linear systems of equations
Smoktunowicz, Alicja 한국산업정보응용수학회 1999 한국산업정보응용수학회 Vol.3 No.1
We investigate some techniques of iterative refinement of solutions of a nonsingular system Ax = b with A partitioned into blocks using only single precision arithmetic. We prove that iterative refinement improves a blockwise measure of backward stability. Some applications of the results for the least squares problem (LS) will be also considered.
The strong stability of algorithms for solving the symmetric eigenproblem
SMOKTUNOWICZ, ALICJA 한국산업정보응용수학회 2003 한국산업정보응용수학회 Vol.7 No.1
The concepts of stability of algorithms for solving the symmetric and generalized symmetric-definite eigenproblems are discussed. An algorithm for solving the symmetric eigenproblem Ax = λx is stable if the computed solution z is the exact solution of some slightly perturbed system (A+E)z = λz. We use both normwise approach and componentwise way of measuring the size of the perturbations in data. If E preserves symmetry we say that an algorithm is strongly stable (in a normwise or componentwise sense, respectively). The relations between the stability and strong stability are investigated for some classes of matrices.
On Numerical Properties of Complex Symmetric Householder Matrices
SMOKTUNOWICZ, ALICJA,GRABARSKI, ADAM 한국산업정보응용수학회 2003 한국산업정보응용수학회 Vol.7 No.2
Analysis is given of construction and stability of complex symmetric analogues of Householder matrices, with applications to the eigenproblem for such matrices. We investigate numerical properties of the deflation of complex symmetric matrices by using complex symmetric Householder transformations. The proposed method is very similar to the well-known deflation technique for real symmetric matrices (Cf. [16], pp. 586-595). In this paper we present an error analysis of one step of the deflation of complex symmetric matrices.
Accuracy of Iterative Refinement of Eigenvalue Problems
Jolanta, Gluchowska-Jastrzebska,Alicja, Smoktunowicz 한국산업정보응용수학회 2000 한국산업정보응용수학회 Vol.4 No.1
We investigate numerical properties of Newton's algorithm for improving an eigenpair of a real matrix A using only fixed precision arithmetic. We show that under natural assumptions it produces an eigenpair of a componentwise small relative perturbation of the data matrix A.