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• 1
Electronic Resource
Springer
Numerische Mathematik 64 (1993), S. 295-321
ISSN: 0945-3245
Keywords: 65F05 ; 15A57 ; 42C05
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Summary The solution of systems of linear equations with Hankel coefficient matrices can be computed with onlyO(n 2) arithmetic operations, as compared toO(n 3) operations for the general cases. However, the classical Hankel solvers require the nonsingularity of all leading principal submatrices of the Hankel matrix. The known extensions of these algorithms to general Hankel systems can handle only exactly singular submatrices, but not ill-conditioned ones, and hence they are numerically unstable. In this paper, a stable procedure for solving general nonsingular Hankel systems is presented, using a look-ahead technique to skip over singular or ill-conditioned submatrices. The proposed approach is based on a look-ahead variant of the nonsymmetric Lanczos process that was recently developed by Freund, Gutknecht, and Nachtigal. We first derive a somewhat more general formulation of this look-ahead Lanczos algorithm in terms of formally orthogonal polynomials, which then yields the look-ahead Hankel solver as a special case. We prove some general properties of the resulting look-ahead algorithm for formally orthogonal polynomials. These results are then utilized in the implementation of the Hankel solver. We report some numerical experiments for Hankel systems with ill-conditioned submatrices.
Type of Medium: Electronic Resource
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• 2
Electronic Resource
Springer
Numerische Mathematik 72 (1996), S. 391-417
ISSN: 0945-3245
Keywords: Mathematics Subject Classification (1991):65F15
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Summary. We present a numerical algorithm for computing a few extreme generalized singular values and corresponding vectors of a sparse or structured matrix pair $\{A,B\}$ . The algorithm is based on the CS decomposition and the Lanczos bidiagonalization process. At each iteration step of the Lanczos process, the solution to a linear least squares problem with $(A^{\rm T},B^{\rm T})^{\rm T}$ as the coefficient matrix is approximately computed, and this consists the only interface of the algorithm with the matrix pair $\{A,B\}$ . Numerical results are also given to demonstrate the feasibility and efficiency of the algorithm.
Type of Medium: Electronic Resource
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• 3
Electronic Resource
Springer
BIT 35 (1995), S. 448-452
ISSN: 1572-9125
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract A finite step algorithm is given such that for any two vectorsa, λ ∈R n witha majorized by λ, it computes a symmetric matrixH ∈R n x n with the elements ofa and λ as its diagonal entries and eigenvalues, respectively.
Type of Medium: Electronic Resource
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• 4
Electronic Resource
Springer
BIT 36 (1996), S. 14-40
ISSN: 1572-9125
Keywords: Orthogonal decomposition ; downdating ; error analysis ; subspaces
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract An alternative to performing the singular value decomposition is to factor a matrixA into $$A = U\left( {\begin{array}{*{20}c} C \\ 0 \\ \end{array} } \right)V^T$$ , whereU andV are orthogonal matrices andC is a lower triangular matrix which indicates a separation between two subspaces by the size of its columns. These subspaces are denoted byV = (V 1,V 2), where the columns ofC are partitioned conformally intoC = (C 1,C 2) with ‖C 2 ‖ F ≤ ε. Here ε is some tolerance. In recent years, this has been called the ULV decomposition (ULVD). If the matrixA results from statistical observations, it is often desired to remove old observations, thus deleting a row fromA and its ULVD. In matrix terms, this is called a downdate. A downdating algorithm is proposed that preserves the structure in the downdated matrix $$\bar C$$ to the extent possible. Strong stability results are proven for these algorithms based upon a new perturbation theory.
Type of Medium: Electronic Resource
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• 5
Electronic Resource
Springer
BIT 31 (1991), S. 375-379
ISSN: 1572-9125
Keywords: 65F20 ; 65F25
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract We present a numerical algorithm for computing the implicit QR factorization of a product of three matrices, and we illustrate the technique by applying it to the generalized total least squares and the restricted total least squares problems. We also demonstrate how to take advantage of the block structures of the underlying matrices in order to reduce the computational work.
Type of Medium: Electronic Resource
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• 6
Electronic Resource
Springer
BIT 31 (1991), S. 711-726
ISSN: 1572-9125
Keywords: 65F25 ; 65F30 ; 65F35 ; singular value decomposition ; matrix product ; implicit Kogbetliantz technique
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract A new decomposition of a matrix triplet (A, B, C) corresponding to the singular value decomposition of the matrix productABC is developed in this paper, which will be termed theProduct-Product Singular Value Decomposition (PPSVD). An orthogonal variant of the decomposition which is more suitable for the purpose of numerical computation is also proposed. Some geometric and algebraic issues of the PPSVD, such as the variational and geometric interpretations, and uniqueness properties are discussed. A numerical algorithm for stably computing the PPSVD is given based on the implicit Kogbetliantz technique. A numerical example is outlined to demonstrate the accuracy of the proposed algorithm.
Type of Medium: Electronic Resource
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• 7
Electronic Resource
Springer
Numerical algorithms 4 (1993), S. 101-133
ISSN: 1572-9265
Source: Springer Online Journal Archives 1860-2000
Topics: Computer Science , Mathematics
Notes: Abstract Solving Total Least Squares (TLS) problemsAX≈B requires the computation of the noise subspace of the data matrix [A;B]. The widely used tool for doing this is the Singular Value Decomposition (SVD). However, the SVD has the drawback that it is computationally expensive. Therefore, we consider here a different so-called rank-revealing two-sided orthogonal decomposition which decomposes the matrix into a product of a unitary matrix, a triangular matrix and another unitary matrix in such a way that the effective rank of the matrix is obvious and at the same time the noise subspace is exhibited explicity. We show how this decompsition leads to an efficient and reliable TLS algorithm that can be parallelized in an efficient way.
Type of Medium: Electronic Resource
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• 8
Book
Title: Restricted Singular Value Decomposition of Matrix Triplets. /; Preprint SC 89-02
Author: Zha, Hongyuan
Year of publication: 1989
Series Statement: Preprint / Konrad-Zuse-Zentrum für Informationstechnik Berlin Preprint SC 89-02
ISSN: 0933-7911
Type of Medium: Book
Language: English
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• 9
Title: ¬A¬ Numerical Algorithm for Computing the Restricted Singular Value Decomposition of Matrix Triplets. /; Preprint SC 89-01
Author: Zha, Hongyuan
Year of publication: 1989
Series Statement: Preprint / Konrad-Zuse-Zentrum für Informationstechnik Berlin Preprint SC 89-01
ISSN: 0933-7911
Type of Medium: Book
Language: English
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• 10
Publication Date: 2014-02-26
Description: This paper presents a numerical algorithm for computing the restricted singular value decomposition of matrix triplets (RSVD). It is shown that one can use unitary transformations to separate the regular part from a general matrix triplet. After preprocessing on the regular part, one obtains a matrix triplet consisting of three upper triangular matrices of the same dimensions. The RSVD of this special matrix triplet is computed using the implicit Kogbetliantz technique. The algorithm is well suited for parallel computation. {\bf Keywords:} Restricted singular values, matrix triplets, unitary transformations, implicit Kogbetliantz technique.
Keywords: ddc:000
Language: English
Type: reportzib , doc-type:preprint
Format: application/pdf
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