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• 1
Electronic Resource
Springer
Numerische Mathematik 76 (1997), S. 383-402
ISSN: 0945-3245
Keywords: Mathematics Subject Classification (1991):65F05
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Summary. A new $O(mn)$ algorithm for triangularizing an $m \times n$ Toeplitz matrix is presented. The algorithm is based on the previously developed recursive algorithms that exploit the Toeplitz structure and compute each row of the triangular factor via updating and downdating steps. A forward error analysis for this existing recursive algorithm is presented, which allows us to monitor the conditioning of the problem, and use the method of corrected semi-normal equations to obtain higher accuracy for certain ill-conditioned matrices. Numerical experiments show that the new algorithm improves the accuracy significantly while the computational complexity stays in $O(mn)$ .
Type of Medium: Electronic Resource
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• 2
Electronic Resource
Springer
Numerische Mathematik 82 (1999), S. 599-619
ISSN: 0945-3245
Keywords: Mathematics Subject Classification (1991):65F20, 15A24
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Summary. An orthogonal Procrustes problem on the Stiefel manifold is studied, where a matrix Q with orthonormal columns is to be found that minimizes $\| AQ-B\|_{\rm F}$ for an $l \times m$ matrix A and an $l \times n$ matrix B with $l \geq m$ and $m 〉 n$ . Based on the normal and secular equations and the properties of the Stiefel manifold, necessary conditions for a global minimum, as well as necessary and sufficient conditions for a local minimum, are derived.
Type of Medium: Electronic Resource
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• 3
Electronic Resource
New York, NY [u.a.] : Wiley-Blackwell
ISSN: 1070-5325
Keywords: bidiagonalization ; least squares ; minimum norm solution ; rank-deficient ; regularization ; Riley-Golub iteration ; singular value decomposition ; Engineering ; Numerical Methods and Modeling
Source: Wiley InterScience Backfile Collection 1832-2000
Topics: Mathematics
Notes: In this paper we consider the solution of linear least squares problems          minx∥Ax - b∥22 where the matrix A ∊ Rm × n is rank deficient. Put p = min{m, n}, let σi, i = 1, 2,…, p, denote the singular values of A, and let ui and vi denote the corresponding left and right singular vectors. Then the minimum norm solution of the least squares problem has the form x* = ∫ri = 1(uTib/σi)vi, where r ≤ p is the rank of A.The Riley-Golub iteration,          xk + 1 = arg minx{∥Ax - b∥22 + λ∥x - xk∥22} converges to the minimum norm solution if x0 is chosen equal to zero. The iteration is implemented so that it takes advantage of a bidiagonal decomposition of A. Thus modified, the iteration requires only O(p) flops (floating point operations). A further gain of using the bidiagonalization of A is that both the singular values σi and the scalar products uTib can be computed at marginal extra cost. Moreover, we determine the regularization parameter, λ, and the number of iterations, k, in a way that minimizes the difference x* - xk with respect to a certain norm. Explicit rules are derived for calculating these parameters.One advantage of our approach is that the numerical rank can be easily determined by using the singular values. Furthermore, by the iterative procedure, x* is approximated without computing the singular vectors of A. This gives a fast and reliable method for approximating minimum norm solutions of well-conditioned rank-deficient least squares problems. Numerical experiments illustrate the viability of our ideas, and demonstrate that the new method gives more accurate approximations than an approach based on a QR decomposition with column pivoting. © 1998 John Wiley & Sons, Ltd.
Type of Medium: Electronic Resource
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• 4
Electronic Resource
Springer
BIT 24 (1984), S. 467-472
ISSN: 1572-9125
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Type of Medium: Electronic Resource
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• 5
Electronic Resource
Springer
BIT 22 (1982), S. 487-502
ISSN: 1572-9125
Keywords: Least squares ; pseudoinverse ; weight matrix ; constraint ; generalized singular values
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract The weighted pseudoinverse providing the minimum semi-norm solution of the weighted linear least squares problem is studied. It is shown that it has properties analogous to those of the Moore-Penrose pseudoinverse. The relation between the weighted pseudoinverse and generalized singular values is explained. The weighted pseudoinverse theory is used to analyse least squares problems with linear and quadratic constraints. A numerical algorithm for the computation of the weighted pseudoinverse is briefly described.
Type of Medium: Electronic Resource
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• 6
Electronic Resource
Springer
BIT 17 (1977), S. 134-145
ISSN: 1572-9125
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract Two regularization methods for ill-conditioned least squares problems are studied from the point of view of numerical efficiency. The regularization methods are formulated as quadratically constrained least squares problems, and it is shown that if they are transformed into a certain standard form, very efficient algorithms can be used for their solution. New algorithms are given, both for the transformation and for the regularization methods in standard form. A comparison to previous algorithms is made and it is shown that the overall efficiency (in terms of the number of arithmetic operations) of the new algorithms is better.
Type of Medium: Electronic Resource
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• 7
Electronic Resource
Springer
BIT 30 (1990), S. 466-483
ISSN: 1572-9125
Keywords: 15A06 ; 65F30 ; 65K10 ; ill-posed ; least squares ; constraint ; functional ; algorithm
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract We study a linear, discrete ill-posed problem, by which we mean a very ill-conditioned linear least squares problem. In particular we consider the case when one is primarily interested in computing a functional defined on the solution rather than the solution itself. In order to alleviate the ill-conditioning we require the norm of the solution to be smaller than a given constant. Thus we are lead to minimizing a linear functional subject to two quadratic constraints. We study existence and uniqueness for this problem and show that it is essentially equivalent to a least squares problem with a linear and a quadratic constraint, which is easier to handle computationally. Efficient algorithms are suggested for this problem.
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