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  • 65F20  (3)
  • 1
    Electronic Resource
    Electronic Resource
    Quadratically constrained least squares and quadratic problems (1991)
    Springer
    Numerische Mathematik 59 (1991), S. 561-580 
    Details
    ISSN: 0945-3245
    Keywords: 65F20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We consider the following problem: Compute a vectorx such that ∥Ax−b∥2=min, subject to the constraint ∥x∥2=α. A new approach to this problem based on Gauss quadrature is given. The method is especially well suited when the dimensions ofA are large and the matrix is sparse. It is also possible to extend this technique to a constrained quadratic form: For a symmetric matrixA we consider the minimization ofx T A x−2b T x subject to the constraint ∥x∥2=α. Some numerical examples are given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385796
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Least-squares fitting of circles and ellipses (1994)
    Springer
    BIT 34 (1994), S. 558-578 
    Details
    ISSN: 1572-9125
    Keywords: 65H10 ; 65F20 ; Least squares ; curve fitting ; singular value decomposition
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Fitting circles and ellipses to given points in the plane is a problem that arises in many application areas, e.g., computer graphics, coordinate meteorology, petroleum engineering, statistics. In the past, algorithms have been given which fit circles and ellipses insome least-squares sense without minimizing the geometric distance to the given points. In this paper we present several algorithms which compute the ellipse for which thesum of the squares of the distances to the given points is minimal. These algorithms are compared with classical simple and iterative methods. Circles and ellipses may be represented algebraically, i.e., by an equation of the formF(x)=0. If a point is on the curve, then its coordinates x are a zero of the functionF. Alternatively, curves may be represented in parametric form, which is well suited for minimizing the sum of the squares of the distances.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01934268
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  • 3
    Electronic Resource
    Electronic Resource
    Adaptive Lanczos methods for recursive condition estimation (1991)
    Springer
    Numerical algorithms 1 (1991), S. 1-19 
    Details
    ISSN: 1572-9265
    Keywords: AMS(MOS) ; 15A18 ; 65F10 ; 65F20 ; 65F35 ; Adaptive methods ; condition estimation ; control ; downdating ; eigenvalues ; Lanczos methods ; matrix modifications ; recursive least squares ; signal processing ; singular values ; updating
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Estimates for the condition number of a matrix are useful in many areas of scientific computing, including: recursive least squares computations, optimization, eigenanalysis, and general nonlinear problems solved by linearization techniques where matrix modification techniques are used. The purpose of this paper is to propose anadaptiveLanczosestimator scheme, which we callale, for tracking the condition number of the modified matrix over time. Applications to recursive least squares (RLS) computations using the covariance method with sliding data windows are considered.ale is fast for relatively smalln-parameter problems arising in RLS methods in control and signal processing, and is adaptive over time, i.e., estimates at timet are used to produce estimates at timet+1. Comparisons are made with other adaptive and non-adaptive condition estimators for recursive least squares problems. Numerical experiments are reported indicating thatale yields a very accurate recursive condition estimator.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02145580
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