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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 54 (1988), S. 19-32 
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F20 ; CR: G1.3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A direct method is developed for solving linear least squares problems $$\mathop {\min \left\| {Ax - b} \right\|_2 }\limits_x $$ , whereA is large and sparse and the solution is subject to lower and upper boundsl≦x≦u. The problem is initially transformed to upper triangular form by a sparseQR-factorization. An active set algorithm is then used. The key step is the stable updating of theR-factor associated with the columns ofA corresponding to the free variables, when theQ-factor is not available. For this a new method is developed, which uses the semi-normal equations and iterative refinement.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 21 (1973), S. 130-137 
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Two compact algorithms are developed for solving systems of linear equationsV x=b andV T a=f, whereV=V(α 0,α 1, ...,α n ) is a confluent Vandermonde matrix of Hermite type. The solution is obtained by one forward and one backward vector recursion, starting with the right hand side. The total amount of storage is only ⋟2n. The number of arithmetic operations needed isO(n 2) and compares favourably with other proposed methods.
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    BIT 7 (1967), S. 257-278 
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An iterative procedure is developed for reducing the rounding errors in the computed least squares solution to an overdetermined system of equationsAx =b, whereA is anm ×n matrix (m ≧n) of rankn. The method relies on computing accurate residuals to a certain augmented system of linear equations, by using double precision accumulation of inner products. To determine the corrections, two methods are given, based on a matrix decomposition ofA obtained either by orthogonal Householder transformations or by a modified Gram-Schmidt orthogonalization. It is shown that the rate of convergence in the iteration is independent of the right hand side,b, and depends linearly on the condition number, ℳ2135;(A), of the rectangular matrixA. The limiting accuracy achieved will be approximately the same as that obtained by a double precision factorization. In a second part of this paper the case whenx is subject to linear constraints and/orA has rank less thann is covered. Here also ALGOL-programs embodying the derived algorithms will be given.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    BIT 8 (1968), S. 8-30 
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    BIT 34 (1994), S. 510-534 
    ISSN: 1572-9125
    Keywords: 65F20 ; Ill-posed problems ; Lanczos algorithm ; regularization ; least squares
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Iterative methods based on Lanczos bidiagonalization with full reorthogonalization (LBDR) are considered for solving large-scale discrete ill-posed linear least-squares problems of the form min x ‖Ax−b‖2. Methods for regularization in the Krylov subspaces are discussed which use generalized cross validation (GCV) for determining the regularization parameter. These methods have the advantage that no a priori information about the noise level is required. To improve convergence of the Lanczos process we apply a variant of the implicitly restarted Lanczos algorithm by Sorensen using zero shifts. Although this restarted method simply corresponds to using LBDR with a starting vector (AA T) p b, it is shown that carrying out the process implicitly is essential for numerical stability. An LBDR algorithm is presented which incorporates implicit restarts to ensure that the global minimum of the CGV curve corresponds to a minimum on the curve for the truncated SVD solution. Numerical results are given comparing the performance of this algorithm with non-restarted LBDR.
    Type of Medium: Electronic Resource
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  • 6
    Electronic Resource
    Electronic Resource
    Springer
    BIT 13 (1973), S. 253-264 
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A hybrid method of order five (eight) which uses three (four) function evaluations per step is presented together with an algorithm which ensures global convergence. The method is compared with its closest competitors (among globally convergent methods which do not use derivatives) in a series of numerical examples. An ALGOL procedure of the method is included.
    Type of Medium: Electronic Resource
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  • 7
    Electronic Resource
    Electronic Resource
    Springer
    BIT 28 (1988), S. 659-670 
    ISSN: 1572-9125
    Keywords: 65F20 ; ill-posed problems ; Lanczos algorithm ; regularization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An iterative method based on Lanczos bidiagonalization is developed for computing regularized solutions of large and sparse linear systems, which arise from discretizations of ill-posed problems in partial differential or integral equations. Determination of the regularization parameter and termination criteria are discussed. Comments are given on the computational implementation of the algorithm.
    Type of Medium: Electronic Resource
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  • 8
    Electronic Resource
    Electronic Resource
    Springer
    BIT 23 (1983), S. 329-345 
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract When a stiff differential systemy'(t)=f(y, t),y εR n, is solved by an implicit multistep method, then in each time step one has to solve a set of nonlinear equations by a modified Newton iteration. A fixed approximate JacobianW=(1/hλ)I − J, J=∂f/∂y is normally used for many time steps. The cost of factorizingW and of solving the resulting linear systems can be high. For the case that onlyk ≪n eigenvalues ofJ are stiff, we derive an approximation ofJ which is more easily factorized and still often gives almost the same rate of convergence in the Newton iterations. The approximation is based on a block Schur factorization ofJ, which can be efficiently computed by a modified version of theQR algorithm. Limited numerical experiences indicate that typically just a few iterations in the blockQR algorithm suffice to give a good approximation toJ. It is shown that for sparse Jacobians a similar scheme can be realized by using a slight modification of orthogonal iteration.
    Type of Medium: Electronic Resource
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  • 9
    Electronic Resource
    Electronic Resource
    Springer
    BIT 7 (1967), S. 1-21 
    ISSN: 1572-9125
    Keywords: Least square ; linear equations ; orthogonalization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A general analysis of the condition of the linear least squares problem is given. The influence of rounding errors is studied in detail for a modified version of the Gram-Schmidt orthogonalization to obtain a factorizationA=QR of a givenm×n matrixA, whereR is upper triangular andQ T Q=I. Letx be the vector which minimizes ‖b−Ax‖2 andr=b−Ax. It is shown that if inner-products are accumulated in double precision then the errors in the computedx andr are less than the errors resulting from some simultaneous initial perturbation δA, δb such that $$\parallel \delta A\parallel _E /\parallel A\parallel _E \approx \parallel \delta b\parallel _2 /\parallel b\parallel _2 \approx 2 \cdot n^{3/2} machine units.$$ No reorthogonalization is needed and the result is independent of the pivoting strategy used.
    Type of Medium: Electronic Resource
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  • 10
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Iterative methods are developed for computing the Moore-Penrose pseudoinverse solution of a linear systemAx=b, whereA is anm ×n sparse matrix. The methods do not require the explicit formation ofA T A orAA T and therefore are advantageous to use when these matrices are much less sparse thanA itself. The methods are based on solving the two related systems (i)x=A T y,AA T y=b, and (ii)A T Ax=A T b. First it is shown how theSOR-andSSOR-methods for these two systems can be implemented efficiently. Further, the acceleration of theSSOR-method by Chebyshev semi-iteration and the conjugate gradient method is discussed. In particular it is shown that theSSOR-cg method for (i) and (ii) can be implemented in such a way that each step requires only two sweeps through successive rows and columns ofA respectively. In the general rank deficient and inconsistent case it is shown how the pseudoinverse solution can be computed by a two step procedure. Some possible applications are mentioned and numerical results are given for some problems from picture reconstruction.
    Type of Medium: Electronic Resource
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