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  • 1
    Electronic Resource
    Electronic Resource
    An algebraic approach to approximate evaluation of a polynomial on a set of real points (1995)
    Springer
    Advances in computational mathematics 3 (1995), S. 41-58 
    Details
    ISSN: 1572-9044
    Keywords: Polynomial evaluation ; interpolation ; approximation algorithms ; Chebyshev nodes ; rational algorithms ; algebraic and symbolic computing ; numerical stability ; computational complexity ; Chinese remainder algorithm ; 68Q25 ; 65D05 ; 65D15 ; 65Y20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The previous best algorithm for approximate evaluation of a polynomial on a real set was due to Rokhlin and required of the order ofmu+nu 3 infinite precision arithmetic operations to approximate [on a fixed bounded setX(m) ofm+1 real points] a degreen polynomial $$p\left( z \right) = \sum\nolimits_{i = 0}^n {p_i x^i } $$ within the error bound $$2^{ - u} \sum\nolimits_{i = 0}^n {\left| {p_i } \right|} $$ . We develop an approximation algorithm which exploits algebraic computational techniques and decreases Rokhlin's record estimate toO(mlog2 u+nmin-u, logn}). For logu=o(logn), this result may also be favorably compared with the record boundO(m+n)log2 n) on the complexity of the exact multipoint polynomial evaluation. The new algorithm can be performed in the fields (or rings) generated by the input values, which enables us to decrease the precision of the computations [by using modular (residue) arithmetic] and to simplify our computations further in the case whereu=O(logn). Our algorithm allows NC and simultaneously processor efficient parallel implementation. Because of the fundamental nature of the multipoint polynomial evaluation, our results have further applications to numerical and algebraic computational problems. In passing, we also show a substantial improvement in the Chinese remainder algorithm for integers based on incorporating Kaminski's fast residue computation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02431995
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Bisection acceleration for the symmetric tridiagonal eigenvalue problem (1999)
    Springer
    Numerical algorithms 22 (1999), S. 13-39 
    Details
    ISSN: 1572-9265
    Keywords: symmetric eigenvalue problem ; bisection algorithm ; divide-and-conquer eigenvalue algorithms ; Newton's iteration ; convergence acceleration ; approximating polynomial zeros ; 65F15 ; 65Y20 ; 65B99
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We present new algorithms that accelerate the bisection method for the symmetric tridiagonal eigenvalue problem. The algorithms rely on some new techniques, including a new variant of Newton's iteration that reaches cubic convergence (right from the start) to the well separated eigenvalues and can be further applied to acceleration of some other iterative processes, in particular, of the divide-and-conquer methods for the symmetric tridiagonal eigenvalue problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1019146505291
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  • 3
    Book
    Book
    Polynomial and matrix computations /; Progress in theoretical computer science (1994)
    Boston [u.a.] :Birkhäuser,
    Details
    Title: Polynomial and matrix computations /; Progress in theoretical computer science
    Author: Bini, Dario
    Contributer: Pan, Victor Y.
    Publisher: Boston [u.a.] :Birkhäuser,
    Year of publication: 1994
    Type of Medium: Book
    Language: English
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  • 4
    Book
    Book
    Fundamental algorithms /; vol. 1 (1994)
    Details
    Title: Fundamental algorithms /; vol. 1
    Author: Bini, Dario
    Contributer: Pan, Victor
    Year of publication: 1994
    Series Statement: Polynomial and matrix computations vol. 1
    ISBN: 0-8176-3786-9 , 3-7643-3786-9
    Type of Medium: Book
    Language: English
    URL: Zentralblatt Math
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