id: 05872583
dt: j
an: 05872583
au: Ozaki, Katsuhisa; Ogita, Takeshi; Oishi, Shin’ichi
ti: An algorithm for automatically selecting a suitable verification method for
    linear systems.
so: Numer. Algorithms 56, No. 3, 363-382 (2011).
py: 2011
pu: Springer, Dordrecht
la: EN
cc: 
ut: linear systems; verified computation
ci: Zbl 0999.65015
li: doi:10.1007/s11075-010-9389-6
ab: Summary: Several methods are proposed to calculate a rigorous error bound
    of an approximate solution of a linear system by floating-point
    arithmetic. These methods are called ‘verification methods’. The
    applicable ranges of these methods are different. Whether such methods
    succeed to work or not depends mainly on the condition number and the
    dimension of the coefficient matrix. In general, however, the condition
    number is not known in advance. If the dimension or the condition
    number is large to some extent, then Oishi-Rump’s method [{\it S.
    Oishi} and {\it S. M. Rump}, Numer. Math. 90, No.~4, 755‒773 (2002;
    Zbl 0999.65015)], which is known as the fastest verification method for
    this purpose, may fail. There are more robust verification methods
    whose computational cost are larger than the one of Oishi-Rump’s
    method. It is not that efficient to apply such robust methods to
    well-conditioned problems. The aim of this paper is to choose a
    suitable verification method whose computational cost is minimal to
    succeed. First in this paper, four fast verification methods for linear
    systems are briefly reviewed. Next, a method combined of Oishi-Rump’s
    and Ogita-Oishi’s one [{\it T. Ogita} and {\it S. Oishi}, Trans. Inf.
    Process. Soc. Japan 46 (SIG10 TOM12), 10‒18 (2005)] is developed.
    Then, an algorithm which automatically and efficiently chooses an
    appropriate verification method from five verification methods is
    proposed. The proposed algorithm does as much work as necessary to
    calculate error bounds for the approximate solutions of linear systems.
    Finally, numerical results are presented.
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