Beresford N. Parlett. Analysis of algorithms for reflections in bisectors. SIAM Review, 13(2):197--208, 1971. Home/Search Document Not in Database Summary Related Articles Check |
This paper is cited in the following contexts:
Stability of Householder QR Factorization for Weighted Least.. - Cox, Higham (1997) (5 citations) (Correct)
....= oee 1 . Since v 1 = x 1 Gamma oe, textbooks usually recommend the choice oe = Gammasign(x 1 )kxk 2 ; 5.1) in order to avoid cancellation. The other choice of oe, oe = sign(x 1 )kxk 2 ; 5. 2) can be used in a numerically stable way provided that the computation of v 1 is rearranged as follows [12]: v 1 = x 1 Gamma sign(x 1 )kxk 2 = 1 Gamma kxk Gamma(x 2 Delta Delta Delta x n ) In practice, we might want to take a positive oe at each step of the factorization in order to produce a matrix R normalized to have nonnegative diagonal elements, in which case we would need ....
Beresford N. Parlett. Analysis of algorithms for reflections in bisectors. SIAM Review, 13(2):197--208, 1971.
Notes on Accuracy and Stability of Algorithms in Numerical Linear .. - Higham (1998) (Correct)
....choosing the sign to avoid cancellation in the computation of v 1 = x 1 Gamma oe: sign(oe) Gamma sign(x 1 ) 4.2) This has led to the myth that the other choice of sign is unsuitable. In fact, the other sign is perfectly satisfactory provided that the formula for v 1 is suitably rearranged [43], 44, Section 6.3.1] v 1 = x 1 Gamma sign(x 1 )kxk 2 = 1 Gamma kxk Gamma(x 2 Delta Delta Delta x m ) 4.3) The computation of a QR factorization by Householder transformations can be described as follows. Algorithm 4.1 Given A 2 R with m n this algorithm computes ....
Beresford N. Parlett. Analysis of algorithms for reflections in bisectors. SIAM Review, 13(2):197--208, 1971.
Stability of Householder QR Factorization for Weighted Least.. - Cox, Higham (1997) (5 citations) (Correct)
....= oee 1 . Since v 1 = x 1 Gamma oe, textbooks usually recommend the choice oe = Gammasign(x 1 )kxk 2 ; 5.1) in order to avoid cancellation. The other choice of oe, oe = sign(x 1 )kxk 2 ; 5. 2) can be used in a numerically stable way provided that the computation of v 1 is rearranged as follows [12]: v 1 = x 1 Gamma sign(x 1 )kxk 2 = x 2 1 Gamma kxk 2 2 x 1 sign(x 1 )kxk 2 = Gamma(x 2 2 Delta Delta Delta x 2 n ) x 1 sign(x 1 )kxk 2 : In practice, we might want to take a positive oe at each step of the factorization in order to produce a matrix R normalized to have ....
Beresford N. Parlett. Analysis of algorithms for reflections in bisectors. SIAM Review, 13(2):197--208, 1971.
Notes on Accuracy and Stability of Algorithms in Numerical Linear .. - Higham (1998) (Correct)
....in the computation of v 1 = x 1 Gamma oe: sign(oe) Gamma sign(x 1 ) 4.2) 4.1 Householder QR Factorization 19 This has led to the myth that the other choice of sign is unsuitable. In fact, the other sign is perfectly satisfactory provided that the formula for v 1 is suitably rearranged [43], 44, Section 6.3.1] v 1 = x 1 Gamma sign(x 1 )kxk 2 = x 2 1 Gamma kxk 2 2 x 1 sign(x 1 )kxk 2 = Gamma(x 2 2 Delta Delta Delta x 2 m ) x 1 sign(x 1 )kxk 2 : 4.3) The computation of a QR factorization by Householder transformations can be described as follows. Algorithm ....
Beresford N. Parlett. Analysis of algorithms for reflections in bisectors. SIAM Review, 13(2):197--208, 1971.
On Orthogonal Block Elimination - Bischof, Sun (Correct)
....d i j joe j d j j such that oe j d j 6= 0; 1 i k; 1 j k ) For the case k = 1, U = d = Sigma1 and S = 1 Sigma g 1 d) Gamma1 . The LAPACK selection d = sign(g 1 ) see (18) for a nontrivial Householder matrix results in the smaller scaling factor S among the two choices. Parlett [12] showed, however, that the alternate can be computed in an equally stable fashion. In their computational procedures for block reflectors, Schreiber and Parlett [13] use Higham s algorithm [10] to compute the polar decomposition of G 1 , and hence implicitly chose D = I. In this case, S = V r (I ....
Beresford N. Parlett. Analysis of algorithms for reflections in bisectors. SIAM Review, 13:197--208, 1971.
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