James R. Bunch. Partial pivoting strategies for symmetric matrices. SIAM J. Numer. Anal., 11(3):521--528, 1974.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....be used to determine the inertia, while an LDL factorization yields the inertia of A directly from the diagonal of D, but can fail to exist and its computation can be numerically unstable when it does exist. A method that promises to combine the benefits of GEPP and LDL was proposed by Bunch [3], but has received little attention in the literature. Bunch s idea is to compute a block LDL factorization without interchanges, with a particular strategy for choosing the pivot size (1 or 2) at each stage of the factorization. Bunch s Department of Mathematics, University of Manchester, ....

....for choosing the pivot size (1 or 2) at each stage of the factorization. Bunch s Department of Mathematics, University of Manchester, Manchester, M13 9PL, England (higham ma.man.ac.uk, http: www.ma.man.ac. uk higham ) method requires less storage but slightly more computation than GEPP (see [3] for the details) The purpose of this work is to examine the numerical stability of block LDL with Bunch s pivoting strategy. In Section 2 we define the pivoting strategy and explain how Bunch s derivation of it yields a bound of order 1 for the growth factor. In Section 3 we show that kLk=kAk ....

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James R. Bunch. Partial pivoting strategies for symmetric matrices. SIAM J. Numer. Anal., 11(3):521--528, 1974.

....section we noted that solving a symmetric tridiagonal system by LU factorization with partial pivoting does not take advantage of the symmetry of A. On the other hand, any attempt to compute the symmetrypreserving with a diagonal D can fail, since the factorization does not always exist. Bunch [10] suggested a way to avoid both difficulties: compute a block LDL factorization without interchanges (in the same way as in Section 3.1) with a particular strategy for choosing the pivot size (1 or 2) at each stage of the factorization. Bunch s strategy [10] for choosing the pivot size is fully ....

....does not always exist. Bunch [10] suggested a way to avoid both difficulties: compute a block LDL factorization without interchanges (in the same way as in Section 3.1) with a particular strategy for choosing the pivot size (1 or 2) at each stage of the factorization. Bunch s strategy [10] for choosing the pivot size is fully defined by describing the choice of the first pivot. Algorithm 3.7 (Bunch s pivoting strategy) This algorithm determines the pivot size, s, for the first stage of block LDL factorization applied to a symmetric tridiagonal matrix A 2 R . oe : maxf ja ....

James R. Bunch. Partial pivoting strategies for symmetric matrices. SIAM J. Numer. Anal., 11(3):521--528, 1974.

....Factorization of a Tridiagonal Matrix. In this section we analyze the possible breakdown in the factorization without pivoting of an nonsingular tridiagonal matrix, and show how the problem can be corrected by the occasional use of 2 Theta 2 block pivots. This idea is similar to one used by Bunch [7] for the case of symmetric indefinite matrices. Most of the analysis is elementary, and is included mainly for completeness. Let Tn be the n Theta n nonsingular tridiagonal matrix given by Tn = 2 6 6 6 6 6 4 ff 1 fi 1 fl 1 ff 2 fi 2 . fl n Gamma2 ff n Gamma1 fi n Gamma1 fl ....

....nonsingular tridiagonal matrix given in (2) Then Tn can be factored as Tn = LnDnUn (3) where Ln is unit lower block bidiagonal, Un is unit upper block bidiagonal, and Dn is block diagonal, with 1 Theta 1 and 2 Theta 2 diagonal blocks. Proof. Theorems related to Theorem 2. 2 appear in Bunch [7] and Gutknecht [15] The proof is by induction. The cases n = 1 and n = 2 are clear. There are two possibilities for the induction step. First, suppose ff 1 6= 0. Then one has Tn = 1 0 c n Gamma1 I n Gamma1 ff 1 0 0 Tn Gamma1 1 r t n Gamma1 0 I n Gamma1 where c t n Gamma1 = fl 1 ....

J. R. Bunch, Partial pivoting strategies for symmetric matrices, SIAM J. Numer. Anal., 11 (1974), pp. 521--528.

....the inertia, while an LDL T factorization yields the inertia of A directly from the diagonal of D, but can fail to exist and its computation can be numerically unstable when it does exist. A method that promises to combine the benefits of GEPP and LDL T factorization was proposed by Bunch [3], but has received little attention in the literature. Bunch s idea is to compute a block LDL T factorization without interchanges, with a particular strategy for choosing the pivot size (1 or 2) at each stage of the factorization. Bunch s Department of Mathematics, University of Manchester, ....

....for choosing the pivot size (1 or 2) at each stage of the factorization. Bunch s Department of Mathematics, University of Manchester, Manchester, M13 9PL, England (higham ma.man.ac.uk, http: www.ma.man.ac. uk higham ) method requires less storage but slightly more computation than GEPP (see [3] for the details) The purpose of this work is to examine the numerical stability of block LDL T factorization with Bunch s pivoting strategy. In Section 2 we define the pivoting strategy and explain how Bunch s derivation of it yields a bound of order 1 for the growth factor. In Section 3 we ....

[Article contains additional citation context not shown here]

James R. Bunch. Partial pivoting strategies for symmetric matrices. SIAM J. Numer. Anal., 11(3):521--528, 1974.

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