V. Eijkhout, Qualitative properties of the conjugate gradient and Lanczos method in a matrix framework, Tech. Rep. working note 51, LAPACK, 1992.  Home/Search   Document Details and Download   Summary   Related Articles

....0 1 : 0 0 . 0 0 : 1 0 1 C C C C C C A ; E k = 0 B B B B 0 : 0 1 0 : 0 0 : 0 1 0 : 0 . 0 : 0 1 0 : 0 1 C C C C A ; where E k has ones on the k th column. We also need the following lemma due to Eijkhout [10]. Lemma 1 If AR = RH and r 0 = ffx 0 for some ff 6= 0, then H is an irreducible upper Hessenberg matrix if and only if R = XU where AX = XJ and U is a nonsingular upper triangular matrix satisfying H = U Gamma1 JU . The main idea in the proof of this lemma is that if AR = RH then an upper ....

....is a solution to (1) hence r = b Gamma Ax = 0 and by the ansatz r = r 0 Gamma ARz = r 0 Gamma RHz = 0; or RHz = r 0 . Multiplication with R T gives R T RHz = Hz = R T r 0 = kr 0 k 2 2 e 1 : The most important property of the Arnoldi process is that r k is minimized in the 2 norm. In [10] a proof of this minimization is given formulated in matrix form for the case S T R = I . We can do some simplification in the case S = R and get the following proof: Let X be the Krylov sequence from AX = XJ and assume that r 0 = ffx 0 , for some ff 2 IR, and AR = RH . Thus H is an upper ....

V. Eijkhout, Qualitative properties of the conjugate gradient and Lanczos method in a matrix framework, Tech. Rep. working note 51, LAPACK, 1992.

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