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By Ortega R., Tarallo M.

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QH+1 Q +1 = IM , and H +1, is an upper triangular matrix. Further, the coefficients Hi, ∈ CM ×M are chosen such that Q +1 is orthogonal, i. , QH i Q +1 H +1, ! QH i Qj Hj, = 0M ×M . 11) j=0 9 Two matrices are denoted to be orthogonal/orthonormal if each column of the one matrix is orthogonal/orthonormal to all the columns of the second matrix. 2 Block Arnoldi Algorithm 39 If we recall that the matrices Q0 , Q1 , . , and Q are already orthonormal to each other (cf. 10), compute as Hi, = QH i AQ , i ∈ {0, 1, .

Since Hi, = 0M ×M for i < − 1 (cf. 10) simplifies to Q +1 H +1, = AQ − Q H , −Q H = IN − Q Q − Q or to the modified Gram–Schmidt version, i. 21) which are both the fundamental three-term iteration formulas of the block Lanczos procedure. Again, the left-hand sides are obtained by QR factorization of the right-hand sides. 2. 1. in an adequate way. 1. 1. 2.. Therefore, the amount of computer storage is reduced tremendously compared to the block Arnoldi algorithm. 2. is designed to compute the basis Q.

And qM −1 based on a Gram–Schmidt procedure. The resulting vector is defined to be qM . , q0 , q1 , . , qM −1 , and qM . At the (m + 1)th step, the vector qm+M is the part of Aqm orthonormal to the last at most 2M vectors qi , i ∈ {m − M, m − M + 1, . . , m + M − 1} ∩ N0 . This procedure is repeated until the vector qD−1 is obtained. 3.. Note that we exploit the fact that A is Hermitian in Lines 6 to 10 in order to reduce computational complexity. There, we compute hi,m only for i ≥ m and use the already computed hm,i to get hi,m = h∗m,i in the cases where i < m.

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Masseras theorem for quasi-periodic differential equations by Ortega R., Tarallo M.

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