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Robust Selective Gram-Schmidt Reorthogonalization - Giraud, Langou (2002) (Correct)
....columns of Q. Giraud, Langou and Rozloznk [15] have shown that either for CGS or MGS, two projections were enough when the initial matrix A is numerically nonsingular. This confirms what was already experimentally well known and generalized the result with n = 2 vectors of Kahan and Parlett [10]. In this paper, we focus only on Gram Schmidt algorithms where the number of projections for each column of A is either one or two. We call the resulting algorithm classical (resp. modified) Gram Schmidt algorithm with reorthogonalization (CGS2, resp. MGS2) or Gram Schmidt algorithm with ....
....resulting algorithm as GS2(K) Other criterion like super orthogonalization [8] have also been tested but the K criterion is the most commonly used reorthogonalization criterion. We give below the modified Gram Schmidt algorithm with reorthogonalization and K criterion (MGS2(K) Kahan and Parlett [10] have shown that for two vectors the orthogonality obAlgorithm 1 MGS2(K) for j = 1 to n do = q if K = #a 2 # K then k j , 1 else k j = q r , 1 end if end for tained ( q 1 q 2 ) is ....
PARLETT B. N. The Symmetric Eigenvalue Problem. Englewood Cliffs, N.J., Prentice-Hall, 1980.
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....i = V y i ; z i = Wy i ; i = 1 : k (local rows) 7. r i = z i i x i ; i = 1 : k (local rows) 8. Correction equation: Solve eq. 1) for each i end while Block methods improve robustness for difficult eigenproblems where the sought eigenvalues occur in multiplicities or are clustered together [21]. In general, the total number of outer JD iterations reduces with larger block sizes, but the total number of matrix vector products increases [12, 22, 28] However, larger blocks yield better cache efficiency, and better computation to communication ratio (coarser granularity) in parallel ....
B. N. Parlett. The Symmetric Eigenvalue Problem. SIAM, Philadelphia, PA, 1998.
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....time, as measured also by the total number of matrix vector multiplications, slowly increases. Block methods are known to accelerate convergence in linear systems with multiple right hand sides [13] and to improve robustness when solving for eigenvalues that occur close together (clustered) [14]. However, it is known that larger block sizes increase the total number of floating point operations. This is manifested as an increase in the number of total matrix vector multiplications performed by both inner and outer iterations. Despite improved cache efficiency and a relatively coarser ....
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