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This can be easily veriﬁed by replacing x ∈ Cn in xH Ax > 0 by [0T , x ,T , 0T ]T with x ∈ Cm , such that xH Ax = x ,H A x > 0. 20) i,i , H = Di+1,i+1 − li+1,i li+1,i =: Di+1 . 21) deﬁnes the CF of the Hermitian and positive deﬁnite (N −i−1)×(N −i−1) matrix Di+1 = Li+1,i+1 LH i+1,i+1 . In other words, at each step i, one additional column of the lower triangular matrix L is computed and the CF problem is reduced by one dimension. Finally, at the last step, i. , for i = N − 1, the reduced-dimension CF is DN −1 = LN −1,N −1 LH N −1,N −1 where both DN −1 and LN −1,N −1 are real-valued scalars.

For large N , the direct inversion as performed in Line 8 is computationally cheaper (cf. 1). 4. 41) Instead of computing all n2 elements of the Hermitian n × n matrix, it suﬃces to compute the n(n + 1)/2 elements of its lower triangular part. 26 2 Eﬃcient Matrix Wiener Filter Implementations if it is applied to estimate a signal of length S. The term with the S-fraction is due to the fact that the inverse of the N × N auto-covariance matrix Cy [n − 1] which requires ζHI (N ) = N 3 + N 2 + N FLOPs (cf.

Paige and Michael A. Saunders in 1982 in order to solve numerically efﬁcient LS problems based on normal equations. Note that there exists much more Krylov subspace algorithms for non-symmetric as well as symmetric and non-positive deﬁnite systems which are not listed in this book due to its concentration on symmetric (or Hermitian) and positive deﬁnite problems. If the reader is interested in solving non-symmetric or symmetric and non-positive deﬁnite systems, it is recommended to read the surveys [92, 206, 252] or the books [204, 237].