![The new algorithms are based on the following result. For and any vector b of order n, let X = Xc(#) denote the matrix whose rows x! are defined in the following way: The algorithms presented here for testing whether W has full row rank achieve substantially the same bounds. However, they can be considered as useful competitors on the ground of the numerical accuracy. In particular, our linear time algorithm avoids the numerically very dangerous computation of matrix powers, but eventually need: computing the determinant of a matrix with no special structure. On the other hand. the classical algorithm reduces to the computation of the rank of a nonnegative-definite matrix, which is known to be stable. ‘eae ?. oq e 4 es at 7 . rT. computing the determinant of a matrix with no special structure. On the other hand, LICPICoCMtiiie UO UIpPUl) ath 1ACU “4. Assume n2>m. The computation of the row rank of W can be done in O(log” n) parallel] time by known algorithms, which use O(nM(n)) processors. As an example, one could apply the customary algorithm which consists in computing S = WW! = 5-"") A'BBT A and then checking the nonsingularity of the nonnegative-definite matrix S, say with the LU factorization, or by computing its determinant. The computation of S can be performed in time O(n) with O(nm?) processors, or, in light of Corollary 5, in time O(log? n), with O(nM(n)) processors. The computation of det S can be performed in O(n) steps with O(n’) processors by means of either LU or OR (using Givens’ rotations) factorization, or in O(log” n) steps with O(nM(n)) processors in the light of Theorem 6.](https://figures.academia-assets.com/43075771/figure_002.jpg)
Figure 2 The new algorithms are based on the following result. For and any vector b of order n, let X = Xc(#) denote the matrix whose rows x! are defined in the following way: The algorithms presented here for testing whether W has full row rank achieve substantially the same bounds. However, they can be considered as useful competitors on the ground of the numerical accuracy. In particular, our linear time algorithm avoids the numerically very dangerous computation of matrix powers, but eventually need: computing the determinant of a matrix with no special structure. On the other hand. the classical algorithm reduces to the computation of the rank of a nonnegative-definite matrix, which is known to be stable. ‘eae ?. oq e 4 es at 7 . rT. computing the determinant of a matrix with no special structure. On the other hand, LICPICoCMtiiie UO UIpPUl) ath 1ACU “4. Assume n2>m. The computation of the row rank of W can be done in O(log” n) parallel] time by known algorithms, which use O(nM(n)) processors. As an example, one could apply the customary algorithm which consists in computing S = WW! = 5-"") A'BBT A and then checking the nonsingularity of the nonnegative-definite matrix S, say with the LU factorization, or by computing its determinant. The computation of S can be performed in time O(n) with O(nm?) processors, or, in light of Corollary 5, in time O(log? n), with O(nM(n)) processors. The computation of det S can be performed in O(n) steps with O(n’) processors by means of either LU or OR (using Givens’ rotations) factorization, or in O(log” n) steps with O(nM(n)) processors in the light of Theorem 6.
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