T. Y. Li, N. H. Rhee, and Z. Zeng. An efficient and accurate parallel algorithm for the singular value problems of bidiagonal matrices. Submitted for publication, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of oe i . Let ffl be the the machine precision. Then, based on the assumption that the eigenvalue algorithm can achieve a relative accuracy of 3nffl when finding the eigenvalues of T , the following goal of relative accuracy is set as j oe i Gammaoe i j oe i 3nffl. It has been shown [12] that this degree of relative accuracy can be obtained by using a threshold to determine how specific eigenvalues are to be found. The threshold, which is calculated using the 1 norm and the size of the matrix G T G, is 7kG T Gk1 12n . The smallest singular values of G are computed by ....

T. Y. Li, N. H. Rhee, and Z. Zeng. An efficient and accurate parallel algorithm for the singular value problems of bidiagonal matrices. Submitted for publication, 1993.

....have been used to solve the SVD problem of bidiagonal matrices [2, 3, 7, 9] and space limitations do not allow us to review them here. Many algorithms use an approach that first finds the eigenvalues of a symmetric tridiagonal (ST) matrix that is related to the bidiagonal matrix B. Li et al. [10] recently proposed a new SVD algorithm that combines two eigenvalue algorithms in order to find all the singular values of a bidiagonal matrix both accurately and efficiently. In this paper, we report the results of parallelizing Li s algorithm. Two versions of the new algorithm were studied. This ....

....of B with high relative accuracy if the eigenvalue algorithm used on T introduces small relative Parallel Singular Value Algorithm 3 perturbations entrywise on T . The drawback of this method is that, since the size of the matrix is doubled, the SVD is more expensive to calculate. Li s algorithm [10] uses a novel approach that combines both methods. Let oe i be an actual singular value, and let oe 0 i be an approximation of oe i . Let be the machine precision. Then, based on the assumption that the eigenvalue algorithm can achieve a relative accuracy of 3n when finding the eigenvalues of ....

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T. Y. Li, N. H. Rhee, and Z. Zeng, An efficient and accurate parallel algorithm for the singular value problem of bidiagonal matrices, Submitted for publication, (1993).

....theory of Demmel and Kahan ( 6] Theorem 2, p875) leads to an important extension of our algorithm in the evaluation of singular values of bidiagonal matrices. Based on this algorithm and a hybrid strategy, the singular values of bidiagonal matrices can be evaluated within a tiny relative error [18]. Remark 2.2. Wilkinson [28, p303] analyzed the three term recurrence (2.4) and proved that ae n = f( where f( is the exact characteristic polynomial of S ffiS with ffiS = 2 6 6 6 6 4 ffiff 1 ffifi 1 ffifi 1 . ffifi n Gamma1 ffifi n Gamma1 ffiff n 3 7 7 7 7 5 ....

....singular value evaluation which is equivalent to the symmetric tridiagonal eigenvalue problem on matrices with zero diagonal entries, rank two tearing has another advantage of keeping the structure of zero diagonal. Thus the eigenvalues of split matrices can be evaluated within tiny relative error [18]. 6.2. On parallelization and vectorization. Our algorithm can be parallelized in a similar way as in D C [7] There are two levels of parallelism. First, when the matrix T is split in a tree as in Fig. 3.3, then solving the eigenproblem of each submatrix is independent of the others and thus can ....

T. Y. Li, N. H. Rhee, and Z. Zeng, An efficient and accurate parallel algorithm for the singular value problem of bidiagonal matrices. preprint, submitted to Numerische Mathematik.

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