Z. Drmac. Computing the Singular and the Generalized Singular Values. PhD thesis, Fernuniversit at - Hagen, Hagen, Germany, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the size of relative perturbation of H as described below, and inversely proportional to a relative distance between the eigenvalues from T and the rest of the spectrum of H. Relative perturbation bounds for eigenvalue and singular value problems have been actively researched in the past years [3, 1, 4, 21, 16, 6, 5, 10, 11, 7, 8]. We consider perturbations ffiH which satisfy H x; 4) for all x and some j 2 [0; 1) Here = QjjQ ; 5) is a spectral absolute value of H, that is, a positive definite polar factor of H. This inequality implies that the perturbations which satisfy (4) are inertia preserving. Such ....

Z. Drmac, Computing the Singular and the Generalized Singular Values, PhD thesis, Fernuniversitat, Hagen, 1994.

....H ffiH j f H = D (A ffiA)D: Our bound is a relative variant of the well known sin Theta theorems by Davis and Kahan [2] 24, Section V.3. 3] The development of relative perturbation results for eigenvalue and singular value problems has been very active area of research in the past years [3,1,4,29,21,6,5,15,16,7,13] (see also the review article [12] We shall first describe the relative perturbation and state the existing eigenvalue perturbation results. Let ffiH be the Hermitian relative perturbation which satisfies jx ffiHxj jx H x; 8x; j 1; 2) where H = p H 2 = U jjU is a spectral ....

Z. Drmac, Computing the Singular and the Generalized Singular Values, PhD thesis, Fernuniversitat, Hagen, 1994.

....the size of relative perturbation of H as described below, and inversely proportional to a relative distance between the eigenvalues from T and the rest of the spectrum of H. Relative perturbation bounds for eigenvalue and singular value problems have been actively researched in the past years [3, 1, 4, 21, 16, 6, 5, 10, 11, 7, 8]. We consider perturbations ffiH which satisfy jx ffiHxj jx H x; 8x; j 1; 4) where H = p H 2 = QjjQ ; 5) is a spectral absolute value of H (that is, a positive definite polar factor of H) This inequality implies that the perturbations which satisfy (4) are inertia preserving. Such ....

Z. Drmac, Computing the Singular and the Generalized Singular Values, PhD thesis, Fernuniversitat, Hagen, 1994.

.... Gamma1 2 k; guarantees a small relative error. This means, before requesting high relative accuracy you d better be sure to have a small relative perturbation. Several theses have been written on the subject of relative error bounds in the context of Jacobi methods for computing singular values (Drmac 1994), eigendecompositions of Hermitian matrices (Slapnicar 1992) and eigenvalues of skew symmetric matrices (Pietzsch 1993) as well as fast algorithms for computing eigendecompositions of real symmetric tridiagonal matrices (Dhillon 1997) We have omitted the following issues in our discussion of ....

.... matrices (Pietzsch 1993) as well as fast algorithms for computing eigendecompositions of real symmetric tridiagonal matrices (Dhillon 1997) We have omitted the following issues in our discussion of relative error bounds: ffl generalised eigenvalue problems (Barlow and Demmel 1990, Hari and Drmac 1997, Li 1994a, Veseli c and Slapnicar 1993) ffl sensitivity of eigenvalues and singular values to perturbations in the factors of a matrix (Dhillon 1997, Demmel et al. 1997, Parlett 1997, Veseli c and Slapnicar 1993) ffl relative errors in the form of derivatives when the matrix elements 48 I.C.F. Ipsen ....

Z. Drmac (1994), Computing the Singular and the Generalized Singular Values, PhD thesis, Fachbereich Mathematik, Fernuniversitat Gesamthochschule Hagen, Germany.

.... ffiH j f H = D (A ffiA)D: Our bound is a relative variant of the well known sin Theta theorems by Davis and Kahan [2] 23, Section V.3. 3] The development of relative perturbation results for eigenvalue and singular value problems has been very active area of research in the past years [3,1,4,26,21,6,5,14,15,7,12] (see also the review article [11] We shall first describe the relative perturbation and state the existing eigenvalue perturbation results. Let ffiH be the Hermitian relative perturbation which satisfies jx ffiHxj jx H x; 8x; j 1; 2) where H = p H 2 = U jjU is a spectral ....

Z. Drmac, Computing the Singular and the Generalized Singular Values, PhD thesis, Fernuniversitat, Hagen, 1994.

....I.5.3] Since kffiP k = k sin Thetak [18, Theorem I.5.5] our bound is, in fact, relative variant of the well known sin Theta theorem [2, Section 2] 18, Theorem V.3. 6] Relative perturbation bounds for eigenvalue and singular value problems have been actively researched in the past years [3, 1, 4, 21, 16, 6, 5, 10, 11, 7, 8]. We consider perturbations ffiH which satisfy jx ffiHxj jx H x; 8x; j 1; 2) where H = p H 2 = QjjQ ; 3) is a spectral absolute value of H (that is, a positive definite polar factor of H) Under such perturbations the relative change in eigenvalues is bounded by [21] 1 Gamma ....

Z. Drmac, Computing the Singular and the Generalized Singular Values, PhD thesis, Fernuniversitat, Hagen, 1994.

....X and f X . Since [25] kffiP k = k sin Thetak = sin 1 ; our bounds are in fact relative variants of the well known sin Theta theorems [2] The development of relative perturbation results for eigenvalue and singular value problems has been very active area of research in the past years [5,1,6,28,21,3,12,9,7,15,16,10]. In this paper we give bounds for three types of relative perturbations of Hermitian matrices, thus generalizing some of these results. The first, most general, case is for H being perturbed into f H = H ffiH, Pennsylvania State University, Department of Computer Science and Engineering, ....

....can clearly be interpreted as the perturbation of a graded matrix, and vice versa. These results, related results from [1,6,28,21] and other works, and the results of this paper enable us to estimate accuracy of highly accurate algorithms for computing eigenvalue or singular value decompositions [1,6,27,20,7,8,4]. We would like to point out a major difference between positive definite and indefinite case: if H is positive definite, then small perturbations of the type (1) cause small relative changes in eigenvalues if and only if ( b A) is small [6] while if H is indefinite, then such perturbations can ....

Z. Drmac, Computing the Singular and the Generalized Singular Values, PhD thesis, Fernuniversitat, Hagen, 1994.

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Z. Drmac. Computing the Singular and the Generalized Singular Values. PhD thesis, Fernuniversit at - Hagen, Hagen, Germany, 1994.

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Z. Drmac. Computing the Singular and the Generalized Singular Values. PhD thesis, Lehrgebiet Mathematische Physik, Fernuniversitat Hagen, 1994.

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Z. Drmac. Computing the Singular and the Generalized Singular Values. PhD thesis, Fernuniversit at - Hagen, Hagen, Germany, 1994.

....We present an algorithm for computing the SVD to high relative accuracy from an RRD G = XDY T . After presenting Algorithm 3.1 in detail, we discuss some other algorithms for this problem briefly. None of our algorithms yet incorporates all known tricks for accelerating high accuracy algorithms [48, 39, 21]. Our goal here is simplicity and accuracy. A future paper will address speed issues. Algorithm 3.1 Computing the SVD G = U SigmaV T given an RRD G = XDY T . 1) Perform QR factorization with pivoting on XD to get XD = QRP , where P is a permutation. Thus G = QRPY T . 2) Multiply to get W ....

....0 ) factor in the error bound, but it is likely to be slower. A very similar algorithm appeared in [45] which essentially applied a Jacobi like iteration to the pencil F T F Gamma 0 I GammaI 0 # ; where F = 0 XD 1=2 Y D 1=2 0 # : Finally, another algorithm appeared in [21, 22]. 3.1 Numerical Experiments In this section, we present results of numerical experiments with Algorithm 3.1, assuming we that are given an RRD G = XDY T . We used Sun FORTRAN on a Sun SPARC 20 Workstation, with IEEE arithmetic. Our single precision procedure, SGEPSV, is implemented as follows. ....

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Z. Drmac. Computing the Singular and the Generalized Singular Values. PhD thesis, Fernuniversit at - Hagen, Hagen, Germany, 1994.

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Z. Drmac, Computing the Singular and the Generalized Singular Values, PhD thesis, Fernuniversitat, Hagen, 1994.

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Zlatko Drmac. Computing the Singular and the Generalized Singular Values. PhD thesis, Lehrgebiet Mathematische Physik, Fernuniversitat Hagen, 1994. 193 pp.

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Zlatko Drmac. Computing the Singular and the Generalized Singular Values. PhD thesis, Lehrgebiet Mathematische Physik, Fernuniversitat Hagen, 1994. 193 pp.

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Zlatko Drmac. Computing the Singular and the Generalized Singular Values. PhD thesis, Lehrgebiet Mathematische Physik, Fernuniversitat Hagen, 1994. 193 pp.

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