S. C. Eisenstat and I. C. F. Ipsen, Relative perturbation results for eigenvalues and eigenvectors of diagonalisable matrices, Bit 38:3:502--509 (1998)  Home/Search   Document Details and Download   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 ....

S. C. Eisenstat and I. C. F. Ipsen, Relative perturbation results for eigenvalues and eigenvectors of diagonalisable matrices, BIT, 38:502--509 (1998).

....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 ....

S. C. Eisenstat and I. C. F. Ipsen, Relative perturbation results for eigenvalues and eigenvectors of diagonalisable matrices, BIT, 38:502--509 (1998).

.... suitable way to compute the eigenvalue decomposition of the difference of two outer products [24, 14] The condition number of X appears in other relative perturbation results [20, 22] Relative perturbation bounds for eigenvalue problem have been the topic of many articles in past years, such as [1, 3, 23, 10, 11, 4, 16, 12, 13, 5] (see also the review article [8] Some of the recent works include [9, 20, 22] These works covered positive definite, indefinite, and diagonalizable matrices. In this paper we give relative perturbation bounds for eigenvalues and perturbation bounds for eigenvectors of the problem (1) under ....

S. C. Eisenstat and I. C. F. Ipsen, Relative perturbation results for eigenvalues and eigenvectors of diagonalisable matrices, BIT, 38:502--509 (1998).

....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 ....

S. C. Eisenstat and I. C. F. Ipsen, Relative perturbation results for eigenvalues and eigenvectors of diagonalisable matrices, BIT, 38:502--509 (1998).

....singular vectors are small. In the rest of this chapter, we show how to exploit this property of bidiagonal matrices to compute numerically orthogonal eigenvectors without any explicit orthogonalization. The relative perturbation results for bidiagonal matrices mentioned above have appeared in [89, 29, 35, 49, 50, 51, 100, 101]. We state the precise results in the next section. 4.3 Relative Perturbation Theory for Bidiagonals In 1966, Kahan proved the remarkable result that a real, symmetric tridiagonal with zero entries on the diagonal determines its eigenvalues to high relative accuracy with respect to small relative ....

S.C. Eisenstat and I.C.F. Ipsen. Relative perturbation results for eigenvalues and eigenvectors of diagonalisable matrices. Technical Report CRSC-TR96-6, Center for Research in Scientific Computation, Department of Mathematics, North Carolina State University, 1996. (14 pages).

.... 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 ....

S. C. Eisenstat and I. C. F. Ipsen, Relative perturbation results for eigenvalues and eigenvectors of diagonalisable matrices, BIT, 38:502--509 (1998).

....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 ....

S. C. Eisenstat and I. C. F. Ipsen, Relative perturbation results for eigenvalues and eigenvectors of diagonalisable matrices, BIT, 38:502--509 (1998).

.... G is a suitable way to compute the eigenvalue decomposition of the difference of two outer products [24, 14] The condition of X appears in other relative perturbation results [20, 22] Relative perturbation bounds for eigenvalue problem have been the topic of many articles in the past years, like [1, 3, 23, 10, 11, 4, 16, 12, 13, 5] (see also the review article [8] Some of the recent works include [9, 20, 22] These works covered positive definite, indefinite, and diagonalizable matrices. In this paper we give relative perturbation bounds for eigenvalues and perturbation bounds for eigenvectors of the problem (1) under ....

S. C. Eisenstat and I. C. F. Ipsen, Relative perturbation results for eigenvalues and eigenvectors of diagonalisable matrices, BIT, 38:502--509 (1998).

....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, ....

.... subspaces of the hyperbolic singular value decomposition [17] iv) the bounds are applicable to engineering problems which can be directly formulated through factors [4] Our bound is also closely related the bounds for perturbations of invariant subspaces under multiplicative perturbations from [10,16]. This and some other relationships are discussed in Section 5. Theorem 8 Let H = GAG , where G has full column rank and A is nonsingular Hermitian matrix. Let T = f i ; Delta Delta Delta ; i k Gamma1 g be the set of non zero neighboring eigenvalues of the same sign, that is, i 0 ....

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S. C. Eisenstat and I. C. F. Ipsen, Relative perturbation results for eigenvalues and eigenvectors of diagonalisable matrices, Technical Report CRSC-TR96-6, Department of Mathematics, North Carolina State University, Rayleigh, 1996.

....The relative error bound below for the eigenvalue of the perturbed matrix D 1 AD 2 measures the error relative to the perturbed eigenvalue rather than an exact eigenvalue. Theorem 5.1 If A is diagonalisable then min 1in j i Gamma j j j (X) kI Gamma D Gamma1 1 D Gamma1 2 k: Proof. (Eisenstat and Ipsen 1996, Theorem 6.1) The idea is to concoct a residual that contains the factor and then to use the absolute bound (5:1) From (D 1 AD 2 )x = x follows A z = D Gamma1 1 D Gamma1 2 z; where z j D 2 x=kD 2 xk: The residual for and z is f j Az Gamma z = D Gamma1 1 D Gamma1 ....

S. Eisenstat and I. Ipsen (1996), Relative perturbation results for eigenvalues and eigenvectors of diagonalisable matrices, Technical Report CRSC-TR96-6, Center for Research in Scientific Computation, Department of Mathematics, North Carolina State University.

No context found.

S. Eisenstat and I. Ipsen, Relative perturbation results for eigenvalues and eigenvectors of diagonalisable matrices, Tech. Rep. CRSC-TR96-6, Center for Research in Scientific Computation, Department of Mathematics, North Carolina State University, 1996.

No context found.

S. C. Eisenstat and I. C. F. Ipsen, Relative perturbation results for eigenvalues and eigenvectors of diagonalisable matrices, Bit 38:3:502--509 (1998)

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